↑

Properties

Label 2790.2.a.bc.1.1
Level $2790$
Weight $2$
Character 2790.1
Self dual yes
Analytic conductor $22.278$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(0\)
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} +4.00000 q^{11} +2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +4.00000 q^{19} +1.00000 q^{20} +4.00000 q^{22} -4.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +4.00000 q^{28} +2.00000 q^{29} -1.00000 q^{31} +1.00000 q^{32} -2.00000 q^{34} +4.00000 q^{35} -6.00000 q^{37} +4.00000 q^{38} +1.00000 q^{40} -10.0000 q^{41} -4.00000 q^{43} +4.00000 q^{44} -4.00000 q^{46} -8.00000 q^{47} +9.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} +6.00000 q^{53} +4.00000 q^{55} +4.00000 q^{56} +2.00000 q^{58} -8.00000 q^{59} +10.0000 q^{61} -1.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} -12.0000 q^{67} -2.00000 q^{68} +4.00000 q^{70} +14.0000 q^{73} -6.00000 q^{74} +4.00000 q^{76} +16.0000 q^{77} -8.00000 q^{79} +1.00000 q^{80} -10.0000 q^{82} -4.00000 q^{83} -2.00000 q^{85} -4.00000 q^{86} +4.00000 q^{88} +6.00000 q^{89} +8.00000 q^{91} -4.00000 q^{92} -8.00000 q^{94} +4.00000 q^{95} -6.00000 q^{97} +9.00000 q^{98} +O(q^{100})\)

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
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 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
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 4.00000 1.06904
\(15\) 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
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 4.00000 0.755929
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 − 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 4.00000 0.648886
\(39\) 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
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 4.00000 0.534522
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 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
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 16.0000 1.82337
\(78\) 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
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −18.0000 −1.79107 −0.895533 − 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 − 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 4.00000 0.386695 0.193347 − 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 − 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) 4.00000 0.377964
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) −8.00000 −0.736460
\(119\) −8.00000 −0.733359
\(120\) 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
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) 8.00000 0.698963 0.349482 − 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 − 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 4.00000 0.338062
\(141\) 0 0
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 4.00000 0.324443
\(153\) 0 0
\(154\) 16.0000 1.28932
\(155\) −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
\(163\) 12.0000 0.939913 0.469956 − 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 20.0000 1.54765 0.773823 − 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 20.0000 1.49487 0.747435 − 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 − 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 8.00000 0.592999
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) −24.0000 −1.73658 −0.868290 − 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 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
\(196\) 9.00000 0.642857
\(197\) 14.0000 0.997459 0.498729 − 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 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
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 − 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 6.00000 0.406371
\(219\) 0 0
\(220\) 4.00000 0.269680
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 4.00000 0.267261
\(225\) 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
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) 10.0000 0.655122 0.327561 − 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) −8.00000 −0.520756
\(237\) 0 0
\(238\) −8.00000 −0.518563
\(239\) −8.00000 −0.517477 −0.258738 − 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 − 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 5.00000 0.321412
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) 9.00000 0.574989
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) −1.00000 −0.0635001
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 28.0000 1.76734 0.883672 − 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 0 0
\(253\) −16.0000 −1.00591
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 − 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) −24.0000 −1.49129
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) 8.00000 0.494242
\(263\) 12.0000 0.739952 0.369976 − 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 16.0000 0.981023
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 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
\(277\) −22.0000 −1.32185 −0.660926 − 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 16.0000 0.959616
\(279\) 0 0
\(280\) 4.00000 0.239046
\(281\) 30.0000 1.78965 0.894825 − 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 − 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 8.00000 0.473050
\(287\) −40.0000 −2.36113
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 2.00000 0.117444
\(291\) 0 0
\(292\) 14.0000 0.819288
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 − 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 16.0000 0.911685
\(309\) 0 0
\(310\) −1.00000 −0.0567962
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −34.0000 −1.92179 −0.960897 − 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −2.00000 −0.112331 −0.0561656 − 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −16.0000 −0.891645
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 12.0000 0.664619
\(327\) 0 0
\(328\) −10.0000 −0.552158
\(329\) −32.0000 −1.76422
\(330\) 0 0
\(331\) 24.0000 1.31916 0.659580 − 0.751635i \(-0.270734\pi\)
0.659580 + 0.751635i \(0.270734\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) 20.0000 1.09435
\(335\) −12.0000 −0.655630
\(336\) 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
\(340\) −2.00000 −0.108465
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) −20.0000 −1.07366 −0.536828 − 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 0 0
\(349\) 30.0000 1.60586 0.802932 − 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) −10.0000 −0.532246 −0.266123 − 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 20.0000 1.05703
\(359\) −16.0000 −0.844448 −0.422224 − 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −6.00000 −0.315353
\(363\) 0 0
\(364\) 8.00000 0.419314
\(365\) 14.0000 0.732793
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) −6.00000 −0.311925
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 − 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 − 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 4.00000 0.205196
\(381\) 0 0
\(382\) −24.0000 −1.22795
\(383\) 4.00000 0.204390 0.102195 − 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 0 0
\(385\) 16.0000 0.815436
\(386\) −14.0000 −0.712581
\(387\) 0 0
\(388\) −6.00000 −0.304604
\(389\) 10.0000 0.507020 0.253510 − 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) 14.0000 0.705310
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 − 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −34.0000 −1.69788 −0.848939 − 0.528490i \(-0.822758\pi\)
−0.848939 + 0.528490i \(0.822758\pi\)
\(402\) 0 0
\(403\) −2.00000 −0.0996271
\(404\) −18.0000 −0.895533
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) 18.0000 0.890043 0.445021 − 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) −10.0000 −0.493865
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) −32.0000 −1.57462
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) 16.0000 0.782586
\(419\) −8.00000 −0.390826 −0.195413 − 0.980721i \(-0.562605\pi\)
−0.195413 + 0.980721i \(0.562605\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 − 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) −12.0000 −0.584151
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 40.0000 1.93574
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) 24.0000 1.15604 0.578020 − 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 − 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) −16.0000 −0.765384
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 − 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 4.00000 0.190693
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) −28.0000 −1.33032 −0.665160 − 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 16.0000 0.757622
\(447\) 0 0
\(448\) 4.00000 0.188982
\(449\) 14.0000 0.660701 0.330350 − 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 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
\(457\) −18.0000 −0.842004 −0.421002 − 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) −14.0000 −0.654177
\(459\) 0 0
\(460\) −4.00000 −0.186501
\(461\) −38.0000 −1.76984 −0.884918 − 0.465746i \(-0.845786\pi\)
−0.884918 + 0.465746i \(0.845786\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 − 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) 12.0000 0.555294 0.277647 − 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) −48.0000 −2.21643
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) −8.00000 −0.368230
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) −8.00000 −0.366679
\(477\) 0 0
\(478\) −8.00000 −0.365911
\(479\) 16.0000 0.731059 0.365529 − 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) −14.0000 −0.637683
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) 24.0000 1.08754 0.543772 − 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) 10.0000 0.452679
\(489\) 0 0
\(490\) 9.00000 0.406579
\(491\) 20.0000 0.902587 0.451294 − 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 0 0
\(498\) 0 0
\(499\) −24.0000 −1.07439 −0.537194 − 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 28.0000 1.24970
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) −16.0000 −0.711287
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) −22.0000 −0.975133 −0.487566 − 0.873086i \(-0.662115\pi\)
−0.487566 + 0.873086i \(0.662115\pi\)
\(510\) 0 0
\(511\) 56.0000 2.47729
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) 4.00000 0.176261
\(516\) 0 0
\(517\) −32.0000 −1.40736
\(518\) −24.0000 −1.05450
\(519\) 0 0
\(520\) 2.00000 0.0877058
\(521\) 30.0000 1.31432 0.657162 − 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) 36.0000 1.57417 0.787085 − 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 2.00000 0.0871214
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) 16.0000 0.693688
\(533\) −20.0000 −0.866296
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) −12.0000 −0.518321
\(537\) 0 0
\(538\) 18.0000 0.776035
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 − 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 8.00000 0.343629
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 − 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 6.00000 0.256307
\(549\) 0 0
\(550\) 4.00000 0.170561
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) −42.0000 −1.77960 −0.889799 − 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) 30.0000 1.26547
\(563\) 36.0000 1.51722 0.758610 − 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 20.0000 0.840663
\(567\) 0 0
\(568\) 0 0
\(569\) −26.0000 −1.08998 −0.544988 − 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 8.00000 0.334497
\(573\) 0 0
\(574\) −40.0000 −1.66957
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) −30.0000 −1.24892 −0.624458 − 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) 2.00000 0.0830455
\(581\) −16.0000 −0.663792
\(582\) 0 0
\(583\) 24.0000 0.993978
\(584\) 14.0000 0.579324
\(585\) 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
\(589\) −4.00000 −0.164817
\(590\) −8.00000 −0.329355
\(591\) 0 0
\(592\) −6.00000 −0.246598
\(593\) 18.0000 0.739171 0.369586 − 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) −8.00000 −0.327968
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) −8.00000 −0.327144
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 34.0000 1.38689 0.693444 − 0.720510i \(-0.256092\pi\)
0.693444 + 0.720510i \(0.256092\pi\)
\(602\) −16.0000 −0.652111
\(603\) 0 0
\(604\) 8.00000 0.325515
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) 28.0000 1.13648 0.568242 − 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) 10.0000 0.404888
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) 26.0000 1.05013 0.525065 − 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 16.0000 0.644658
\(617\) 18.0000 0.724653 0.362326 − 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −32.0000 −1.28619 −0.643094 − 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) −1.00000 −0.0401610
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) 24.0000 0.961540
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −34.0000 −1.35891
\(627\) 0 0
\(628\) −22.0000 −0.877896
\(629\) 12.0000 0.478471
\(630\) 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
\(634\) −2.00000 −0.0794301
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 18.0000 0.713186
\(638\) 8.00000 0.316723
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −2.00000 −0.0789953 −0.0394976 − 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) −20.0000 −0.788723 −0.394362 − 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 28.0000 1.10079 0.550397 − 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 0 0
\(649\) −32.0000 −1.25611
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) −18.0000 −0.704394 −0.352197 − 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) 8.00000 0.312586
\(656\) −10.0000 −0.390434
\(657\) 0 0
\(658\) −32.0000 −1.24749
\(659\) 8.00000 0.311636 0.155818 − 0.987786i \(-0.450199\pi\)
0.155818 + 0.987786i \(0.450199\pi\)
\(660\) 0 0
\(661\) 46.0000 1.78919 0.894596 − 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) 24.0000 0.932786
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 16.0000 0.620453
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) 20.0000 0.773823
\(669\) 0 0
\(670\) −12.0000 −0.463600
\(671\) 40.0000 1.54418
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 − 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) −2.00000 −0.0766965
\(681\) 0 0
\(682\) −4.00000 −0.153168
\(683\) −28.0000 −1.07139 −0.535695 − 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) 8.00000 0.305441
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) 16.0000 0.606915
\(696\) 0 0
\(697\) 20.0000 0.757554
\(698\) 30.0000 1.13552
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) 46.0000 1.73740 0.868698 − 0.495342i \(-0.164957\pi\)
0.868698 + 0.495342i \(0.164957\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −10.0000 −0.376355
\(707\) −72.0000 −2.70784
\(708\) 0 0
\(709\) 42.0000 1.57734 0.788672 − 0.614815i \(-0.210769\pi\)
0.788672 + 0.614815i \(0.210769\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 20.0000 0.747435
\(717\) 0 0
\(718\) −16.0000 −0.597115
\(719\) −24.0000 −0.895049 −0.447524 − 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) −6.00000 −0.222988
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) 4.00000 0.148352 0.0741759 − 0.997245i \(-0.476367\pi\)
0.0741759 + 0.997245i \(0.476367\pi\)
\(728\) 8.00000 0.296500
\(729\) 0 0
\(730\) 14.0000 0.518163
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) 18.0000 0.664845 0.332423 − 0.943131i \(-0.392134\pi\)
0.332423 + 0.943131i \(0.392134\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −48.0000 −1.76810
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 − 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) −6.00000 −0.220564
\(741\) 0 0
\(742\) 24.0000 0.881068
\(743\) 4.00000 0.146746 0.0733729 − 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) −14.0000 −0.512576
\(747\) 0 0
\(748\) −8.00000 −0.292509
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 − 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −54.0000 −1.96266 −0.981332 − 0.192323i \(-0.938398\pi\)
−0.981332 + 0.192323i \(0.938398\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) −10.0000 −0.362500 −0.181250 − 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) 24.0000 0.868858
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) 4.00000 0.144526
\(767\) −16.0000 −0.577727
\(768\) 0 0
\(769\) −46.0000 −1.65880 −0.829401 − 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 16.0000 0.576600
\(771\) 0 0
\(772\) −14.0000 −0.503871
\(773\) −26.0000 −0.935155 −0.467578 − 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) −6.00000 −0.215387
\(777\) 0 0
\(778\) 10.0000 0.358517
\(779\) −40.0000 −1.43315
\(780\) 0 0
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) −22.0000 −0.785214
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 − 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 14.0000 0.498729
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) −24.0000 −0.853342
\(792\) 0 0
\(793\) 20.0000 0.710221
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −34.0000 −1.20058
\(803\) 56.0000 1.97620
\(804\) 0 0
\(805\) −16.0000 −0.563926
\(806\) −2.00000 −0.0704470
\(807\) 0 0
\(808\) −18.0000 −0.633238
\(809\) 38.0000 1.33601 0.668004 − 0.744157i \(-0.267149\pi\)
0.668004 + 0.744157i \(0.267149\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 − 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 8.00000 0.280745
\(813\) 0 0
\(814\) −24.0000 −0.841200
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 18.0000 0.629355
\(819\) 0 0
\(820\) −10.0000 −0.349215
\(821\) 10.0000 0.349002 0.174501 − 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) 40.0000 1.39431 0.697156 − 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) −32.0000 −1.11342
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 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
\(832\) 2.00000 0.0693375
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) 20.0000 0.692129
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) −8.00000 −0.276355
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 14.0000 0.482472
\(843\) 0 0
\(844\) −12.0000 −0.413057
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 20.0000 0.687208
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 − 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 40.0000 1.36877
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) −22.0000 −0.751506 −0.375753 − 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 − 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) −4.00000 −0.136162 −0.0680808 − 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) −26.0000 −0.883516
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 6.00000 0.203186
\(873\) 0 0
\(874\) −16.0000 −0.541208
\(875\) 4.00000 0.135225
\(876\) 0 0
\(877\) −38.0000 −1.28317 −0.641584 − 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) 32.0000 1.07995
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) 30.0000 1.01073 0.505363 − 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) −52.0000 −1.74994 −0.874970 − 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −28.0000 −0.940678
\(887\) −40.0000 −1.34307 −0.671534 − 0.740973i \(-0.734364\pi\)
−0.671534 + 0.740973i \(0.734364\pi\)
\(888\) 0 0
\(889\) −32.0000 −1.07325
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) 16.0000 0.535720
\(893\) −32.0000 −1.07084
\(894\) 0 0
\(895\) 20.0000 0.668526
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) 14.0000 0.467186
\(899\) −2.00000 −0.0667037
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) −40.0000 −1.33185
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 − 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −4.00000 −0.132745
\(909\) 0 0
\(910\) 8.00000 0.265197
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 32.0000 1.05673
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 − 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) −4.00000 −0.131876
\(921\) 0 0
\(922\) −38.0000 −1.25146
\(923\) 0 0
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) −24.0000 −0.788689
\(927\) 0 0
\(928\) 2.00000 0.0656532
\(929\) 46.0000 1.50921 0.754606 − 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(930\) 0 0
\(931\) 36.0000 1.17985
\(932\) 10.0000 0.327561
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) −14.0000 −0.457360 −0.228680 − 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) −48.0000 −1.56726
\(939\) 0 0
\(940\) −8.00000 −0.260931
\(941\) 18.0000 0.586783 0.293392 − 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) 40.0000 1.30258
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) −52.0000 −1.68977 −0.844886 − 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 0 0
\(949\) 28.0000 0.908918
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) −8.00000 −0.259281
\(953\) 38.0000 1.23094 0.615470 − 0.788160i \(-0.288966\pi\)
0.615470 + 0.788160i \(0.288966\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) −8.00000 −0.258738
\(957\) 0 0
\(958\) 16.0000 0.516937
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −12.0000 −0.386896
\(963\) 0 0
\(964\) −14.0000 −0.450910
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) 48.0000 1.54358 0.771788 − 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) −6.00000 −0.192648
\(971\) −40.0000 −1.28366 −0.641831 − 0.766846i \(-0.721825\pi\)
−0.641831 + 0.766846i \(0.721825\pi\)
\(972\) 0 0
\(973\) 64.0000 2.05175
\(974\) 24.0000 0.769010
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 10.0000 0.319928 0.159964 − 0.987123i \(-0.448862\pi\)
0.159964 + 0.987123i \(0.448862\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 9.00000 0.287494
\(981\) 0 0
\(982\) 20.0000 0.638226
\(983\) 36.0000 1.14822 0.574111 − 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) 14.0000 0.446077
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 − 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 0 0
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) 0 0
\(997\) 18.0000 0.570066 0.285033 − 0.958518i \(-0.407995\pi\)
0.285033 + 0.958518i \(0.407995\pi\)
\(998\) −24.0000 −0.759707
\(999\) 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.bc.1.1 1
3.2 odd 2 930.2.a.g.1.1 ✓ 1
12.11 even 2 7440.2.a.a.1.1 1
15.2 even 4 4650.2.d.c.3349.1 2
15.8 even 4 4650.2.d.c.3349.2 2
15.14 odd 2 4650.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.g.1.1 ✓ 1 3.2 odd 2
2790.2.a.bc.1.1 1 1.1 even 1 trivial
4650.2.a.w.1.1 1 15.14 odd 2
4650.2.d.c.3349.1 2 15.2 even 4
4650.2.d.c.3349.2 2 15.8 even 4
7440.2.a.a.1.1 1 12.11 even 2