Properties

Label 4840.2.a.bf.1.3
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $0$
Dimension $6$
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] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.45753625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.444728\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.719585 q^{3} -1.00000 q^{5} +4.18418 q^{7} -2.48220 q^{9} +O(q^{10})\) \(q+0.719585 q^{3} -1.00000 q^{5} +4.18418 q^{7} -2.48220 q^{9} -6.05056 q^{13} -0.719585 q^{15} +6.52180 q^{17} +0.739455 q^{19} +3.01088 q^{21} -5.13093 q^{23} +1.00000 q^{25} -3.94491 q^{27} +2.08036 q^{29} -1.12698 q^{31} -4.18418 q^{35} +7.11676 q^{37} -4.35390 q^{39} -4.02170 q^{41} +6.29891 q^{43} +2.48220 q^{45} +9.24147 q^{47} +10.5074 q^{49} +4.69299 q^{51} +7.77015 q^{53} +0.532101 q^{57} +6.16416 q^{59} +14.6869 q^{61} -10.3860 q^{63} +6.05056 q^{65} -5.53268 q^{67} -3.69214 q^{69} +8.87252 q^{71} -8.09351 q^{73} +0.719585 q^{75} -0.570482 q^{79} +4.60789 q^{81} -15.2224 q^{83} -6.52180 q^{85} +1.49700 q^{87} -1.33867 q^{89} -25.3167 q^{91} -0.810957 q^{93} -0.739455 q^{95} +9.96643 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} - 6 q^{13} - 2 q^{15} + 11 q^{17} - 11 q^{19} + 2 q^{21} + 18 q^{23} + 6 q^{25} - q^{27} - 6 q^{29} + q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} - 4 q^{41} + 3 q^{43} - 10 q^{45} + 14 q^{47} + 8 q^{49} + 31 q^{51} + 14 q^{53} - 5 q^{57} + 2 q^{59} - 4 q^{61} - 16 q^{63} + 6 q^{65} + 11 q^{67} + 8 q^{69} + 7 q^{71} - 9 q^{73} + 2 q^{75} + 36 q^{79} + 30 q^{81} - 45 q^{83} - 11 q^{85} + 25 q^{87} + q^{89} - 8 q^{91} + 55 q^{93} + 11 q^{95} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.719585 0.415453 0.207726 0.978187i \(-0.433394\pi\)
0.207726 + 0.978187i \(0.433394\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.18418 1.58147 0.790736 0.612157i \(-0.209698\pi\)
0.790736 + 0.612157i \(0.209698\pi\)
\(8\) 0 0
\(9\) −2.48220 −0.827399
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −6.05056 −1.67812 −0.839062 0.544035i \(-0.816896\pi\)
−0.839062 + 0.544035i \(0.816896\pi\)
\(14\) 0 0
\(15\) −0.719585 −0.185796
\(16\) 0 0
\(17\) 6.52180 1.58177 0.790885 0.611965i \(-0.209621\pi\)
0.790885 + 0.611965i \(0.209621\pi\)
\(18\) 0 0
\(19\) 0.739455 0.169642 0.0848212 0.996396i \(-0.472968\pi\)
0.0848212 + 0.996396i \(0.472968\pi\)
\(20\) 0 0
\(21\) 3.01088 0.657027
\(22\) 0 0
\(23\) −5.13093 −1.06987 −0.534936 0.844893i \(-0.679664\pi\)
−0.534936 + 0.844893i \(0.679664\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −3.94491 −0.759198
\(28\) 0 0
\(29\) 2.08036 0.386313 0.193157 0.981168i \(-0.438127\pi\)
0.193157 + 0.981168i \(0.438127\pi\)
\(30\) 0 0
\(31\) −1.12698 −0.202411 −0.101206 0.994866i \(-0.532270\pi\)
−0.101206 + 0.994866i \(0.532270\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.18418 −0.707256
\(36\) 0 0
\(37\) 7.11676 1.16999 0.584994 0.811037i \(-0.301097\pi\)
0.584994 + 0.811037i \(0.301097\pi\)
\(38\) 0 0
\(39\) −4.35390 −0.697182
\(40\) 0 0
\(41\) −4.02170 −0.628085 −0.314042 0.949409i \(-0.601683\pi\)
−0.314042 + 0.949409i \(0.601683\pi\)
\(42\) 0 0
\(43\) 6.29891 0.960575 0.480288 0.877111i \(-0.340532\pi\)
0.480288 + 0.877111i \(0.340532\pi\)
\(44\) 0 0
\(45\) 2.48220 0.370024
\(46\) 0 0
\(47\) 9.24147 1.34801 0.674003 0.738728i \(-0.264573\pi\)
0.674003 + 0.738728i \(0.264573\pi\)
\(48\) 0 0
\(49\) 10.5074 1.50106
\(50\) 0 0
\(51\) 4.69299 0.657151
\(52\) 0 0
\(53\) 7.77015 1.06731 0.533656 0.845702i \(-0.320818\pi\)
0.533656 + 0.845702i \(0.320818\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.532101 0.0704785
\(58\) 0 0
\(59\) 6.16416 0.802505 0.401253 0.915967i \(-0.368575\pi\)
0.401253 + 0.915967i \(0.368575\pi\)
\(60\) 0 0
\(61\) 14.6869 1.88046 0.940230 0.340541i \(-0.110610\pi\)
0.940230 + 0.340541i \(0.110610\pi\)
\(62\) 0 0
\(63\) −10.3860 −1.30851
\(64\) 0 0
\(65\) 6.05056 0.750480
\(66\) 0 0
\(67\) −5.53268 −0.675924 −0.337962 0.941160i \(-0.609738\pi\)
−0.337962 + 0.941160i \(0.609738\pi\)
\(68\) 0 0
\(69\) −3.69214 −0.444481
\(70\) 0 0
\(71\) 8.87252 1.05297 0.526487 0.850183i \(-0.323509\pi\)
0.526487 + 0.850183i \(0.323509\pi\)
\(72\) 0 0
\(73\) −8.09351 −0.947274 −0.473637 0.880720i \(-0.657059\pi\)
−0.473637 + 0.880720i \(0.657059\pi\)
\(74\) 0 0
\(75\) 0.719585 0.0830906
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −0.570482 −0.0641842 −0.0320921 0.999485i \(-0.510217\pi\)
−0.0320921 + 0.999485i \(0.510217\pi\)
\(80\) 0 0
\(81\) 4.60789 0.511988
\(82\) 0 0
\(83\) −15.2224 −1.67088 −0.835439 0.549584i \(-0.814786\pi\)
−0.835439 + 0.549584i \(0.814786\pi\)
\(84\) 0 0
\(85\) −6.52180 −0.707389
\(86\) 0 0
\(87\) 1.49700 0.160495
\(88\) 0 0
\(89\) −1.33867 −0.141899 −0.0709495 0.997480i \(-0.522603\pi\)
−0.0709495 + 0.997480i \(0.522603\pi\)
\(90\) 0 0
\(91\) −25.3167 −2.65391
\(92\) 0 0
\(93\) −0.810957 −0.0840923
\(94\) 0 0
\(95\) −0.739455 −0.0758664
\(96\) 0 0
\(97\) 9.96643 1.01194 0.505969 0.862552i \(-0.331135\pi\)
0.505969 + 0.862552i \(0.331135\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 11.8701 1.18112 0.590562 0.806992i \(-0.298906\pi\)
0.590562 + 0.806992i \(0.298906\pi\)
\(102\) 0 0
\(103\) −10.3935 −1.02410 −0.512050 0.858955i \(-0.671114\pi\)
−0.512050 + 0.858955i \(0.671114\pi\)
\(104\) 0 0
\(105\) −3.01088 −0.293832
\(106\) 0 0
\(107\) 8.34807 0.807039 0.403519 0.914971i \(-0.367787\pi\)
0.403519 + 0.914971i \(0.367787\pi\)
\(108\) 0 0
\(109\) −6.25046 −0.598686 −0.299343 0.954146i \(-0.596767\pi\)
−0.299343 + 0.954146i \(0.596767\pi\)
\(110\) 0 0
\(111\) 5.12112 0.486075
\(112\) 0 0
\(113\) 16.4937 1.55160 0.775798 0.630982i \(-0.217348\pi\)
0.775798 + 0.630982i \(0.217348\pi\)
\(114\) 0 0
\(115\) 5.13093 0.478461
\(116\) 0 0
\(117\) 15.0187 1.38848
\(118\) 0 0
\(119\) 27.2884 2.50152
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −2.89396 −0.260940
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 16.5082 1.46487 0.732434 0.680838i \(-0.238384\pi\)
0.732434 + 0.680838i \(0.238384\pi\)
\(128\) 0 0
\(129\) 4.53261 0.399074
\(130\) 0 0
\(131\) 12.1354 1.06027 0.530137 0.847912i \(-0.322141\pi\)
0.530137 + 0.847912i \(0.322141\pi\)
\(132\) 0 0
\(133\) 3.09401 0.268285
\(134\) 0 0
\(135\) 3.94491 0.339524
\(136\) 0 0
\(137\) 6.18352 0.528294 0.264147 0.964482i \(-0.414910\pi\)
0.264147 + 0.964482i \(0.414910\pi\)
\(138\) 0 0
\(139\) −5.20245 −0.441266 −0.220633 0.975357i \(-0.570812\pi\)
−0.220633 + 0.975357i \(0.570812\pi\)
\(140\) 0 0
\(141\) 6.65003 0.560033
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −2.08036 −0.172765
\(146\) 0 0
\(147\) 7.56096 0.623618
\(148\) 0 0
\(149\) −11.2912 −0.925013 −0.462506 0.886616i \(-0.653050\pi\)
−0.462506 + 0.886616i \(0.653050\pi\)
\(150\) 0 0
\(151\) −0.156619 −0.0127455 −0.00637275 0.999980i \(-0.502029\pi\)
−0.00637275 + 0.999980i \(0.502029\pi\)
\(152\) 0 0
\(153\) −16.1884 −1.30875
\(154\) 0 0
\(155\) 1.12698 0.0905210
\(156\) 0 0
\(157\) −13.6990 −1.09330 −0.546650 0.837361i \(-0.684097\pi\)
−0.546650 + 0.837361i \(0.684097\pi\)
\(158\) 0 0
\(159\) 5.59129 0.443418
\(160\) 0 0
\(161\) −21.4687 −1.69197
\(162\) 0 0
\(163\) 5.85320 0.458458 0.229229 0.973372i \(-0.426379\pi\)
0.229229 + 0.973372i \(0.426379\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.2892 −0.950965 −0.475483 0.879725i \(-0.657727\pi\)
−0.475483 + 0.879725i \(0.657727\pi\)
\(168\) 0 0
\(169\) 23.6093 1.81610
\(170\) 0 0
\(171\) −1.83547 −0.140362
\(172\) 0 0
\(173\) −19.0157 −1.44574 −0.722868 0.690987i \(-0.757176\pi\)
−0.722868 + 0.690987i \(0.757176\pi\)
\(174\) 0 0
\(175\) 4.18418 0.316294
\(176\) 0 0
\(177\) 4.43564 0.333403
\(178\) 0 0
\(179\) −0.166428 −0.0124394 −0.00621969 0.999981i \(-0.501980\pi\)
−0.00621969 + 0.999981i \(0.501980\pi\)
\(180\) 0 0
\(181\) 4.08469 0.303613 0.151806 0.988410i \(-0.451491\pi\)
0.151806 + 0.988410i \(0.451491\pi\)
\(182\) 0 0
\(183\) 10.5684 0.781242
\(184\) 0 0
\(185\) −7.11676 −0.523235
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −16.5062 −1.20065
\(190\) 0 0
\(191\) 14.7337 1.06609 0.533047 0.846086i \(-0.321047\pi\)
0.533047 + 0.846086i \(0.321047\pi\)
\(192\) 0 0
\(193\) 0.314976 0.0226725 0.0113362 0.999936i \(-0.496391\pi\)
0.0113362 + 0.999936i \(0.496391\pi\)
\(194\) 0 0
\(195\) 4.35390 0.311789
\(196\) 0 0
\(197\) 1.41533 0.100838 0.0504191 0.998728i \(-0.483944\pi\)
0.0504191 + 0.998728i \(0.483944\pi\)
\(198\) 0 0
\(199\) 22.3049 1.58115 0.790575 0.612365i \(-0.209782\pi\)
0.790575 + 0.612365i \(0.209782\pi\)
\(200\) 0 0
\(201\) −3.98124 −0.280815
\(202\) 0 0
\(203\) 8.70461 0.610944
\(204\) 0 0
\(205\) 4.02170 0.280888
\(206\) 0 0
\(207\) 12.7360 0.885211
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −24.3488 −1.67624 −0.838120 0.545486i \(-0.816345\pi\)
−0.838120 + 0.545486i \(0.816345\pi\)
\(212\) 0 0
\(213\) 6.38453 0.437461
\(214\) 0 0
\(215\) −6.29891 −0.429582
\(216\) 0 0
\(217\) −4.71548 −0.320108
\(218\) 0 0
\(219\) −5.82397 −0.393547
\(220\) 0 0
\(221\) −39.4606 −2.65441
\(222\) 0 0
\(223\) 6.22145 0.416619 0.208310 0.978063i \(-0.433204\pi\)
0.208310 + 0.978063i \(0.433204\pi\)
\(224\) 0 0
\(225\) −2.48220 −0.165480
\(226\) 0 0
\(227\) −16.0658 −1.06632 −0.533162 0.846013i \(-0.678996\pi\)
−0.533162 + 0.846013i \(0.678996\pi\)
\(228\) 0 0
\(229\) −15.6832 −1.03638 −0.518189 0.855266i \(-0.673394\pi\)
−0.518189 + 0.855266i \(0.673394\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.0236 −0.918718 −0.459359 0.888251i \(-0.651921\pi\)
−0.459359 + 0.888251i \(0.651921\pi\)
\(234\) 0 0
\(235\) −9.24147 −0.602847
\(236\) 0 0
\(237\) −0.410510 −0.0266655
\(238\) 0 0
\(239\) 10.8094 0.699200 0.349600 0.936899i \(-0.386317\pi\)
0.349600 + 0.936899i \(0.386317\pi\)
\(240\) 0 0
\(241\) 19.7912 1.27486 0.637431 0.770507i \(-0.279997\pi\)
0.637431 + 0.770507i \(0.279997\pi\)
\(242\) 0 0
\(243\) 15.1505 0.971905
\(244\) 0 0
\(245\) −10.5074 −0.671292
\(246\) 0 0
\(247\) −4.47412 −0.284681
\(248\) 0 0
\(249\) −10.9538 −0.694171
\(250\) 0 0
\(251\) 3.82947 0.241714 0.120857 0.992670i \(-0.461436\pi\)
0.120857 + 0.992670i \(0.461436\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −4.69299 −0.293887
\(256\) 0 0
\(257\) 19.0304 1.18709 0.593543 0.804802i \(-0.297729\pi\)
0.593543 + 0.804802i \(0.297729\pi\)
\(258\) 0 0
\(259\) 29.7778 1.85030
\(260\) 0 0
\(261\) −5.16387 −0.319635
\(262\) 0 0
\(263\) 26.6567 1.64372 0.821862 0.569686i \(-0.192935\pi\)
0.821862 + 0.569686i \(0.192935\pi\)
\(264\) 0 0
\(265\) −7.77015 −0.477317
\(266\) 0 0
\(267\) −0.963290 −0.0589524
\(268\) 0 0
\(269\) 0.454544 0.0277141 0.0138570 0.999904i \(-0.495589\pi\)
0.0138570 + 0.999904i \(0.495589\pi\)
\(270\) 0 0
\(271\) 16.2608 0.987771 0.493885 0.869527i \(-0.335576\pi\)
0.493885 + 0.869527i \(0.335576\pi\)
\(272\) 0 0
\(273\) −18.2175 −1.10257
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −11.4338 −0.686989 −0.343495 0.939155i \(-0.611611\pi\)
−0.343495 + 0.939155i \(0.611611\pi\)
\(278\) 0 0
\(279\) 2.79738 0.167475
\(280\) 0 0
\(281\) −5.69217 −0.339566 −0.169783 0.985481i \(-0.554307\pi\)
−0.169783 + 0.985481i \(0.554307\pi\)
\(282\) 0 0
\(283\) −0.580914 −0.0345318 −0.0172659 0.999851i \(-0.505496\pi\)
−0.0172659 + 0.999851i \(0.505496\pi\)
\(284\) 0 0
\(285\) −0.532101 −0.0315189
\(286\) 0 0
\(287\) −16.8275 −0.993298
\(288\) 0 0
\(289\) 25.5339 1.50199
\(290\) 0 0
\(291\) 7.17170 0.420413
\(292\) 0 0
\(293\) −1.17741 −0.0687850 −0.0343925 0.999408i \(-0.510950\pi\)
−0.0343925 + 0.999408i \(0.510950\pi\)
\(294\) 0 0
\(295\) −6.16416 −0.358891
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 31.0450 1.79538
\(300\) 0 0
\(301\) 26.3558 1.51912
\(302\) 0 0
\(303\) 8.54158 0.490701
\(304\) 0 0
\(305\) −14.6869 −0.840967
\(306\) 0 0
\(307\) −19.1615 −1.09360 −0.546802 0.837262i \(-0.684155\pi\)
−0.546802 + 0.837262i \(0.684155\pi\)
\(308\) 0 0
\(309\) −7.47900 −0.425466
\(310\) 0 0
\(311\) 18.6728 1.05884 0.529419 0.848361i \(-0.322410\pi\)
0.529419 + 0.848361i \(0.322410\pi\)
\(312\) 0 0
\(313\) −3.58974 −0.202904 −0.101452 0.994840i \(-0.532349\pi\)
−0.101452 + 0.994840i \(0.532349\pi\)
\(314\) 0 0
\(315\) 10.3860 0.585183
\(316\) 0 0
\(317\) 2.50027 0.140429 0.0702146 0.997532i \(-0.477632\pi\)
0.0702146 + 0.997532i \(0.477632\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 6.00715 0.335287
\(322\) 0 0
\(323\) 4.82258 0.268335
\(324\) 0 0
\(325\) −6.05056 −0.335625
\(326\) 0 0
\(327\) −4.49774 −0.248726
\(328\) 0 0
\(329\) 38.6680 2.13184
\(330\) 0 0
\(331\) −6.58629 −0.362015 −0.181008 0.983482i \(-0.557936\pi\)
−0.181008 + 0.983482i \(0.557936\pi\)
\(332\) 0 0
\(333\) −17.6652 −0.968047
\(334\) 0 0
\(335\) 5.53268 0.302283
\(336\) 0 0
\(337\) −15.2151 −0.828821 −0.414410 0.910090i \(-0.636012\pi\)
−0.414410 + 0.910090i \(0.636012\pi\)
\(338\) 0 0
\(339\) 11.8686 0.644615
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 14.6755 0.792405
\(344\) 0 0
\(345\) 3.69214 0.198778
\(346\) 0 0
\(347\) 7.66135 0.411283 0.205641 0.978627i \(-0.434072\pi\)
0.205641 + 0.978627i \(0.434072\pi\)
\(348\) 0 0
\(349\) −8.99401 −0.481438 −0.240719 0.970595i \(-0.577383\pi\)
−0.240719 + 0.970595i \(0.577383\pi\)
\(350\) 0 0
\(351\) 23.8689 1.27403
\(352\) 0 0
\(353\) 10.3310 0.549862 0.274931 0.961464i \(-0.411345\pi\)
0.274931 + 0.961464i \(0.411345\pi\)
\(354\) 0 0
\(355\) −8.87252 −0.470904
\(356\) 0 0
\(357\) 19.6363 1.03927
\(358\) 0 0
\(359\) 16.2627 0.858310 0.429155 0.903231i \(-0.358811\pi\)
0.429155 + 0.903231i \(0.358811\pi\)
\(360\) 0 0
\(361\) −18.4532 −0.971221
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 8.09351 0.423634
\(366\) 0 0
\(367\) −34.5427 −1.80311 −0.901557 0.432660i \(-0.857575\pi\)
−0.901557 + 0.432660i \(0.857575\pi\)
\(368\) 0 0
\(369\) 9.98266 0.519676
\(370\) 0 0
\(371\) 32.5117 1.68792
\(372\) 0 0
\(373\) 31.2168 1.61634 0.808172 0.588946i \(-0.200457\pi\)
0.808172 + 0.588946i \(0.200457\pi\)
\(374\) 0 0
\(375\) −0.719585 −0.0371592
\(376\) 0 0
\(377\) −12.5874 −0.648282
\(378\) 0 0
\(379\) −33.4663 −1.71905 −0.859525 0.511094i \(-0.829241\pi\)
−0.859525 + 0.511094i \(0.829241\pi\)
\(380\) 0 0
\(381\) 11.8791 0.608584
\(382\) 0 0
\(383\) −0.134288 −0.00686178 −0.00343089 0.999994i \(-0.501092\pi\)
−0.00343089 + 0.999994i \(0.501092\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −15.6351 −0.794779
\(388\) 0 0
\(389\) −8.53807 −0.432897 −0.216449 0.976294i \(-0.569447\pi\)
−0.216449 + 0.976294i \(0.569447\pi\)
\(390\) 0 0
\(391\) −33.4629 −1.69229
\(392\) 0 0
\(393\) 8.73245 0.440494
\(394\) 0 0
\(395\) 0.570482 0.0287040
\(396\) 0 0
\(397\) 38.0250 1.90842 0.954211 0.299135i \(-0.0966982\pi\)
0.954211 + 0.299135i \(0.0966982\pi\)
\(398\) 0 0
\(399\) 2.22641 0.111460
\(400\) 0 0
\(401\) −11.7343 −0.585982 −0.292991 0.956115i \(-0.594651\pi\)
−0.292991 + 0.956115i \(0.594651\pi\)
\(402\) 0 0
\(403\) 6.81885 0.339671
\(404\) 0 0
\(405\) −4.60789 −0.228968
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −28.9837 −1.43315 −0.716575 0.697510i \(-0.754291\pi\)
−0.716575 + 0.697510i \(0.754291\pi\)
\(410\) 0 0
\(411\) 4.44957 0.219481
\(412\) 0 0
\(413\) 25.7920 1.26914
\(414\) 0 0
\(415\) 15.2224 0.747239
\(416\) 0 0
\(417\) −3.74361 −0.183325
\(418\) 0 0
\(419\) 12.5318 0.612221 0.306110 0.951996i \(-0.400972\pi\)
0.306110 + 0.951996i \(0.400972\pi\)
\(420\) 0 0
\(421\) −11.4281 −0.556973 −0.278486 0.960440i \(-0.589833\pi\)
−0.278486 + 0.960440i \(0.589833\pi\)
\(422\) 0 0
\(423\) −22.9391 −1.11534
\(424\) 0 0
\(425\) 6.52180 0.316354
\(426\) 0 0
\(427\) 61.4525 2.97389
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3.78747 0.182436 0.0912181 0.995831i \(-0.470924\pi\)
0.0912181 + 0.995831i \(0.470924\pi\)
\(432\) 0 0
\(433\) 25.6848 1.23433 0.617166 0.786833i \(-0.288281\pi\)
0.617166 + 0.786833i \(0.288281\pi\)
\(434\) 0 0
\(435\) −1.49700 −0.0717756
\(436\) 0 0
\(437\) −3.79409 −0.181496
\(438\) 0 0
\(439\) 29.1218 1.38991 0.694955 0.719054i \(-0.255424\pi\)
0.694955 + 0.719054i \(0.255424\pi\)
\(440\) 0 0
\(441\) −26.0814 −1.24197
\(442\) 0 0
\(443\) 12.1115 0.575437 0.287718 0.957715i \(-0.407103\pi\)
0.287718 + 0.957715i \(0.407103\pi\)
\(444\) 0 0
\(445\) 1.33867 0.0634592
\(446\) 0 0
\(447\) −8.12500 −0.384299
\(448\) 0 0
\(449\) 29.6886 1.40109 0.700546 0.713607i \(-0.252940\pi\)
0.700546 + 0.713607i \(0.252940\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −0.112701 −0.00529516
\(454\) 0 0
\(455\) 25.3167 1.18686
\(456\) 0 0
\(457\) 0.751497 0.0351535 0.0175768 0.999846i \(-0.494405\pi\)
0.0175768 + 0.999846i \(0.494405\pi\)
\(458\) 0 0
\(459\) −25.7279 −1.20088
\(460\) 0 0
\(461\) −9.62920 −0.448477 −0.224238 0.974534i \(-0.571989\pi\)
−0.224238 + 0.974534i \(0.571989\pi\)
\(462\) 0 0
\(463\) −3.33661 −0.155065 −0.0775327 0.996990i \(-0.524704\pi\)
−0.0775327 + 0.996990i \(0.524704\pi\)
\(464\) 0 0
\(465\) 0.810957 0.0376072
\(466\) 0 0
\(467\) 1.69675 0.0785161 0.0392581 0.999229i \(-0.487501\pi\)
0.0392581 + 0.999229i \(0.487501\pi\)
\(468\) 0 0
\(469\) −23.1497 −1.06896
\(470\) 0 0
\(471\) −9.85761 −0.454214
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0.739455 0.0339285
\(476\) 0 0
\(477\) −19.2870 −0.883093
\(478\) 0 0
\(479\) 22.1315 1.01121 0.505607 0.862764i \(-0.331269\pi\)
0.505607 + 0.862764i \(0.331269\pi\)
\(480\) 0 0
\(481\) −43.0604 −1.96339
\(482\) 0 0
\(483\) −15.4486 −0.702935
\(484\) 0 0
\(485\) −9.96643 −0.452552
\(486\) 0 0
\(487\) −19.9414 −0.903631 −0.451815 0.892111i \(-0.649223\pi\)
−0.451815 + 0.892111i \(0.649223\pi\)
\(488\) 0 0
\(489\) 4.21188 0.190468
\(490\) 0 0
\(491\) 26.2451 1.18443 0.592213 0.805782i \(-0.298254\pi\)
0.592213 + 0.805782i \(0.298254\pi\)
\(492\) 0 0
\(493\) 13.5677 0.611059
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 37.1242 1.66525
\(498\) 0 0
\(499\) −8.34480 −0.373565 −0.186782 0.982401i \(-0.559806\pi\)
−0.186782 + 0.982401i \(0.559806\pi\)
\(500\) 0 0
\(501\) −8.84311 −0.395081
\(502\) 0 0
\(503\) 30.3824 1.35469 0.677343 0.735667i \(-0.263131\pi\)
0.677343 + 0.735667i \(0.263131\pi\)
\(504\) 0 0
\(505\) −11.8701 −0.528214
\(506\) 0 0
\(507\) 16.9889 0.754505
\(508\) 0 0
\(509\) 17.2197 0.763248 0.381624 0.924318i \(-0.375365\pi\)
0.381624 + 0.924318i \(0.375365\pi\)
\(510\) 0 0
\(511\) −33.8647 −1.49809
\(512\) 0 0
\(513\) −2.91708 −0.128792
\(514\) 0 0
\(515\) 10.3935 0.457992
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −13.6834 −0.600635
\(520\) 0 0
\(521\) −21.4975 −0.941822 −0.470911 0.882181i \(-0.656075\pi\)
−0.470911 + 0.882181i \(0.656075\pi\)
\(522\) 0 0
\(523\) −0.0139271 −0.000608991 0 −0.000304496 1.00000i \(-0.500097\pi\)
−0.000304496 1.00000i \(0.500097\pi\)
\(524\) 0 0
\(525\) 3.01088 0.131405
\(526\) 0 0
\(527\) −7.34993 −0.320168
\(528\) 0 0
\(529\) 3.32640 0.144626
\(530\) 0 0
\(531\) −15.3007 −0.663992
\(532\) 0 0
\(533\) 24.3336 1.05400
\(534\) 0 0
\(535\) −8.34807 −0.360919
\(536\) 0 0
\(537\) −0.119759 −0.00516798
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −42.7268 −1.83697 −0.918483 0.395459i \(-0.870585\pi\)
−0.918483 + 0.395459i \(0.870585\pi\)
\(542\) 0 0
\(543\) 2.93929 0.126137
\(544\) 0 0
\(545\) 6.25046 0.267740
\(546\) 0 0
\(547\) 17.8319 0.762438 0.381219 0.924485i \(-0.375504\pi\)
0.381219 + 0.924485i \(0.375504\pi\)
\(548\) 0 0
\(549\) −36.4557 −1.55589
\(550\) 0 0
\(551\) 1.53833 0.0655352
\(552\) 0 0
\(553\) −2.38700 −0.101506
\(554\) 0 0
\(555\) −5.12112 −0.217379
\(556\) 0 0
\(557\) −17.8786 −0.757541 −0.378771 0.925491i \(-0.623653\pi\)
−0.378771 + 0.925491i \(0.623653\pi\)
\(558\) 0 0
\(559\) −38.1120 −1.61196
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −10.6011 −0.446781 −0.223391 0.974729i \(-0.571713\pi\)
−0.223391 + 0.974729i \(0.571713\pi\)
\(564\) 0 0
\(565\) −16.4937 −0.693894
\(566\) 0 0
\(567\) 19.2803 0.809695
\(568\) 0 0
\(569\) −22.8718 −0.958836 −0.479418 0.877587i \(-0.659152\pi\)
−0.479418 + 0.877587i \(0.659152\pi\)
\(570\) 0 0
\(571\) −20.5139 −0.858480 −0.429240 0.903191i \(-0.641218\pi\)
−0.429240 + 0.903191i \(0.641218\pi\)
\(572\) 0 0
\(573\) 10.6022 0.442912
\(574\) 0 0
\(575\) −5.13093 −0.213974
\(576\) 0 0
\(577\) 24.2174 1.00819 0.504093 0.863650i \(-0.331827\pi\)
0.504093 + 0.863650i \(0.331827\pi\)
\(578\) 0 0
\(579\) 0.226652 0.00941935
\(580\) 0 0
\(581\) −63.6934 −2.64245
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −15.0187 −0.620946
\(586\) 0 0
\(587\) −31.9904 −1.32039 −0.660194 0.751095i \(-0.729526\pi\)
−0.660194 + 0.751095i \(0.729526\pi\)
\(588\) 0 0
\(589\) −0.833349 −0.0343375
\(590\) 0 0
\(591\) 1.01845 0.0418935
\(592\) 0 0
\(593\) −42.5206 −1.74611 −0.873056 0.487620i \(-0.837865\pi\)
−0.873056 + 0.487620i \(0.837865\pi\)
\(594\) 0 0
\(595\) −27.2884 −1.11872
\(596\) 0 0
\(597\) 16.0503 0.656894
\(598\) 0 0
\(599\) −4.29039 −0.175301 −0.0876503 0.996151i \(-0.527936\pi\)
−0.0876503 + 0.996151i \(0.527936\pi\)
\(600\) 0 0
\(601\) 19.9972 0.815704 0.407852 0.913048i \(-0.366278\pi\)
0.407852 + 0.913048i \(0.366278\pi\)
\(602\) 0 0
\(603\) 13.7332 0.559259
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −23.2114 −0.942121 −0.471060 0.882101i \(-0.656129\pi\)
−0.471060 + 0.882101i \(0.656129\pi\)
\(608\) 0 0
\(609\) 6.26371 0.253818
\(610\) 0 0
\(611\) −55.9161 −2.26212
\(612\) 0 0
\(613\) −5.94916 −0.240284 −0.120142 0.992757i \(-0.538335\pi\)
−0.120142 + 0.992757i \(0.538335\pi\)
\(614\) 0 0
\(615\) 2.89396 0.116696
\(616\) 0 0
\(617\) 39.7576 1.60058 0.800291 0.599612i \(-0.204678\pi\)
0.800291 + 0.599612i \(0.204678\pi\)
\(618\) 0 0
\(619\) −11.8287 −0.475436 −0.237718 0.971334i \(-0.576399\pi\)
−0.237718 + 0.971334i \(0.576399\pi\)
\(620\) 0 0
\(621\) 20.2410 0.812245
\(622\) 0 0
\(623\) −5.60125 −0.224410
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 46.4141 1.85065
\(630\) 0 0
\(631\) 14.8112 0.589623 0.294812 0.955555i \(-0.404743\pi\)
0.294812 + 0.955555i \(0.404743\pi\)
\(632\) 0 0
\(633\) −17.5210 −0.696398
\(634\) 0 0
\(635\) −16.5082 −0.655109
\(636\) 0 0
\(637\) −63.5756 −2.51896
\(638\) 0 0
\(639\) −22.0233 −0.871230
\(640\) 0 0
\(641\) −8.51035 −0.336139 −0.168069 0.985775i \(-0.553753\pi\)
−0.168069 + 0.985775i \(0.553753\pi\)
\(642\) 0 0
\(643\) 34.4340 1.35794 0.678971 0.734165i \(-0.262426\pi\)
0.678971 + 0.734165i \(0.262426\pi\)
\(644\) 0 0
\(645\) −4.53261 −0.178471
\(646\) 0 0
\(647\) 5.32491 0.209344 0.104672 0.994507i \(-0.466621\pi\)
0.104672 + 0.994507i \(0.466621\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −3.39319 −0.132990
\(652\) 0 0
\(653\) 28.2099 1.10394 0.551970 0.833864i \(-0.313876\pi\)
0.551970 + 0.833864i \(0.313876\pi\)
\(654\) 0 0
\(655\) −12.1354 −0.474169
\(656\) 0 0
\(657\) 20.0897 0.783773
\(658\) 0 0
\(659\) −19.9228 −0.776082 −0.388041 0.921642i \(-0.626848\pi\)
−0.388041 + 0.921642i \(0.626848\pi\)
\(660\) 0 0
\(661\) −16.6144 −0.646225 −0.323112 0.946361i \(-0.604729\pi\)
−0.323112 + 0.946361i \(0.604729\pi\)
\(662\) 0 0
\(663\) −28.3953 −1.10278
\(664\) 0 0
\(665\) −3.09401 −0.119981
\(666\) 0 0
\(667\) −10.6742 −0.413306
\(668\) 0 0
\(669\) 4.47687 0.173086
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 32.7246 1.26144 0.630720 0.776010i \(-0.282760\pi\)
0.630720 + 0.776010i \(0.282760\pi\)
\(674\) 0 0
\(675\) −3.94491 −0.151840
\(676\) 0 0
\(677\) 41.0308 1.57694 0.788472 0.615071i \(-0.210873\pi\)
0.788472 + 0.615071i \(0.210873\pi\)
\(678\) 0 0
\(679\) 41.7014 1.60035
\(680\) 0 0
\(681\) −11.5607 −0.443007
\(682\) 0 0
\(683\) 5.71534 0.218691 0.109346 0.994004i \(-0.465124\pi\)
0.109346 + 0.994004i \(0.465124\pi\)
\(684\) 0 0
\(685\) −6.18352 −0.236260
\(686\) 0 0
\(687\) −11.2854 −0.430566
\(688\) 0 0
\(689\) −47.0138 −1.79108
\(690\) 0 0
\(691\) −38.8709 −1.47872 −0.739358 0.673312i \(-0.764871\pi\)
−0.739358 + 0.673312i \(0.764871\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.20245 0.197340
\(696\) 0 0
\(697\) −26.2288 −0.993485
\(698\) 0 0
\(699\) −10.0912 −0.381684
\(700\) 0 0
\(701\) 3.37005 0.127285 0.0636426 0.997973i \(-0.479728\pi\)
0.0636426 + 0.997973i \(0.479728\pi\)
\(702\) 0 0
\(703\) 5.26252 0.198480
\(704\) 0 0
\(705\) −6.65003 −0.250455
\(706\) 0 0
\(707\) 49.6668 1.86791
\(708\) 0 0
\(709\) −22.6143 −0.849300 −0.424650 0.905358i \(-0.639603\pi\)
−0.424650 + 0.905358i \(0.639603\pi\)
\(710\) 0 0
\(711\) 1.41605 0.0531059
\(712\) 0 0
\(713\) 5.78244 0.216554
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 7.77827 0.290485
\(718\) 0 0
\(719\) −18.7071 −0.697655 −0.348828 0.937187i \(-0.613420\pi\)
−0.348828 + 0.937187i \(0.613420\pi\)
\(720\) 0 0
\(721\) −43.4883 −1.61959
\(722\) 0 0
\(723\) 14.2415 0.529645
\(724\) 0 0
\(725\) 2.08036 0.0772627
\(726\) 0 0
\(727\) −2.98028 −0.110532 −0.0552662 0.998472i \(-0.517601\pi\)
−0.0552662 + 0.998472i \(0.517601\pi\)
\(728\) 0 0
\(729\) −2.92160 −0.108207
\(730\) 0 0
\(731\) 41.0803 1.51941
\(732\) 0 0
\(733\) −20.1444 −0.744050 −0.372025 0.928223i \(-0.621336\pi\)
−0.372025 + 0.928223i \(0.621336\pi\)
\(734\) 0 0
\(735\) −7.56096 −0.278890
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 23.2196 0.854146 0.427073 0.904217i \(-0.359545\pi\)
0.427073 + 0.904217i \(0.359545\pi\)
\(740\) 0 0
\(741\) −3.21951 −0.118272
\(742\) 0 0
\(743\) 31.4412 1.15346 0.576732 0.816933i \(-0.304328\pi\)
0.576732 + 0.816933i \(0.304328\pi\)
\(744\) 0 0
\(745\) 11.2912 0.413678
\(746\) 0 0
\(747\) 37.7850 1.38248
\(748\) 0 0
\(749\) 34.9299 1.27631
\(750\) 0 0
\(751\) 40.5769 1.48067 0.740337 0.672236i \(-0.234666\pi\)
0.740337 + 0.672236i \(0.234666\pi\)
\(752\) 0 0
\(753\) 2.75563 0.100421
\(754\) 0 0
\(755\) 0.156619 0.00569996
\(756\) 0 0
\(757\) −22.5716 −0.820378 −0.410189 0.912000i \(-0.634537\pi\)
−0.410189 + 0.912000i \(0.634537\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 21.9901 0.797139 0.398570 0.917138i \(-0.369507\pi\)
0.398570 + 0.917138i \(0.369507\pi\)
\(762\) 0 0
\(763\) −26.1531 −0.946805
\(764\) 0 0
\(765\) 16.1884 0.585293
\(766\) 0 0
\(767\) −37.2966 −1.34670
\(768\) 0 0
\(769\) −41.1834 −1.48511 −0.742556 0.669784i \(-0.766387\pi\)
−0.742556 + 0.669784i \(0.766387\pi\)
\(770\) 0 0
\(771\) 13.6940 0.493178
\(772\) 0 0
\(773\) −10.8352 −0.389716 −0.194858 0.980831i \(-0.562425\pi\)
−0.194858 + 0.980831i \(0.562425\pi\)
\(774\) 0 0
\(775\) −1.12698 −0.0404822
\(776\) 0 0
\(777\) 21.4277 0.768714
\(778\) 0 0
\(779\) −2.97387 −0.106550
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −8.20684 −0.293288
\(784\) 0 0
\(785\) 13.6990 0.488938
\(786\) 0 0
\(787\) −31.3348 −1.11696 −0.558482 0.829517i \(-0.688616\pi\)
−0.558482 + 0.829517i \(0.688616\pi\)
\(788\) 0 0
\(789\) 19.1818 0.682890
\(790\) 0 0
\(791\) 69.0126 2.45381
\(792\) 0 0
\(793\) −88.8638 −3.15565
\(794\) 0 0
\(795\) −5.59129 −0.198303
\(796\) 0 0
\(797\) −9.52077 −0.337243 −0.168621 0.985681i \(-0.553932\pi\)
−0.168621 + 0.985681i \(0.553932\pi\)
\(798\) 0 0
\(799\) 60.2710 2.13224
\(800\) 0 0
\(801\) 3.32285 0.117407
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 21.4687 0.756673
\(806\) 0 0
\(807\) 0.327084 0.0115139
\(808\) 0 0
\(809\) −29.0902 −1.02276 −0.511379 0.859355i \(-0.670865\pi\)
−0.511379 + 0.859355i \(0.670865\pi\)
\(810\) 0 0
\(811\) −43.6205 −1.53172 −0.765862 0.643005i \(-0.777687\pi\)
−0.765862 + 0.643005i \(0.777687\pi\)
\(812\) 0 0
\(813\) 11.7010 0.410372
\(814\) 0 0
\(815\) −5.85320 −0.205029
\(816\) 0 0
\(817\) 4.65776 0.162954
\(818\) 0 0
\(819\) 62.8410 2.19584
\(820\) 0 0
\(821\) 26.6375 0.929656 0.464828 0.885401i \(-0.346116\pi\)
0.464828 + 0.885401i \(0.346116\pi\)
\(822\) 0 0
\(823\) 19.9584 0.695706 0.347853 0.937549i \(-0.386911\pi\)
0.347853 + 0.937549i \(0.386911\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 26.6819 0.927819 0.463910 0.885883i \(-0.346446\pi\)
0.463910 + 0.885883i \(0.346446\pi\)
\(828\) 0 0
\(829\) −50.7077 −1.76115 −0.880576 0.473905i \(-0.842844\pi\)
−0.880576 + 0.473905i \(0.842844\pi\)
\(830\) 0 0
\(831\) −8.22758 −0.285412
\(832\) 0 0
\(833\) 68.5271 2.37432
\(834\) 0 0
\(835\) 12.2892 0.425285
\(836\) 0 0
\(837\) 4.44583 0.153670
\(838\) 0 0
\(839\) −2.69881 −0.0931733 −0.0465867 0.998914i \(-0.514834\pi\)
−0.0465867 + 0.998914i \(0.514834\pi\)
\(840\) 0 0
\(841\) −24.6721 −0.850762
\(842\) 0 0
\(843\) −4.09600 −0.141074
\(844\) 0 0
\(845\) −23.6093 −0.812186
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −0.418017 −0.0143463
\(850\) 0 0
\(851\) −36.5156 −1.25174
\(852\) 0 0
\(853\) −17.6391 −0.603951 −0.301976 0.953316i \(-0.597646\pi\)
−0.301976 + 0.953316i \(0.597646\pi\)
\(854\) 0 0
\(855\) 1.83547 0.0627718
\(856\) 0 0
\(857\) −8.72254 −0.297956 −0.148978 0.988840i \(-0.547598\pi\)
−0.148978 + 0.988840i \(0.547598\pi\)
\(858\) 0 0
\(859\) −34.2322 −1.16799 −0.583993 0.811759i \(-0.698510\pi\)
−0.583993 + 0.811759i \(0.698510\pi\)
\(860\) 0 0
\(861\) −12.1089 −0.412669
\(862\) 0 0
\(863\) −40.8286 −1.38982 −0.694911 0.719096i \(-0.744556\pi\)
−0.694911 + 0.719096i \(0.744556\pi\)
\(864\) 0 0
\(865\) 19.0157 0.646552
\(866\) 0 0
\(867\) 18.3738 0.624008
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 33.4758 1.13429
\(872\) 0 0
\(873\) −24.7387 −0.837276
\(874\) 0 0
\(875\) −4.18418 −0.141451
\(876\) 0 0
\(877\) −4.01395 −0.135541 −0.0677707 0.997701i \(-0.521589\pi\)
−0.0677707 + 0.997701i \(0.521589\pi\)
\(878\) 0 0
\(879\) −0.847247 −0.0285769
\(880\) 0 0
\(881\) 21.1713 0.713278 0.356639 0.934242i \(-0.383923\pi\)
0.356639 + 0.934242i \(0.383923\pi\)
\(882\) 0 0
\(883\) 6.66990 0.224460 0.112230 0.993682i \(-0.464201\pi\)
0.112230 + 0.993682i \(0.464201\pi\)
\(884\) 0 0
\(885\) −4.43564 −0.149102
\(886\) 0 0
\(887\) 2.53548 0.0851332 0.0425666 0.999094i \(-0.486447\pi\)
0.0425666 + 0.999094i \(0.486447\pi\)
\(888\) 0 0
\(889\) 69.0735 2.31665
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.83365 0.228679
\(894\) 0 0
\(895\) 0.166428 0.00556306
\(896\) 0 0
\(897\) 22.3395 0.745895
\(898\) 0 0
\(899\) −2.34452 −0.0781942
\(900\) 0 0
\(901\) 50.6754 1.68824
\(902\) 0 0
\(903\) 18.9653 0.631124
\(904\) 0 0
\(905\) −4.08469 −0.135780
\(906\) 0 0
\(907\) −36.3808 −1.20801 −0.604003 0.796982i \(-0.706428\pi\)
−0.604003 + 0.796982i \(0.706428\pi\)
\(908\) 0 0
\(909\) −29.4640 −0.977260
\(910\) 0 0
\(911\) −58.3264 −1.93244 −0.966220 0.257719i \(-0.917029\pi\)
−0.966220 + 0.257719i \(0.917029\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −10.5684 −0.349382
\(916\) 0 0
\(917\) 50.7767 1.67679
\(918\) 0 0
\(919\) 29.7560 0.981560 0.490780 0.871283i \(-0.336712\pi\)
0.490780 + 0.871283i \(0.336712\pi\)
\(920\) 0 0
\(921\) −13.7883 −0.454341
\(922\) 0 0
\(923\) −53.6837 −1.76702
\(924\) 0 0
\(925\) 7.11676 0.233998
\(926\) 0 0
\(927\) 25.7987 0.847340
\(928\) 0 0
\(929\) −26.0033 −0.853140 −0.426570 0.904455i \(-0.640278\pi\)
−0.426570 + 0.904455i \(0.640278\pi\)
\(930\) 0 0
\(931\) 7.76973 0.254643
\(932\) 0 0
\(933\) 13.4367 0.439897
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −6.46058 −0.211058 −0.105529 0.994416i \(-0.533654\pi\)
−0.105529 + 0.994416i \(0.533654\pi\)
\(938\) 0 0
\(939\) −2.58312 −0.0842970
\(940\) 0 0
\(941\) −29.8484 −0.973030 −0.486515 0.873672i \(-0.661732\pi\)
−0.486515 + 0.873672i \(0.661732\pi\)
\(942\) 0 0
\(943\) 20.6351 0.671970
\(944\) 0 0
\(945\) 16.5062 0.536947
\(946\) 0 0
\(947\) 36.1069 1.17332 0.586658 0.809834i \(-0.300443\pi\)
0.586658 + 0.809834i \(0.300443\pi\)
\(948\) 0 0
\(949\) 48.9703 1.58964
\(950\) 0 0
\(951\) 1.79916 0.0583417
\(952\) 0 0
\(953\) −34.5190 −1.11818 −0.559091 0.829107i \(-0.688850\pi\)
−0.559091 + 0.829107i \(0.688850\pi\)
\(954\) 0 0
\(955\) −14.7337 −0.476772
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 25.8730 0.835482
\(960\) 0 0
\(961\) −29.7299 −0.959030
\(962\) 0 0
\(963\) −20.7216 −0.667743
\(964\) 0 0
\(965\) −0.314976 −0.0101394
\(966\) 0 0
\(967\) −28.4086 −0.913559 −0.456780 0.889580i \(-0.650997\pi\)
−0.456780 + 0.889580i \(0.650997\pi\)
\(968\) 0 0
\(969\) 3.47026 0.111481
\(970\) 0 0
\(971\) −30.9718 −0.993932 −0.496966 0.867770i \(-0.665553\pi\)
−0.496966 + 0.867770i \(0.665553\pi\)
\(972\) 0 0
\(973\) −21.7680 −0.697850
\(974\) 0 0
\(975\) −4.35390 −0.139436
\(976\) 0 0
\(977\) −38.7235 −1.23887 −0.619437 0.785046i \(-0.712639\pi\)
−0.619437 + 0.785046i \(0.712639\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 15.5149 0.495352
\(982\) 0 0
\(983\) 11.1197 0.354663 0.177331 0.984151i \(-0.443254\pi\)
0.177331 + 0.984151i \(0.443254\pi\)
\(984\) 0 0
\(985\) −1.41533 −0.0450962
\(986\) 0 0
\(987\) 27.8249 0.885677
\(988\) 0 0
\(989\) −32.3193 −1.02769
\(990\) 0 0
\(991\) −11.5682 −0.367475 −0.183737 0.982975i \(-0.558820\pi\)
−0.183737 + 0.982975i \(0.558820\pi\)
\(992\) 0 0
\(993\) −4.73940 −0.150400
\(994\) 0 0
\(995\) −22.3049 −0.707112
\(996\) 0 0
\(997\) 28.1683 0.892097 0.446049 0.895009i \(-0.352831\pi\)
0.446049 + 0.895009i \(0.352831\pi\)
\(998\) 0 0
\(999\) −28.0750 −0.888253
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 4840.2.a.bf.1.3 6
4.3 odd 2 9680.2.a.cx.1.4 6
11.3 even 5 440.2.y.b.361.2 12
11.4 even 5 440.2.y.b.401.2 yes 12
11.10 odd 2 4840.2.a.be.1.3 6
44.3 odd 10 880.2.bo.j.801.2 12
44.15 odd 10 880.2.bo.j.401.2 12
44.43 even 2 9680.2.a.cy.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.b.361.2 12 11.3 even 5
440.2.y.b.401.2 yes 12 11.4 even 5
880.2.bo.j.401.2 12 44.15 odd 10
880.2.bo.j.801.2 12 44.3 odd 10
4840.2.a.be.1.3 6 11.10 odd 2
4840.2.a.bf.1.3 6 1.1 even 1 trivial
9680.2.a.cx.1.4 6 4.3 odd 2
9680.2.a.cy.1.4 6 44.43 even 2