Properties

Label 4840.2.a.bb.1.6
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.25903625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 1.6
Root \(-1.71280\) 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)
 
\(f(q)\) \(=\) \(q+1.71280 q^{3} -1.00000 q^{5} +3.70505 q^{7} -0.0663151 q^{9} +O(q^{10})\) \(q+1.71280 q^{3} -1.00000 q^{5} +3.70505 q^{7} -0.0663151 q^{9} -1.17750 q^{13} -1.71280 q^{15} -5.40027 q^{17} -3.84752 q^{19} +6.34602 q^{21} -8.68237 q^{23} +1.00000 q^{25} -5.25199 q^{27} -6.87597 q^{29} +8.75541 q^{31} -3.70505 q^{35} +2.25468 q^{37} -2.01682 q^{39} +1.59733 q^{41} +4.11979 q^{43} +0.0663151 q^{45} -12.6800 q^{47} +6.72743 q^{49} -9.24959 q^{51} -12.3795 q^{53} -6.59004 q^{57} -0.393999 q^{59} +1.80085 q^{61} -0.245701 q^{63} +1.17750 q^{65} -14.3809 q^{67} -14.8712 q^{69} -6.85527 q^{71} +10.0551 q^{73} +1.71280 q^{75} -1.11383 q^{79} -8.79666 q^{81} +0.0173212 q^{83} +5.40027 q^{85} -11.7772 q^{87} -8.49434 q^{89} -4.36270 q^{91} +14.9963 q^{93} +3.84752 q^{95} +10.5155 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} - 6 q^{5} + 7 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} - 6 q^{5} + 7 q^{7} + 5 q^{9} - q^{13} + 3 q^{15} - 6 q^{17} + 7 q^{19} - 14 q^{21} - 9 q^{23} + 6 q^{25} - 21 q^{27} - 10 q^{29} + q^{31} - 7 q^{35} + 3 q^{37} + q^{39} + 6 q^{41} - 18 q^{43} - 5 q^{45} - 3 q^{47} + 17 q^{49} - 15 q^{51} - 23 q^{53} - 9 q^{57} - 2 q^{59} - 6 q^{61} + 49 q^{63} + q^{65} - 22 q^{67} + 2 q^{69} - 13 q^{71} - 10 q^{73} - 3 q^{75} + 22 q^{79} + 10 q^{81} - 10 q^{83} + 6 q^{85} + 3 q^{87} - 25 q^{89} + 12 q^{91} + 19 q^{93} - 7 q^{95} - 33 q^{97}+O(q^{100}) \) 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
\(3\) 1.71280 0.988886 0.494443 0.869210i \(-0.335372\pi\)
0.494443 + 0.869210i \(0.335372\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.70505 1.40038 0.700189 0.713957i \(-0.253099\pi\)
0.700189 + 0.713957i \(0.253099\pi\)
\(8\) 0 0
\(9\) −0.0663151 −0.0221050
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −1.17750 −0.326580 −0.163290 0.986578i \(-0.552211\pi\)
−0.163290 + 0.986578i \(0.552211\pi\)
\(14\) 0 0
\(15\) −1.71280 −0.442243
\(16\) 0 0
\(17\) −5.40027 −1.30976 −0.654879 0.755734i \(-0.727280\pi\)
−0.654879 + 0.755734i \(0.727280\pi\)
\(18\) 0 0
\(19\) −3.84752 −0.882682 −0.441341 0.897339i \(-0.645497\pi\)
−0.441341 + 0.897339i \(0.645497\pi\)
\(20\) 0 0
\(21\) 6.34602 1.38481
\(22\) 0 0
\(23\) −8.68237 −1.81040 −0.905200 0.424986i \(-0.860279\pi\)
−0.905200 + 0.424986i \(0.860279\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.25199 −1.01075
\(28\) 0 0
\(29\) −6.87597 −1.27684 −0.638418 0.769690i \(-0.720411\pi\)
−0.638418 + 0.769690i \(0.720411\pi\)
\(30\) 0 0
\(31\) 8.75541 1.57252 0.786259 0.617898i \(-0.212015\pi\)
0.786259 + 0.617898i \(0.212015\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.70505 −0.626268
\(36\) 0 0
\(37\) 2.25468 0.370667 0.185334 0.982676i \(-0.440663\pi\)
0.185334 + 0.982676i \(0.440663\pi\)
\(38\) 0 0
\(39\) −2.01682 −0.322950
\(40\) 0 0
\(41\) 1.59733 0.249461 0.124730 0.992191i \(-0.460193\pi\)
0.124730 + 0.992191i \(0.460193\pi\)
\(42\) 0 0
\(43\) 4.11979 0.628262 0.314131 0.949380i \(-0.398287\pi\)
0.314131 + 0.949380i \(0.398287\pi\)
\(44\) 0 0
\(45\) 0.0663151 0.00988567
\(46\) 0 0
\(47\) −12.6800 −1.84956 −0.924782 0.380497i \(-0.875753\pi\)
−0.924782 + 0.380497i \(0.875753\pi\)
\(48\) 0 0
\(49\) 6.72743 0.961061
\(50\) 0 0
\(51\) −9.24959 −1.29520
\(52\) 0 0
\(53\) −12.3795 −1.70046 −0.850229 0.526413i \(-0.823536\pi\)
−0.850229 + 0.526413i \(0.823536\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.59004 −0.872872
\(58\) 0 0
\(59\) −0.393999 −0.0512944 −0.0256472 0.999671i \(-0.508165\pi\)
−0.0256472 + 0.999671i \(0.508165\pi\)
\(60\) 0 0
\(61\) 1.80085 0.230575 0.115288 0.993332i \(-0.463221\pi\)
0.115288 + 0.993332i \(0.463221\pi\)
\(62\) 0 0
\(63\) −0.245701 −0.0309554
\(64\) 0 0
\(65\) 1.17750 0.146051
\(66\) 0 0
\(67\) −14.3809 −1.75691 −0.878455 0.477826i \(-0.841425\pi\)
−0.878455 + 0.477826i \(0.841425\pi\)
\(68\) 0 0
\(69\) −14.8712 −1.79028
\(70\) 0 0
\(71\) −6.85527 −0.813571 −0.406785 0.913524i \(-0.633350\pi\)
−0.406785 + 0.913524i \(0.633350\pi\)
\(72\) 0 0
\(73\) 10.0551 1.17686 0.588430 0.808548i \(-0.299746\pi\)
0.588430 + 0.808548i \(0.299746\pi\)
\(74\) 0 0
\(75\) 1.71280 0.197777
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.11383 −0.125316 −0.0626580 0.998035i \(-0.519958\pi\)
−0.0626580 + 0.998035i \(0.519958\pi\)
\(80\) 0 0
\(81\) −8.79666 −0.977406
\(82\) 0 0
\(83\) 0.0173212 0.00190125 0.000950626 1.00000i \(-0.499697\pi\)
0.000950626 1.00000i \(0.499697\pi\)
\(84\) 0 0
\(85\) 5.40027 0.585742
\(86\) 0 0
\(87\) −11.7772 −1.26265
\(88\) 0 0
\(89\) −8.49434 −0.900399 −0.450199 0.892928i \(-0.648647\pi\)
−0.450199 + 0.892928i \(0.648647\pi\)
\(90\) 0 0
\(91\) −4.36270 −0.457335
\(92\) 0 0
\(93\) 14.9963 1.55504
\(94\) 0 0
\(95\) 3.84752 0.394747
\(96\) 0 0
\(97\) 10.5155 1.06768 0.533842 0.845584i \(-0.320748\pi\)
0.533842 + 0.845584i \(0.320748\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.17862 0.714299 0.357150 0.934047i \(-0.383749\pi\)
0.357150 + 0.934047i \(0.383749\pi\)
\(102\) 0 0
\(103\) −3.75016 −0.369515 −0.184757 0.982784i \(-0.559150\pi\)
−0.184757 + 0.982784i \(0.559150\pi\)
\(104\) 0 0
\(105\) −6.34602 −0.619308
\(106\) 0 0
\(107\) −6.35128 −0.614001 −0.307001 0.951709i \(-0.599325\pi\)
−0.307001 + 0.951709i \(0.599325\pi\)
\(108\) 0 0
\(109\) 1.32407 0.126823 0.0634115 0.997987i \(-0.479802\pi\)
0.0634115 + 0.997987i \(0.479802\pi\)
\(110\) 0 0
\(111\) 3.86182 0.366548
\(112\) 0 0
\(113\) 5.33169 0.501563 0.250782 0.968044i \(-0.419312\pi\)
0.250782 + 0.968044i \(0.419312\pi\)
\(114\) 0 0
\(115\) 8.68237 0.809636
\(116\) 0 0
\(117\) 0.0780860 0.00721905
\(118\) 0 0
\(119\) −20.0083 −1.83416
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 2.73591 0.246688
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 14.6406 1.29914 0.649571 0.760301i \(-0.274948\pi\)
0.649571 + 0.760301i \(0.274948\pi\)
\(128\) 0 0
\(129\) 7.05637 0.621279
\(130\) 0 0
\(131\) 5.93922 0.518912 0.259456 0.965755i \(-0.416457\pi\)
0.259456 + 0.965755i \(0.416457\pi\)
\(132\) 0 0
\(133\) −14.2553 −1.23609
\(134\) 0 0
\(135\) 5.25199 0.452019
\(136\) 0 0
\(137\) 14.7698 1.26187 0.630934 0.775837i \(-0.282672\pi\)
0.630934 + 0.775837i \(0.282672\pi\)
\(138\) 0 0
\(139\) 1.29252 0.109630 0.0548149 0.998497i \(-0.482543\pi\)
0.0548149 + 0.998497i \(0.482543\pi\)
\(140\) 0 0
\(141\) −21.7183 −1.82901
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 6.87597 0.571018
\(146\) 0 0
\(147\) 11.5227 0.950379
\(148\) 0 0
\(149\) 23.9060 1.95846 0.979229 0.202757i \(-0.0649900\pi\)
0.979229 + 0.202757i \(0.0649900\pi\)
\(150\) 0 0
\(151\) 8.28048 0.673856 0.336928 0.941530i \(-0.390612\pi\)
0.336928 + 0.941530i \(0.390612\pi\)
\(152\) 0 0
\(153\) 0.358119 0.0289522
\(154\) 0 0
\(155\) −8.75541 −0.703251
\(156\) 0 0
\(157\) −7.96444 −0.635631 −0.317816 0.948153i \(-0.602949\pi\)
−0.317816 + 0.948153i \(0.602949\pi\)
\(158\) 0 0
\(159\) −21.2036 −1.68156
\(160\) 0 0
\(161\) −32.1687 −2.53525
\(162\) 0 0
\(163\) −4.08778 −0.320180 −0.160090 0.987102i \(-0.551178\pi\)
−0.160090 + 0.987102i \(0.551178\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.98229 −0.308159 −0.154079 0.988058i \(-0.549241\pi\)
−0.154079 + 0.988058i \(0.549241\pi\)
\(168\) 0 0
\(169\) −11.6135 −0.893346
\(170\) 0 0
\(171\) 0.255149 0.0195117
\(172\) 0 0
\(173\) 17.6573 1.34246 0.671230 0.741250i \(-0.265766\pi\)
0.671230 + 0.741250i \(0.265766\pi\)
\(174\) 0 0
\(175\) 3.70505 0.280076
\(176\) 0 0
\(177\) −0.674842 −0.0507243
\(178\) 0 0
\(179\) 11.9417 0.892562 0.446281 0.894893i \(-0.352748\pi\)
0.446281 + 0.894893i \(0.352748\pi\)
\(180\) 0 0
\(181\) −1.03171 −0.0766863 −0.0383431 0.999265i \(-0.512208\pi\)
−0.0383431 + 0.999265i \(0.512208\pi\)
\(182\) 0 0
\(183\) 3.08450 0.228013
\(184\) 0 0
\(185\) −2.25468 −0.165767
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −19.4589 −1.41543
\(190\) 0 0
\(191\) 19.0153 1.37590 0.687950 0.725758i \(-0.258511\pi\)
0.687950 + 0.725758i \(0.258511\pi\)
\(192\) 0 0
\(193\) 5.40874 0.389329 0.194665 0.980870i \(-0.437638\pi\)
0.194665 + 0.980870i \(0.437638\pi\)
\(194\) 0 0
\(195\) 2.01682 0.144428
\(196\) 0 0
\(197\) 4.07618 0.290416 0.145208 0.989401i \(-0.453615\pi\)
0.145208 + 0.989401i \(0.453615\pi\)
\(198\) 0 0
\(199\) −12.0711 −0.855700 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(200\) 0 0
\(201\) −24.6316 −1.73738
\(202\) 0 0
\(203\) −25.4759 −1.78805
\(204\) 0 0
\(205\) −1.59733 −0.111562
\(206\) 0 0
\(207\) 0.575772 0.0400190
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −9.31289 −0.641126 −0.320563 0.947227i \(-0.603872\pi\)
−0.320563 + 0.947227i \(0.603872\pi\)
\(212\) 0 0
\(213\) −11.7417 −0.804528
\(214\) 0 0
\(215\) −4.11979 −0.280967
\(216\) 0 0
\(217\) 32.4392 2.20212
\(218\) 0 0
\(219\) 17.2224 1.16378
\(220\) 0 0
\(221\) 6.35881 0.427740
\(222\) 0 0
\(223\) −4.74364 −0.317658 −0.158829 0.987306i \(-0.550772\pi\)
−0.158829 + 0.987306i \(0.550772\pi\)
\(224\) 0 0
\(225\) −0.0663151 −0.00442101
\(226\) 0 0
\(227\) 3.51523 0.233314 0.116657 0.993172i \(-0.462782\pi\)
0.116657 + 0.993172i \(0.462782\pi\)
\(228\) 0 0
\(229\) −9.12613 −0.603072 −0.301536 0.953455i \(-0.597499\pi\)
−0.301536 + 0.953455i \(0.597499\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −21.5251 −1.41016 −0.705079 0.709128i \(-0.749089\pi\)
−0.705079 + 0.709128i \(0.749089\pi\)
\(234\) 0 0
\(235\) 12.6800 0.827150
\(236\) 0 0
\(237\) −1.90777 −0.123923
\(238\) 0 0
\(239\) 11.3074 0.731417 0.365709 0.930729i \(-0.380827\pi\)
0.365709 + 0.930729i \(0.380827\pi\)
\(240\) 0 0
\(241\) −26.1144 −1.68218 −0.841089 0.540897i \(-0.818085\pi\)
−0.841089 + 0.540897i \(0.818085\pi\)
\(242\) 0 0
\(243\) 0.689039 0.0442019
\(244\) 0 0
\(245\) −6.72743 −0.429799
\(246\) 0 0
\(247\) 4.53045 0.288266
\(248\) 0 0
\(249\) 0.0296678 0.00188012
\(250\) 0 0
\(251\) −3.71812 −0.234685 −0.117343 0.993091i \(-0.537438\pi\)
−0.117343 + 0.993091i \(0.537438\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 9.24959 0.579231
\(256\) 0 0
\(257\) −7.09278 −0.442436 −0.221218 0.975224i \(-0.571003\pi\)
−0.221218 + 0.975224i \(0.571003\pi\)
\(258\) 0 0
\(259\) 8.35372 0.519075
\(260\) 0 0
\(261\) 0.455981 0.0282245
\(262\) 0 0
\(263\) 23.3578 1.44030 0.720151 0.693817i \(-0.244073\pi\)
0.720151 + 0.693817i \(0.244073\pi\)
\(264\) 0 0
\(265\) 12.3795 0.760468
\(266\) 0 0
\(267\) −14.5491 −0.890391
\(268\) 0 0
\(269\) −17.0818 −1.04149 −0.520746 0.853711i \(-0.674346\pi\)
−0.520746 + 0.853711i \(0.674346\pi\)
\(270\) 0 0
\(271\) −27.4343 −1.66652 −0.833258 0.552885i \(-0.813527\pi\)
−0.833258 + 0.552885i \(0.813527\pi\)
\(272\) 0 0
\(273\) −7.47243 −0.452252
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −24.2250 −1.45554 −0.727771 0.685820i \(-0.759444\pi\)
−0.727771 + 0.685820i \(0.759444\pi\)
\(278\) 0 0
\(279\) −0.580616 −0.0347605
\(280\) 0 0
\(281\) −17.1634 −1.02388 −0.511941 0.859020i \(-0.671073\pi\)
−0.511941 + 0.859020i \(0.671073\pi\)
\(282\) 0 0
\(283\) 4.62908 0.275170 0.137585 0.990490i \(-0.456066\pi\)
0.137585 + 0.990490i \(0.456066\pi\)
\(284\) 0 0
\(285\) 6.59004 0.390360
\(286\) 0 0
\(287\) 5.91819 0.349340
\(288\) 0 0
\(289\) 12.1629 0.715466
\(290\) 0 0
\(291\) 18.0109 1.05582
\(292\) 0 0
\(293\) −18.5006 −1.08082 −0.540409 0.841402i \(-0.681730\pi\)
−0.540409 + 0.841402i \(0.681730\pi\)
\(294\) 0 0
\(295\) 0.393999 0.0229395
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10.2235 0.591240
\(300\) 0 0
\(301\) 15.2640 0.879804
\(302\) 0 0
\(303\) 12.2955 0.706360
\(304\) 0 0
\(305\) −1.80085 −0.103116
\(306\) 0 0
\(307\) −27.2021 −1.55250 −0.776252 0.630422i \(-0.782882\pi\)
−0.776252 + 0.630422i \(0.782882\pi\)
\(308\) 0 0
\(309\) −6.42328 −0.365408
\(310\) 0 0
\(311\) 32.2598 1.82929 0.914644 0.404260i \(-0.132471\pi\)
0.914644 + 0.404260i \(0.132471\pi\)
\(312\) 0 0
\(313\) −0.290748 −0.0164341 −0.00821704 0.999966i \(-0.502616\pi\)
−0.00821704 + 0.999966i \(0.502616\pi\)
\(314\) 0 0
\(315\) 0.245701 0.0138437
\(316\) 0 0
\(317\) 22.8346 1.28252 0.641258 0.767325i \(-0.278413\pi\)
0.641258 + 0.767325i \(0.278413\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −10.8785 −0.607177
\(322\) 0 0
\(323\) 20.7777 1.15610
\(324\) 0 0
\(325\) −1.17750 −0.0653159
\(326\) 0 0
\(327\) 2.26787 0.125413
\(328\) 0 0
\(329\) −46.9800 −2.59009
\(330\) 0 0
\(331\) −14.5262 −0.798433 −0.399217 0.916857i \(-0.630718\pi\)
−0.399217 + 0.916857i \(0.630718\pi\)
\(332\) 0 0
\(333\) −0.149519 −0.00819361
\(334\) 0 0
\(335\) 14.3809 0.785714
\(336\) 0 0
\(337\) 14.0343 0.764498 0.382249 0.924059i \(-0.375150\pi\)
0.382249 + 0.924059i \(0.375150\pi\)
\(338\) 0 0
\(339\) 9.13212 0.495989
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00990 −0.0545297
\(344\) 0 0
\(345\) 14.8712 0.800637
\(346\) 0 0
\(347\) −27.2726 −1.46407 −0.732034 0.681268i \(-0.761429\pi\)
−0.732034 + 0.681268i \(0.761429\pi\)
\(348\) 0 0
\(349\) −24.9117 −1.33349 −0.666747 0.745284i \(-0.732314\pi\)
−0.666747 + 0.745284i \(0.732314\pi\)
\(350\) 0 0
\(351\) 6.18421 0.330089
\(352\) 0 0
\(353\) −22.2916 −1.18646 −0.593232 0.805031i \(-0.702148\pi\)
−0.593232 + 0.805031i \(0.702148\pi\)
\(354\) 0 0
\(355\) 6.85527 0.363840
\(356\) 0 0
\(357\) −34.2702 −1.81377
\(358\) 0 0
\(359\) 1.86792 0.0985850 0.0492925 0.998784i \(-0.484303\pi\)
0.0492925 + 0.998784i \(0.484303\pi\)
\(360\) 0 0
\(361\) −4.19658 −0.220872
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −10.0551 −0.526308
\(366\) 0 0
\(367\) 2.57693 0.134515 0.0672574 0.997736i \(-0.478575\pi\)
0.0672574 + 0.997736i \(0.478575\pi\)
\(368\) 0 0
\(369\) −0.105927 −0.00551434
\(370\) 0 0
\(371\) −45.8668 −2.38128
\(372\) 0 0
\(373\) 7.12031 0.368675 0.184338 0.982863i \(-0.440986\pi\)
0.184338 + 0.982863i \(0.440986\pi\)
\(374\) 0 0
\(375\) −1.71280 −0.0884486
\(376\) 0 0
\(377\) 8.09645 0.416989
\(378\) 0 0
\(379\) −28.5424 −1.46612 −0.733061 0.680163i \(-0.761909\pi\)
−0.733061 + 0.680163i \(0.761909\pi\)
\(380\) 0 0
\(381\) 25.0764 1.28470
\(382\) 0 0
\(383\) −9.33065 −0.476774 −0.238387 0.971170i \(-0.576619\pi\)
−0.238387 + 0.971170i \(0.576619\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.273204 −0.0138877
\(388\) 0 0
\(389\) 0.444940 0.0225594 0.0112797 0.999936i \(-0.496409\pi\)
0.0112797 + 0.999936i \(0.496409\pi\)
\(390\) 0 0
\(391\) 46.8872 2.37119
\(392\) 0 0
\(393\) 10.1727 0.513145
\(394\) 0 0
\(395\) 1.11383 0.0560431
\(396\) 0 0
\(397\) 1.66941 0.0837850 0.0418925 0.999122i \(-0.486661\pi\)
0.0418925 + 0.999122i \(0.486661\pi\)
\(398\) 0 0
\(399\) −24.4164 −1.22235
\(400\) 0 0
\(401\) 33.8566 1.69072 0.845360 0.534197i \(-0.179386\pi\)
0.845360 + 0.534197i \(0.179386\pi\)
\(402\) 0 0
\(403\) −10.3095 −0.513552
\(404\) 0 0
\(405\) 8.79666 0.437109
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −21.7117 −1.07357 −0.536787 0.843718i \(-0.680362\pi\)
−0.536787 + 0.843718i \(0.680362\pi\)
\(410\) 0 0
\(411\) 25.2977 1.24784
\(412\) 0 0
\(413\) −1.45979 −0.0718315
\(414\) 0 0
\(415\) −0.0173212 −0.000850265 0
\(416\) 0 0
\(417\) 2.21382 0.108411
\(418\) 0 0
\(419\) −26.5502 −1.29706 −0.648532 0.761187i \(-0.724617\pi\)
−0.648532 + 0.761187i \(0.724617\pi\)
\(420\) 0 0
\(421\) −10.4845 −0.510984 −0.255492 0.966811i \(-0.582237\pi\)
−0.255492 + 0.966811i \(0.582237\pi\)
\(422\) 0 0
\(423\) 0.840874 0.0408847
\(424\) 0 0
\(425\) −5.40027 −0.261952
\(426\) 0 0
\(427\) 6.67225 0.322893
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.41218 0.260696 0.130348 0.991468i \(-0.458391\pi\)
0.130348 + 0.991468i \(0.458391\pi\)
\(432\) 0 0
\(433\) 9.08123 0.436416 0.218208 0.975902i \(-0.429979\pi\)
0.218208 + 0.975902i \(0.429979\pi\)
\(434\) 0 0
\(435\) 11.7772 0.564672
\(436\) 0 0
\(437\) 33.4056 1.59801
\(438\) 0 0
\(439\) 17.3683 0.828945 0.414473 0.910062i \(-0.363966\pi\)
0.414473 + 0.910062i \(0.363966\pi\)
\(440\) 0 0
\(441\) −0.446130 −0.0212443
\(442\) 0 0
\(443\) −13.6634 −0.649170 −0.324585 0.945857i \(-0.605225\pi\)
−0.324585 + 0.945857i \(0.605225\pi\)
\(444\) 0 0
\(445\) 8.49434 0.402671
\(446\) 0 0
\(447\) 40.9463 1.93669
\(448\) 0 0
\(449\) −15.0222 −0.708942 −0.354471 0.935067i \(-0.615339\pi\)
−0.354471 + 0.935067i \(0.615339\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 14.1828 0.666367
\(454\) 0 0
\(455\) 4.36270 0.204526
\(456\) 0 0
\(457\) −2.62422 −0.122756 −0.0613779 0.998115i \(-0.519549\pi\)
−0.0613779 + 0.998115i \(0.519549\pi\)
\(458\) 0 0
\(459\) 28.3621 1.32383
\(460\) 0 0
\(461\) 39.2715 1.82906 0.914529 0.404521i \(-0.132562\pi\)
0.914529 + 0.404521i \(0.132562\pi\)
\(462\) 0 0
\(463\) −3.26421 −0.151701 −0.0758505 0.997119i \(-0.524167\pi\)
−0.0758505 + 0.997119i \(0.524167\pi\)
\(464\) 0 0
\(465\) −14.9963 −0.695435
\(466\) 0 0
\(467\) −25.1521 −1.16390 −0.581949 0.813225i \(-0.697710\pi\)
−0.581949 + 0.813225i \(0.697710\pi\)
\(468\) 0 0
\(469\) −53.2821 −2.46034
\(470\) 0 0
\(471\) −13.6415 −0.628567
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −3.84752 −0.176536
\(476\) 0 0
\(477\) 0.820949 0.0375887
\(478\) 0 0
\(479\) 29.8987 1.36611 0.683053 0.730369i \(-0.260652\pi\)
0.683053 + 0.730369i \(0.260652\pi\)
\(480\) 0 0
\(481\) −2.65489 −0.121052
\(482\) 0 0
\(483\) −55.0985 −2.50707
\(484\) 0 0
\(485\) −10.5155 −0.477483
\(486\) 0 0
\(487\) 31.2658 1.41679 0.708393 0.705818i \(-0.249420\pi\)
0.708393 + 0.705818i \(0.249420\pi\)
\(488\) 0 0
\(489\) −7.00156 −0.316621
\(490\) 0 0
\(491\) 8.15752 0.368144 0.184072 0.982913i \(-0.441072\pi\)
0.184072 + 0.982913i \(0.441072\pi\)
\(492\) 0 0
\(493\) 37.1321 1.67235
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −25.3991 −1.13931
\(498\) 0 0
\(499\) 8.52746 0.381742 0.190871 0.981615i \(-0.438869\pi\)
0.190871 + 0.981615i \(0.438869\pi\)
\(500\) 0 0
\(501\) −6.82087 −0.304734
\(502\) 0 0
\(503\) 9.83853 0.438678 0.219339 0.975649i \(-0.429610\pi\)
0.219339 + 0.975649i \(0.429610\pi\)
\(504\) 0 0
\(505\) −7.17862 −0.319444
\(506\) 0 0
\(507\) −19.8916 −0.883417
\(508\) 0 0
\(509\) −0.488846 −0.0216677 −0.0108339 0.999941i \(-0.503449\pi\)
−0.0108339 + 0.999941i \(0.503449\pi\)
\(510\) 0 0
\(511\) 37.2547 1.64805
\(512\) 0 0
\(513\) 20.2071 0.892167
\(514\) 0 0
\(515\) 3.75016 0.165252
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 30.2434 1.32754
\(520\) 0 0
\(521\) 11.1536 0.488650 0.244325 0.969693i \(-0.421434\pi\)
0.244325 + 0.969693i \(0.421434\pi\)
\(522\) 0 0
\(523\) 3.36279 0.147045 0.0735223 0.997294i \(-0.476576\pi\)
0.0735223 + 0.997294i \(0.476576\pi\)
\(524\) 0 0
\(525\) 6.34602 0.276963
\(526\) 0 0
\(527\) −47.2816 −2.05962
\(528\) 0 0
\(529\) 52.3836 2.27755
\(530\) 0 0
\(531\) 0.0261281 0.00113386
\(532\) 0 0
\(533\) −1.88085 −0.0814688
\(534\) 0 0
\(535\) 6.35128 0.274590
\(536\) 0 0
\(537\) 20.4537 0.882642
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2.63681 0.113365 0.0566827 0.998392i \(-0.481948\pi\)
0.0566827 + 0.998392i \(0.481948\pi\)
\(542\) 0 0
\(543\) −1.76711 −0.0758340
\(544\) 0 0
\(545\) −1.32407 −0.0567170
\(546\) 0 0
\(547\) 38.2751 1.63652 0.818262 0.574845i \(-0.194938\pi\)
0.818262 + 0.574845i \(0.194938\pi\)
\(548\) 0 0
\(549\) −0.119424 −0.00509687
\(550\) 0 0
\(551\) 26.4555 1.12704
\(552\) 0 0
\(553\) −4.12681 −0.175490
\(554\) 0 0
\(555\) −3.86182 −0.163925
\(556\) 0 0
\(557\) −18.1087 −0.767289 −0.383645 0.923481i \(-0.625331\pi\)
−0.383645 + 0.923481i \(0.625331\pi\)
\(558\) 0 0
\(559\) −4.85105 −0.205177
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.56855 0.150396 0.0751982 0.997169i \(-0.476041\pi\)
0.0751982 + 0.997169i \(0.476041\pi\)
\(564\) 0 0
\(565\) −5.33169 −0.224306
\(566\) 0 0
\(567\) −32.5921 −1.36874
\(568\) 0 0
\(569\) −40.3751 −1.69261 −0.846307 0.532696i \(-0.821179\pi\)
−0.846307 + 0.532696i \(0.821179\pi\)
\(570\) 0 0
\(571\) 36.3873 1.52276 0.761381 0.648305i \(-0.224522\pi\)
0.761381 + 0.648305i \(0.224522\pi\)
\(572\) 0 0
\(573\) 32.5694 1.36061
\(574\) 0 0
\(575\) −8.68237 −0.362080
\(576\) 0 0
\(577\) −1.48422 −0.0617887 −0.0308944 0.999523i \(-0.509836\pi\)
−0.0308944 + 0.999523i \(0.509836\pi\)
\(578\) 0 0
\(579\) 9.26409 0.385002
\(580\) 0 0
\(581\) 0.0641761 0.00266247
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −0.0780860 −0.00322846
\(586\) 0 0
\(587\) −39.3563 −1.62441 −0.812204 0.583374i \(-0.801732\pi\)
−0.812204 + 0.583374i \(0.801732\pi\)
\(588\) 0 0
\(589\) −33.6866 −1.38803
\(590\) 0 0
\(591\) 6.98168 0.287188
\(592\) 0 0
\(593\) 27.2990 1.12104 0.560518 0.828142i \(-0.310602\pi\)
0.560518 + 0.828142i \(0.310602\pi\)
\(594\) 0 0
\(595\) 20.0083 0.820260
\(596\) 0 0
\(597\) −20.6755 −0.846190
\(598\) 0 0
\(599\) 21.1311 0.863393 0.431696 0.902019i \(-0.357915\pi\)
0.431696 + 0.902019i \(0.357915\pi\)
\(600\) 0 0
\(601\) −2.86260 −0.116768 −0.0583840 0.998294i \(-0.518595\pi\)
−0.0583840 + 0.998294i \(0.518595\pi\)
\(602\) 0 0
\(603\) 0.953672 0.0388365
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 31.9944 1.29861 0.649306 0.760527i \(-0.275059\pi\)
0.649306 + 0.760527i \(0.275059\pi\)
\(608\) 0 0
\(609\) −43.6350 −1.76818
\(610\) 0 0
\(611\) 14.9307 0.604030
\(612\) 0 0
\(613\) −35.6008 −1.43790 −0.718952 0.695060i \(-0.755378\pi\)
−0.718952 + 0.695060i \(0.755378\pi\)
\(614\) 0 0
\(615\) −2.73591 −0.110322
\(616\) 0 0
\(617\) −36.9894 −1.48914 −0.744568 0.667546i \(-0.767345\pi\)
−0.744568 + 0.667546i \(0.767345\pi\)
\(618\) 0 0
\(619\) 33.9911 1.36622 0.683109 0.730317i \(-0.260628\pi\)
0.683109 + 0.730317i \(0.260628\pi\)
\(620\) 0 0
\(621\) 45.5997 1.82985
\(622\) 0 0
\(623\) −31.4720 −1.26090
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.1759 −0.485485
\(630\) 0 0
\(631\) 2.88653 0.114911 0.0574555 0.998348i \(-0.481701\pi\)
0.0574555 + 0.998348i \(0.481701\pi\)
\(632\) 0 0
\(633\) −15.9511 −0.634000
\(634\) 0 0
\(635\) −14.6406 −0.580994
\(636\) 0 0
\(637\) −7.92154 −0.313863
\(638\) 0 0
\(639\) 0.454608 0.0179840
\(640\) 0 0
\(641\) 11.6233 0.459092 0.229546 0.973298i \(-0.426276\pi\)
0.229546 + 0.973298i \(0.426276\pi\)
\(642\) 0 0
\(643\) −27.0864 −1.06818 −0.534091 0.845427i \(-0.679346\pi\)
−0.534091 + 0.845427i \(0.679346\pi\)
\(644\) 0 0
\(645\) −7.05637 −0.277844
\(646\) 0 0
\(647\) 1.66070 0.0652888 0.0326444 0.999467i \(-0.489607\pi\)
0.0326444 + 0.999467i \(0.489607\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 55.5620 2.17764
\(652\) 0 0
\(653\) −40.2905 −1.57669 −0.788345 0.615234i \(-0.789062\pi\)
−0.788345 + 0.615234i \(0.789062\pi\)
\(654\) 0 0
\(655\) −5.93922 −0.232065
\(656\) 0 0
\(657\) −0.666804 −0.0260145
\(658\) 0 0
\(659\) 33.7115 1.31321 0.656607 0.754233i \(-0.271991\pi\)
0.656607 + 0.754233i \(0.271991\pi\)
\(660\) 0 0
\(661\) 42.4279 1.65026 0.825128 0.564946i \(-0.191103\pi\)
0.825128 + 0.564946i \(0.191103\pi\)
\(662\) 0 0
\(663\) 10.8914 0.422986
\(664\) 0 0
\(665\) 14.2553 0.552796
\(666\) 0 0
\(667\) 59.6998 2.31158
\(668\) 0 0
\(669\) −8.12491 −0.314127
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −38.1273 −1.46970 −0.734850 0.678230i \(-0.762747\pi\)
−0.734850 + 0.678230i \(0.762747\pi\)
\(674\) 0 0
\(675\) −5.25199 −0.202149
\(676\) 0 0
\(677\) −12.0095 −0.461561 −0.230781 0.973006i \(-0.574128\pi\)
−0.230781 + 0.973006i \(0.574128\pi\)
\(678\) 0 0
\(679\) 38.9604 1.49516
\(680\) 0 0
\(681\) 6.02089 0.230721
\(682\) 0 0
\(683\) 37.6564 1.44088 0.720441 0.693516i \(-0.243939\pi\)
0.720441 + 0.693516i \(0.243939\pi\)
\(684\) 0 0
\(685\) −14.7698 −0.564324
\(686\) 0 0
\(687\) −15.6312 −0.596369
\(688\) 0 0
\(689\) 14.5769 0.555335
\(690\) 0 0
\(691\) −9.93572 −0.377973 −0.188986 0.981980i \(-0.560520\pi\)
−0.188986 + 0.981980i \(0.560520\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.29252 −0.0490280
\(696\) 0 0
\(697\) −8.62601 −0.326733
\(698\) 0 0
\(699\) −36.8683 −1.39449
\(700\) 0 0
\(701\) 10.0853 0.380915 0.190458 0.981695i \(-0.439003\pi\)
0.190458 + 0.981695i \(0.439003\pi\)
\(702\) 0 0
\(703\) −8.67494 −0.327181
\(704\) 0 0
\(705\) 21.7183 0.817957
\(706\) 0 0
\(707\) 26.5972 1.00029
\(708\) 0 0
\(709\) 5.21637 0.195905 0.0979524 0.995191i \(-0.468771\pi\)
0.0979524 + 0.995191i \(0.468771\pi\)
\(710\) 0 0
\(711\) 0.0738640 0.00277012
\(712\) 0 0
\(713\) −76.0177 −2.84689
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 19.3674 0.723288
\(718\) 0 0
\(719\) −7.90947 −0.294973 −0.147487 0.989064i \(-0.547118\pi\)
−0.147487 + 0.989064i \(0.547118\pi\)
\(720\) 0 0
\(721\) −13.8946 −0.517460
\(722\) 0 0
\(723\) −44.7288 −1.66348
\(724\) 0 0
\(725\) −6.87597 −0.255367
\(726\) 0 0
\(727\) 10.2281 0.379338 0.189669 0.981848i \(-0.439258\pi\)
0.189669 + 0.981848i \(0.439258\pi\)
\(728\) 0 0
\(729\) 27.5702 1.02112
\(730\) 0 0
\(731\) −22.2480 −0.822871
\(732\) 0 0
\(733\) −15.3672 −0.567601 −0.283801 0.958883i \(-0.591595\pi\)
−0.283801 + 0.958883i \(0.591595\pi\)
\(734\) 0 0
\(735\) −11.5227 −0.425023
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −28.2071 −1.03762 −0.518808 0.854891i \(-0.673624\pi\)
−0.518808 + 0.854891i \(0.673624\pi\)
\(740\) 0 0
\(741\) 7.75976 0.285062
\(742\) 0 0
\(743\) 44.1059 1.61809 0.809044 0.587748i \(-0.199985\pi\)
0.809044 + 0.587748i \(0.199985\pi\)
\(744\) 0 0
\(745\) −23.9060 −0.875849
\(746\) 0 0
\(747\) −0.00114866 −4.20272e−5 0
\(748\) 0 0
\(749\) −23.5318 −0.859834
\(750\) 0 0
\(751\) −43.1816 −1.57572 −0.787860 0.615854i \(-0.788811\pi\)
−0.787860 + 0.615854i \(0.788811\pi\)
\(752\) 0 0
\(753\) −6.36839 −0.232077
\(754\) 0 0
\(755\) −8.28048 −0.301358
\(756\) 0 0
\(757\) 19.2800 0.700743 0.350371 0.936611i \(-0.386055\pi\)
0.350371 + 0.936611i \(0.386055\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 34.9095 1.26547 0.632733 0.774370i \(-0.281933\pi\)
0.632733 + 0.774370i \(0.281933\pi\)
\(762\) 0 0
\(763\) 4.90576 0.177600
\(764\) 0 0
\(765\) −0.358119 −0.0129478
\(766\) 0 0
\(767\) 0.463934 0.0167517
\(768\) 0 0
\(769\) 10.7367 0.387177 0.193588 0.981083i \(-0.437987\pi\)
0.193588 + 0.981083i \(0.437987\pi\)
\(770\) 0 0
\(771\) −12.1485 −0.437518
\(772\) 0 0
\(773\) 31.6402 1.13802 0.569010 0.822331i \(-0.307327\pi\)
0.569010 + 0.822331i \(0.307327\pi\)
\(774\) 0 0
\(775\) 8.75541 0.314503
\(776\) 0 0
\(777\) 14.3082 0.513306
\(778\) 0 0
\(779\) −6.14576 −0.220195
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 36.1125 1.29056
\(784\) 0 0
\(785\) 7.96444 0.284263
\(786\) 0 0
\(787\) 15.6444 0.557662 0.278831 0.960340i \(-0.410053\pi\)
0.278831 + 0.960340i \(0.410053\pi\)
\(788\) 0 0
\(789\) 40.0072 1.42429
\(790\) 0 0
\(791\) 19.7542 0.702379
\(792\) 0 0
\(793\) −2.12050 −0.0753011
\(794\) 0 0
\(795\) 21.2036 0.752016
\(796\) 0 0
\(797\) −4.97978 −0.176393 −0.0881964 0.996103i \(-0.528110\pi\)
−0.0881964 + 0.996103i \(0.528110\pi\)
\(798\) 0 0
\(799\) 68.4753 2.42248
\(800\) 0 0
\(801\) 0.563303 0.0199033
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 32.1687 1.13380
\(806\) 0 0
\(807\) −29.2576 −1.02992
\(808\) 0 0
\(809\) 30.8902 1.08604 0.543021 0.839719i \(-0.317280\pi\)
0.543021 + 0.839719i \(0.317280\pi\)
\(810\) 0 0
\(811\) 4.47943 0.157294 0.0786471 0.996903i \(-0.474940\pi\)
0.0786471 + 0.996903i \(0.474940\pi\)
\(812\) 0 0
\(813\) −46.9895 −1.64799
\(814\) 0 0
\(815\) 4.08778 0.143189
\(816\) 0 0
\(817\) −15.8510 −0.554555
\(818\) 0 0
\(819\) 0.289313 0.0101094
\(820\) 0 0
\(821\) 12.4808 0.435583 0.217791 0.975995i \(-0.430115\pi\)
0.217791 + 0.975995i \(0.430115\pi\)
\(822\) 0 0
\(823\) 16.5038 0.575288 0.287644 0.957737i \(-0.407128\pi\)
0.287644 + 0.957737i \(0.407128\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −45.1967 −1.57164 −0.785822 0.618453i \(-0.787760\pi\)
−0.785822 + 0.618453i \(0.787760\pi\)
\(828\) 0 0
\(829\) −11.3874 −0.395502 −0.197751 0.980252i \(-0.563364\pi\)
−0.197751 + 0.980252i \(0.563364\pi\)
\(830\) 0 0
\(831\) −41.4927 −1.43936
\(832\) 0 0
\(833\) −36.3299 −1.25876
\(834\) 0 0
\(835\) 3.98229 0.137813
\(836\) 0 0
\(837\) −45.9833 −1.58941
\(838\) 0 0
\(839\) −14.1100 −0.487131 −0.243565 0.969884i \(-0.578317\pi\)
−0.243565 + 0.969884i \(0.578317\pi\)
\(840\) 0 0
\(841\) 18.2790 0.630310
\(842\) 0 0
\(843\) −29.3975 −1.01250
\(844\) 0 0
\(845\) 11.6135 0.399516
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 7.92870 0.272112
\(850\) 0 0
\(851\) −19.5760 −0.671056
\(852\) 0 0
\(853\) −20.9805 −0.718360 −0.359180 0.933268i \(-0.616944\pi\)
−0.359180 + 0.933268i \(0.616944\pi\)
\(854\) 0 0
\(855\) −0.255149 −0.00872590
\(856\) 0 0
\(857\) 25.0649 0.856201 0.428101 0.903731i \(-0.359183\pi\)
0.428101 + 0.903731i \(0.359183\pi\)
\(858\) 0 0
\(859\) −38.0496 −1.29823 −0.649117 0.760689i \(-0.724861\pi\)
−0.649117 + 0.760689i \(0.724861\pi\)
\(860\) 0 0
\(861\) 10.1367 0.345457
\(862\) 0 0
\(863\) −8.58888 −0.292369 −0.146184 0.989257i \(-0.546699\pi\)
−0.146184 + 0.989257i \(0.546699\pi\)
\(864\) 0 0
\(865\) −17.6573 −0.600366
\(866\) 0 0
\(867\) 20.8327 0.707514
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 16.9335 0.573771
\(872\) 0 0
\(873\) −0.697334 −0.0236012
\(874\) 0 0
\(875\) −3.70505 −0.125254
\(876\) 0 0
\(877\) −10.5663 −0.356798 −0.178399 0.983958i \(-0.557092\pi\)
−0.178399 + 0.983958i \(0.557092\pi\)
\(878\) 0 0
\(879\) −31.6879 −1.06881
\(880\) 0 0
\(881\) −35.5069 −1.19626 −0.598129 0.801400i \(-0.704089\pi\)
−0.598129 + 0.801400i \(0.704089\pi\)
\(882\) 0 0
\(883\) −34.0160 −1.14473 −0.572364 0.820000i \(-0.693974\pi\)
−0.572364 + 0.820000i \(0.693974\pi\)
\(884\) 0 0
\(885\) 0.674842 0.0226846
\(886\) 0 0
\(887\) −9.99766 −0.335689 −0.167844 0.985814i \(-0.553681\pi\)
−0.167844 + 0.985814i \(0.553681\pi\)
\(888\) 0 0
\(889\) 54.2442 1.81929
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 48.7865 1.63258
\(894\) 0 0
\(895\) −11.9417 −0.399166
\(896\) 0 0
\(897\) 17.5108 0.584668
\(898\) 0 0
\(899\) −60.2019 −2.00785
\(900\) 0 0
\(901\) 66.8528 2.22719
\(902\) 0 0
\(903\) 26.1442 0.870026
\(904\) 0 0
\(905\) 1.03171 0.0342952
\(906\) 0 0
\(907\) 14.3972 0.478051 0.239025 0.971013i \(-0.423172\pi\)
0.239025 + 0.971013i \(0.423172\pi\)
\(908\) 0 0
\(909\) −0.476051 −0.0157896
\(910\) 0 0
\(911\) −16.1765 −0.535951 −0.267975 0.963426i \(-0.586355\pi\)
−0.267975 + 0.963426i \(0.586355\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −3.08450 −0.101970
\(916\) 0 0
\(917\) 22.0051 0.726674
\(918\) 0 0
\(919\) −18.1790 −0.599671 −0.299836 0.953991i \(-0.596932\pi\)
−0.299836 + 0.953991i \(0.596932\pi\)
\(920\) 0 0
\(921\) −46.5917 −1.53525
\(922\) 0 0
\(923\) 8.07207 0.265696
\(924\) 0 0
\(925\) 2.25468 0.0741335
\(926\) 0 0
\(927\) 0.248692 0.00816813
\(928\) 0 0
\(929\) −24.3272 −0.798150 −0.399075 0.916918i \(-0.630669\pi\)
−0.399075 + 0.916918i \(0.630669\pi\)
\(930\) 0 0
\(931\) −25.8839 −0.848311
\(932\) 0 0
\(933\) 55.2547 1.80896
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 24.1898 0.790246 0.395123 0.918628i \(-0.370702\pi\)
0.395123 + 0.918628i \(0.370702\pi\)
\(938\) 0 0
\(939\) −0.497994 −0.0162514
\(940\) 0 0
\(941\) −6.12261 −0.199591 −0.0997956 0.995008i \(-0.531819\pi\)
−0.0997956 + 0.995008i \(0.531819\pi\)
\(942\) 0 0
\(943\) −13.8686 −0.451624
\(944\) 0 0
\(945\) 19.4589 0.632998
\(946\) 0 0
\(947\) 30.4758 0.990330 0.495165 0.868799i \(-0.335108\pi\)
0.495165 + 0.868799i \(0.335108\pi\)
\(948\) 0 0
\(949\) −11.8399 −0.384338
\(950\) 0 0
\(951\) 39.1110 1.26826
\(952\) 0 0
\(953\) 12.3055 0.398614 0.199307 0.979937i \(-0.436131\pi\)
0.199307 + 0.979937i \(0.436131\pi\)
\(954\) 0 0
\(955\) −19.0153 −0.615321
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 54.7228 1.76709
\(960\) 0 0
\(961\) 45.6571 1.47281
\(962\) 0 0
\(963\) 0.421186 0.0135725
\(964\) 0 0
\(965\) −5.40874 −0.174113
\(966\) 0 0
\(967\) 22.5927 0.726532 0.363266 0.931685i \(-0.381662\pi\)
0.363266 + 0.931685i \(0.381662\pi\)
\(968\) 0 0
\(969\) 35.5880 1.14325
\(970\) 0 0
\(971\) 43.2544 1.38810 0.694050 0.719927i \(-0.255825\pi\)
0.694050 + 0.719927i \(0.255825\pi\)
\(972\) 0 0
\(973\) 4.78884 0.153523
\(974\) 0 0
\(975\) −2.01682 −0.0645900
\(976\) 0 0
\(977\) 15.0741 0.482263 0.241131 0.970492i \(-0.422482\pi\)
0.241131 + 0.970492i \(0.422482\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −0.0878059 −0.00280343
\(982\) 0 0
\(983\) −41.9188 −1.33700 −0.668502 0.743711i \(-0.733064\pi\)
−0.668502 + 0.743711i \(0.733064\pi\)
\(984\) 0 0
\(985\) −4.07618 −0.129878
\(986\) 0 0
\(987\) −80.4673 −2.56130
\(988\) 0 0
\(989\) −35.7695 −1.13740
\(990\) 0 0
\(991\) 29.5068 0.937314 0.468657 0.883380i \(-0.344738\pi\)
0.468657 + 0.883380i \(0.344738\pi\)
\(992\) 0 0
\(993\) −24.8805 −0.789559
\(994\) 0 0
\(995\) 12.0711 0.382681
\(996\) 0 0
\(997\) 48.6155 1.53967 0.769835 0.638243i \(-0.220339\pi\)
0.769835 + 0.638243i \(0.220339\pi\)
\(998\) 0 0
\(999\) −11.8416 −0.374650
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.bb.1.6 6
4.3 odd 2 9680.2.a.dc.1.1 6
11.5 even 5 440.2.y.c.201.3 yes 12
11.9 even 5 440.2.y.c.81.3 12
11.10 odd 2 4840.2.a.ba.1.6 6
44.27 odd 10 880.2.bo.i.641.1 12
44.31 odd 10 880.2.bo.i.81.1 12
44.43 even 2 9680.2.a.dd.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.c.81.3 12 11.9 even 5
440.2.y.c.201.3 yes 12 11.5 even 5
880.2.bo.i.81.1 12 44.31 odd 10
880.2.bo.i.641.1 12 44.27 odd 10
4840.2.a.ba.1.6 6 11.10 odd 2
4840.2.a.bb.1.6 6 1.1 even 1 trivial
9680.2.a.dc.1.1 6 4.3 odd 2
9680.2.a.dd.1.1 6 44.43 even 2