Properties

Label 4840.2.a.bd.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 $1$

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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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.22733568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 8x^{4} - 2x^{3} + 16x^{2} + 8x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.45825\) 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)
 
\(f(q)\) \(=\) \(q-0.873518 q^{3} +1.00000 q^{5} -3.64048 q^{7} -2.23697 q^{9} +O(q^{10})\) \(q-0.873518 q^{3} +1.00000 q^{5} -3.64048 q^{7} -2.23697 q^{9} -1.06751 q^{13} -0.873518 q^{15} +0.252963 q^{17} +3.06751 q^{19} +3.18003 q^{21} -7.04399 q^{23} +1.00000 q^{25} +4.57459 q^{27} +1.61202 q^{29} -0.788991 q^{31} -3.64048 q^{35} -4.91208 q^{37} +0.932490 q^{39} +6.39545 q^{41} -2.63065 q^{43} -2.23697 q^{45} -7.78559 q^{47} +6.25309 q^{49} -0.220968 q^{51} -6.32291 q^{53} -2.67953 q^{57} -5.48214 q^{59} -6.80160 q^{61} +8.14363 q^{63} -1.06751 q^{65} -9.49860 q^{67} +6.15306 q^{69} +7.53405 q^{71} +16.4419 q^{73} -0.873518 q^{75} -3.02596 q^{79} +2.71491 q^{81} +13.1746 q^{83} +0.252963 q^{85} -1.40813 q^{87} +12.4460 q^{89} +3.88625 q^{91} +0.689198 q^{93} +3.06751 q^{95} +8.77719 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} + 6 q^{5} + 4 q^{7} + 4 q^{9} - 2 q^{15} + 8 q^{17} + 12 q^{19} - 8 q^{21} - 8 q^{23} + 6 q^{25} - 14 q^{27} + 16 q^{29} - 4 q^{31} + 4 q^{35} + 8 q^{37} + 12 q^{39} + 32 q^{41} - 4 q^{43} + 4 q^{45} - 6 q^{47} + 16 q^{49} + 40 q^{51} + 8 q^{53} - 16 q^{57} + 4 q^{59} + 16 q^{61} + 28 q^{63} - 2 q^{67} + 8 q^{69} - 28 q^{71} + 16 q^{73} - 2 q^{75} - 10 q^{81} + 12 q^{83} + 8 q^{85} - 24 q^{87} + 18 q^{89} - 24 q^{91} + 20 q^{93} + 12 q^{95}+O(q^{100}) \) 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
\(3\) −0.873518 −0.504326 −0.252163 0.967685i \(-0.581142\pi\)
−0.252163 + 0.967685i \(0.581142\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −3.64048 −1.37597 −0.687986 0.725724i \(-0.741505\pi\)
−0.687986 + 0.725724i \(0.741505\pi\)
\(8\) 0 0
\(9\) −2.23697 −0.745655
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −1.06751 −0.296074 −0.148037 0.988982i \(-0.547295\pi\)
−0.148037 + 0.988982i \(0.547295\pi\)
\(14\) 0 0
\(15\) −0.873518 −0.225541
\(16\) 0 0
\(17\) 0.252963 0.0613526 0.0306763 0.999529i \(-0.490234\pi\)
0.0306763 + 0.999529i \(0.490234\pi\)
\(18\) 0 0
\(19\) 3.06751 0.703735 0.351868 0.936050i \(-0.385547\pi\)
0.351868 + 0.936050i \(0.385547\pi\)
\(20\) 0 0
\(21\) 3.18003 0.693939
\(22\) 0 0
\(23\) −7.04399 −1.46877 −0.734387 0.678731i \(-0.762530\pi\)
−0.734387 + 0.678731i \(0.762530\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.57459 0.880379
\(28\) 0 0
\(29\) 1.61202 0.299344 0.149672 0.988736i \(-0.452178\pi\)
0.149672 + 0.988736i \(0.452178\pi\)
\(30\) 0 0
\(31\) −0.788991 −0.141707 −0.0708535 0.997487i \(-0.522572\pi\)
−0.0708535 + 0.997487i \(0.522572\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.64048 −0.615353
\(36\) 0 0
\(37\) −4.91208 −0.807540 −0.403770 0.914860i \(-0.632300\pi\)
−0.403770 + 0.914860i \(0.632300\pi\)
\(38\) 0 0
\(39\) 0.932490 0.149318
\(40\) 0 0
\(41\) 6.39545 0.998802 0.499401 0.866371i \(-0.333554\pi\)
0.499401 + 0.866371i \(0.333554\pi\)
\(42\) 0 0
\(43\) −2.63065 −0.401170 −0.200585 0.979676i \(-0.564284\pi\)
−0.200585 + 0.979676i \(0.564284\pi\)
\(44\) 0 0
\(45\) −2.23697 −0.333467
\(46\) 0 0
\(47\) −7.78559 −1.13565 −0.567823 0.823151i \(-0.692214\pi\)
−0.567823 + 0.823151i \(0.692214\pi\)
\(48\) 0 0
\(49\) 6.25309 0.893299
\(50\) 0 0
\(51\) −0.220968 −0.0309417
\(52\) 0 0
\(53\) −6.32291 −0.868519 −0.434259 0.900788i \(-0.642990\pi\)
−0.434259 + 0.900788i \(0.642990\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.67953 −0.354912
\(58\) 0 0
\(59\) −5.48214 −0.713714 −0.356857 0.934159i \(-0.616152\pi\)
−0.356857 + 0.934159i \(0.616152\pi\)
\(60\) 0 0
\(61\) −6.80160 −0.870856 −0.435428 0.900224i \(-0.643403\pi\)
−0.435428 + 0.900224i \(0.643403\pi\)
\(62\) 0 0
\(63\) 8.14363 1.02600
\(64\) 0 0
\(65\) −1.06751 −0.132408
\(66\) 0 0
\(67\) −9.49860 −1.16044 −0.580219 0.814460i \(-0.697033\pi\)
−0.580219 + 0.814460i \(0.697033\pi\)
\(68\) 0 0
\(69\) 6.15306 0.740741
\(70\) 0 0
\(71\) 7.53405 0.894127 0.447064 0.894502i \(-0.352470\pi\)
0.447064 + 0.894502i \(0.352470\pi\)
\(72\) 0 0
\(73\) 16.4419 1.92438 0.962192 0.272374i \(-0.0878087\pi\)
0.962192 + 0.272374i \(0.0878087\pi\)
\(74\) 0 0
\(75\) −0.873518 −0.100865
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.02596 −0.340447 −0.170223 0.985406i \(-0.554449\pi\)
−0.170223 + 0.985406i \(0.554449\pi\)
\(80\) 0 0
\(81\) 2.71491 0.301657
\(82\) 0 0
\(83\) 13.1746 1.44610 0.723050 0.690796i \(-0.242740\pi\)
0.723050 + 0.690796i \(0.242740\pi\)
\(84\) 0 0
\(85\) 0.252963 0.0274377
\(86\) 0 0
\(87\) −1.40813 −0.150967
\(88\) 0 0
\(89\) 12.4460 1.31927 0.659634 0.751587i \(-0.270711\pi\)
0.659634 + 0.751587i \(0.270711\pi\)
\(90\) 0 0
\(91\) 3.88625 0.407390
\(92\) 0 0
\(93\) 0.689198 0.0714665
\(94\) 0 0
\(95\) 3.06751 0.314720
\(96\) 0 0
\(97\) 8.77719 0.891188 0.445594 0.895235i \(-0.352992\pi\)
0.445594 + 0.895235i \(0.352992\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 16.1579 1.60777 0.803884 0.594787i \(-0.202763\pi\)
0.803884 + 0.594787i \(0.202763\pi\)
\(102\) 0 0
\(103\) 4.58972 0.452239 0.226119 0.974100i \(-0.427396\pi\)
0.226119 + 0.974100i \(0.427396\pi\)
\(104\) 0 0
\(105\) 3.18003 0.310339
\(106\) 0 0
\(107\) 0.823893 0.0796487 0.0398244 0.999207i \(-0.487320\pi\)
0.0398244 + 0.999207i \(0.487320\pi\)
\(108\) 0 0
\(109\) −6.43124 −0.616001 −0.308000 0.951386i \(-0.599660\pi\)
−0.308000 + 0.951386i \(0.599660\pi\)
\(110\) 0 0
\(111\) 4.29079 0.407264
\(112\) 0 0
\(113\) −5.96800 −0.561423 −0.280711 0.959792i \(-0.590570\pi\)
−0.280711 + 0.959792i \(0.590570\pi\)
\(114\) 0 0
\(115\) −7.04399 −0.656856
\(116\) 0 0
\(117\) 2.38798 0.220769
\(118\) 0 0
\(119\) −0.920908 −0.0844195
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −5.58654 −0.503722
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −9.41320 −0.835287 −0.417643 0.908611i \(-0.637144\pi\)
−0.417643 + 0.908611i \(0.637144\pi\)
\(128\) 0 0
\(129\) 2.29792 0.202321
\(130\) 0 0
\(131\) −14.2507 −1.24509 −0.622545 0.782584i \(-0.713901\pi\)
−0.622545 + 0.782584i \(0.713901\pi\)
\(132\) 0 0
\(133\) −11.1672 −0.968320
\(134\) 0 0
\(135\) 4.57459 0.393718
\(136\) 0 0
\(137\) 21.6893 1.85304 0.926519 0.376247i \(-0.122786\pi\)
0.926519 + 0.376247i \(0.122786\pi\)
\(138\) 0 0
\(139\) 8.26252 0.700818 0.350409 0.936597i \(-0.386043\pi\)
0.350409 + 0.936597i \(0.386043\pi\)
\(140\) 0 0
\(141\) 6.80086 0.572736
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 1.61202 0.133871
\(146\) 0 0
\(147\) −5.46219 −0.450514
\(148\) 0 0
\(149\) 6.82953 0.559497 0.279749 0.960073i \(-0.409749\pi\)
0.279749 + 0.960073i \(0.409749\pi\)
\(150\) 0 0
\(151\) 17.0217 1.38520 0.692602 0.721320i \(-0.256464\pi\)
0.692602 + 0.721320i \(0.256464\pi\)
\(152\) 0 0
\(153\) −0.565870 −0.0457479
\(154\) 0 0
\(155\) −0.788991 −0.0633733
\(156\) 0 0
\(157\) 20.4419 1.63144 0.815722 0.578444i \(-0.196340\pi\)
0.815722 + 0.578444i \(0.196340\pi\)
\(158\) 0 0
\(159\) 5.52318 0.438017
\(160\) 0 0
\(161\) 25.6435 2.02099
\(162\) 0 0
\(163\) 11.3516 0.889129 0.444565 0.895747i \(-0.353358\pi\)
0.444565 + 0.895747i \(0.353358\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.6105 0.898449 0.449225 0.893419i \(-0.351700\pi\)
0.449225 + 0.893419i \(0.351700\pi\)
\(168\) 0 0
\(169\) −11.8604 −0.912340
\(170\) 0 0
\(171\) −6.86192 −0.524744
\(172\) 0 0
\(173\) −14.5576 −1.10680 −0.553398 0.832917i \(-0.686669\pi\)
−0.553398 + 0.832917i \(0.686669\pi\)
\(174\) 0 0
\(175\) −3.64048 −0.275194
\(176\) 0 0
\(177\) 4.78875 0.359944
\(178\) 0 0
\(179\) −15.3520 −1.14746 −0.573730 0.819044i \(-0.694504\pi\)
−0.573730 + 0.819044i \(0.694504\pi\)
\(180\) 0 0
\(181\) 19.4760 1.44764 0.723819 0.689990i \(-0.242385\pi\)
0.723819 + 0.689990i \(0.242385\pi\)
\(182\) 0 0
\(183\) 5.94132 0.439195
\(184\) 0 0
\(185\) −4.91208 −0.361143
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −16.6537 −1.21138
\(190\) 0 0
\(191\) −23.2241 −1.68044 −0.840220 0.542246i \(-0.817574\pi\)
−0.840220 + 0.542246i \(0.817574\pi\)
\(192\) 0 0
\(193\) 18.3389 1.32006 0.660032 0.751237i \(-0.270543\pi\)
0.660032 + 0.751237i \(0.270543\pi\)
\(194\) 0 0
\(195\) 0.932490 0.0667770
\(196\) 0 0
\(197\) −10.5899 −0.754496 −0.377248 0.926112i \(-0.623129\pi\)
−0.377248 + 0.926112i \(0.623129\pi\)
\(198\) 0 0
\(199\) −14.7745 −1.04733 −0.523667 0.851923i \(-0.675437\pi\)
−0.523667 + 0.851923i \(0.675437\pi\)
\(200\) 0 0
\(201\) 8.29720 0.585239
\(202\) 0 0
\(203\) −5.86851 −0.411889
\(204\) 0 0
\(205\) 6.39545 0.446678
\(206\) 0 0
\(207\) 15.7572 1.09520
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 23.1907 1.59652 0.798258 0.602316i \(-0.205755\pi\)
0.798258 + 0.602316i \(0.205755\pi\)
\(212\) 0 0
\(213\) −6.58113 −0.450932
\(214\) 0 0
\(215\) −2.63065 −0.179409
\(216\) 0 0
\(217\) 2.87230 0.194985
\(218\) 0 0
\(219\) −14.3623 −0.970517
\(220\) 0 0
\(221\) −0.270041 −0.0181649
\(222\) 0 0
\(223\) −1.51136 −0.101208 −0.0506041 0.998719i \(-0.516115\pi\)
−0.0506041 + 0.998719i \(0.516115\pi\)
\(224\) 0 0
\(225\) −2.23697 −0.149131
\(226\) 0 0
\(227\) 12.9716 0.860955 0.430478 0.902601i \(-0.358345\pi\)
0.430478 + 0.902601i \(0.358345\pi\)
\(228\) 0 0
\(229\) 24.8265 1.64058 0.820289 0.571949i \(-0.193813\pi\)
0.820289 + 0.571949i \(0.193813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −16.4206 −1.07575 −0.537874 0.843025i \(-0.680772\pi\)
−0.537874 + 0.843025i \(0.680772\pi\)
\(234\) 0 0
\(235\) −7.78559 −0.507876
\(236\) 0 0
\(237\) 2.64323 0.171696
\(238\) 0 0
\(239\) 22.2830 1.44137 0.720683 0.693265i \(-0.243828\pi\)
0.720683 + 0.693265i \(0.243828\pi\)
\(240\) 0 0
\(241\) 22.5887 1.45507 0.727534 0.686071i \(-0.240666\pi\)
0.727534 + 0.686071i \(0.240666\pi\)
\(242\) 0 0
\(243\) −16.0953 −1.03251
\(244\) 0 0
\(245\) 6.25309 0.399495
\(246\) 0 0
\(247\) −3.27460 −0.208358
\(248\) 0 0
\(249\) −11.5083 −0.729306
\(250\) 0 0
\(251\) −13.6602 −0.862227 −0.431113 0.902298i \(-0.641879\pi\)
−0.431113 + 0.902298i \(0.641879\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −0.220968 −0.0138376
\(256\) 0 0
\(257\) −14.2393 −0.888221 −0.444110 0.895972i \(-0.646480\pi\)
−0.444110 + 0.895972i \(0.646480\pi\)
\(258\) 0 0
\(259\) 17.8823 1.11115
\(260\) 0 0
\(261\) −3.60603 −0.223207
\(262\) 0 0
\(263\) 6.28952 0.387828 0.193914 0.981018i \(-0.437882\pi\)
0.193914 + 0.981018i \(0.437882\pi\)
\(264\) 0 0
\(265\) −6.32291 −0.388413
\(266\) 0 0
\(267\) −10.8718 −0.665341
\(268\) 0 0
\(269\) 21.6774 1.32169 0.660847 0.750520i \(-0.270197\pi\)
0.660847 + 0.750520i \(0.270197\pi\)
\(270\) 0 0
\(271\) −3.40417 −0.206789 −0.103394 0.994640i \(-0.532970\pi\)
−0.103394 + 0.994640i \(0.532970\pi\)
\(272\) 0 0
\(273\) −3.39471 −0.205457
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −18.7849 −1.12868 −0.564338 0.825544i \(-0.690868\pi\)
−0.564338 + 0.825544i \(0.690868\pi\)
\(278\) 0 0
\(279\) 1.76495 0.105664
\(280\) 0 0
\(281\) 15.1832 0.905754 0.452877 0.891573i \(-0.350398\pi\)
0.452877 + 0.891573i \(0.350398\pi\)
\(282\) 0 0
\(283\) 10.2021 0.606450 0.303225 0.952919i \(-0.401937\pi\)
0.303225 + 0.952919i \(0.401937\pi\)
\(284\) 0 0
\(285\) −2.67953 −0.158721
\(286\) 0 0
\(287\) −23.2825 −1.37432
\(288\) 0 0
\(289\) −16.9360 −0.996236
\(290\) 0 0
\(291\) −7.66703 −0.449449
\(292\) 0 0
\(293\) −2.84914 −0.166449 −0.0832243 0.996531i \(-0.526522\pi\)
−0.0832243 + 0.996531i \(0.526522\pi\)
\(294\) 0 0
\(295\) −5.48214 −0.319182
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.51954 0.434866
\(300\) 0 0
\(301\) 9.57683 0.551999
\(302\) 0 0
\(303\) −14.1142 −0.810839
\(304\) 0 0
\(305\) −6.80160 −0.389459
\(306\) 0 0
\(307\) 9.11918 0.520459 0.260230 0.965547i \(-0.416202\pi\)
0.260230 + 0.965547i \(0.416202\pi\)
\(308\) 0 0
\(309\) −4.00921 −0.228076
\(310\) 0 0
\(311\) 14.9333 0.846792 0.423396 0.905945i \(-0.360838\pi\)
0.423396 + 0.905945i \(0.360838\pi\)
\(312\) 0 0
\(313\) 13.9590 0.789012 0.394506 0.918893i \(-0.370916\pi\)
0.394506 + 0.918893i \(0.370916\pi\)
\(314\) 0 0
\(315\) 8.14363 0.458841
\(316\) 0 0
\(317\) 15.2502 0.856534 0.428267 0.903652i \(-0.359124\pi\)
0.428267 + 0.903652i \(0.359124\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −0.719686 −0.0401689
\(322\) 0 0
\(323\) 0.775967 0.0431760
\(324\) 0 0
\(325\) −1.06751 −0.0592148
\(326\) 0 0
\(327\) 5.61780 0.310665
\(328\) 0 0
\(329\) 28.3433 1.56262
\(330\) 0 0
\(331\) 17.1961 0.945183 0.472591 0.881282i \(-0.343319\pi\)
0.472591 + 0.881282i \(0.343319\pi\)
\(332\) 0 0
\(333\) 10.9881 0.602147
\(334\) 0 0
\(335\) −9.49860 −0.518964
\(336\) 0 0
\(337\) −27.4167 −1.49348 −0.746742 0.665114i \(-0.768383\pi\)
−0.746742 + 0.665114i \(0.768383\pi\)
\(338\) 0 0
\(339\) 5.21316 0.283140
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 2.71910 0.146818
\(344\) 0 0
\(345\) 6.15306 0.331269
\(346\) 0 0
\(347\) −1.52482 −0.0818565 −0.0409283 0.999162i \(-0.513032\pi\)
−0.0409283 + 0.999162i \(0.513032\pi\)
\(348\) 0 0
\(349\) 8.48480 0.454181 0.227091 0.973874i \(-0.427079\pi\)
0.227091 + 0.973874i \(0.427079\pi\)
\(350\) 0 0
\(351\) −4.88342 −0.260658
\(352\) 0 0
\(353\) 3.14700 0.167498 0.0837490 0.996487i \(-0.473311\pi\)
0.0837490 + 0.996487i \(0.473311\pi\)
\(354\) 0 0
\(355\) 7.53405 0.399866
\(356\) 0 0
\(357\) 0.804430 0.0425749
\(358\) 0 0
\(359\) −22.0212 −1.16223 −0.581117 0.813820i \(-0.697384\pi\)
−0.581117 + 0.813820i \(0.697384\pi\)
\(360\) 0 0
\(361\) −9.59038 −0.504757
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 16.4419 0.860610
\(366\) 0 0
\(367\) −15.8706 −0.828438 −0.414219 0.910177i \(-0.635945\pi\)
−0.414219 + 0.910177i \(0.635945\pi\)
\(368\) 0 0
\(369\) −14.3064 −0.744762
\(370\) 0 0
\(371\) 23.0184 1.19506
\(372\) 0 0
\(373\) 9.75306 0.504994 0.252497 0.967598i \(-0.418748\pi\)
0.252497 + 0.967598i \(0.418748\pi\)
\(374\) 0 0
\(375\) −0.873518 −0.0451083
\(376\) 0 0
\(377\) −1.72084 −0.0886280
\(378\) 0 0
\(379\) −7.56281 −0.388475 −0.194238 0.980955i \(-0.562223\pi\)
−0.194238 + 0.980955i \(0.562223\pi\)
\(380\) 0 0
\(381\) 8.22260 0.421257
\(382\) 0 0
\(383\) −8.24307 −0.421201 −0.210601 0.977572i \(-0.567542\pi\)
−0.210601 + 0.977572i \(0.567542\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 5.88467 0.299135
\(388\) 0 0
\(389\) −24.6089 −1.24772 −0.623860 0.781536i \(-0.714436\pi\)
−0.623860 + 0.781536i \(0.714436\pi\)
\(390\) 0 0
\(391\) −1.78187 −0.0901131
\(392\) 0 0
\(393\) 12.4483 0.627932
\(394\) 0 0
\(395\) −3.02596 −0.152252
\(396\) 0 0
\(397\) −24.6461 −1.23695 −0.618476 0.785804i \(-0.712250\pi\)
−0.618476 + 0.785804i \(0.712250\pi\)
\(398\) 0 0
\(399\) 9.75476 0.488349
\(400\) 0 0
\(401\) −3.71811 −0.185673 −0.0928367 0.995681i \(-0.529593\pi\)
−0.0928367 + 0.995681i \(0.529593\pi\)
\(402\) 0 0
\(403\) 0.842256 0.0419557
\(404\) 0 0
\(405\) 2.71491 0.134905
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −5.95988 −0.294697 −0.147348 0.989085i \(-0.547074\pi\)
−0.147348 + 0.989085i \(0.547074\pi\)
\(410\) 0 0
\(411\) −18.9460 −0.934536
\(412\) 0 0
\(413\) 19.9576 0.982050
\(414\) 0 0
\(415\) 13.1746 0.646716
\(416\) 0 0
\(417\) −7.21746 −0.353441
\(418\) 0 0
\(419\) −7.88194 −0.385058 −0.192529 0.981291i \(-0.561669\pi\)
−0.192529 + 0.981291i \(0.561669\pi\)
\(420\) 0 0
\(421\) 31.5497 1.53764 0.768819 0.639467i \(-0.220845\pi\)
0.768819 + 0.639467i \(0.220845\pi\)
\(422\) 0 0
\(423\) 17.4161 0.846800
\(424\) 0 0
\(425\) 0.252963 0.0122705
\(426\) 0 0
\(427\) 24.7611 1.19827
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.21311 0.395612 0.197806 0.980241i \(-0.436618\pi\)
0.197806 + 0.980241i \(0.436618\pi\)
\(432\) 0 0
\(433\) −26.4771 −1.27241 −0.636203 0.771522i \(-0.719496\pi\)
−0.636203 + 0.771522i \(0.719496\pi\)
\(434\) 0 0
\(435\) −1.40813 −0.0675145
\(436\) 0 0
\(437\) −21.6075 −1.03363
\(438\) 0 0
\(439\) 6.90682 0.329645 0.164822 0.986323i \(-0.447295\pi\)
0.164822 + 0.986323i \(0.447295\pi\)
\(440\) 0 0
\(441\) −13.9880 −0.666093
\(442\) 0 0
\(443\) −15.3657 −0.730048 −0.365024 0.930998i \(-0.618939\pi\)
−0.365024 + 0.930998i \(0.618939\pi\)
\(444\) 0 0
\(445\) 12.4460 0.589995
\(446\) 0 0
\(447\) −5.96572 −0.282169
\(448\) 0 0
\(449\) −28.9971 −1.36846 −0.684230 0.729267i \(-0.739861\pi\)
−0.684230 + 0.729267i \(0.739861\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −14.8687 −0.698594
\(454\) 0 0
\(455\) 3.88625 0.182190
\(456\) 0 0
\(457\) 11.6957 0.547102 0.273551 0.961858i \(-0.411802\pi\)
0.273551 + 0.961858i \(0.411802\pi\)
\(458\) 0 0
\(459\) 1.15720 0.0540136
\(460\) 0 0
\(461\) 14.1531 0.659177 0.329589 0.944125i \(-0.393090\pi\)
0.329589 + 0.944125i \(0.393090\pi\)
\(462\) 0 0
\(463\) 18.1587 0.843906 0.421953 0.906618i \(-0.361345\pi\)
0.421953 + 0.906618i \(0.361345\pi\)
\(464\) 0 0
\(465\) 0.689198 0.0319608
\(466\) 0 0
\(467\) 23.3999 1.08282 0.541408 0.840760i \(-0.317891\pi\)
0.541408 + 0.840760i \(0.317891\pi\)
\(468\) 0 0
\(469\) 34.5794 1.59673
\(470\) 0 0
\(471\) −17.8564 −0.822780
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 3.06751 0.140747
\(476\) 0 0
\(477\) 14.1441 0.647616
\(478\) 0 0
\(479\) 28.3210 1.29402 0.647010 0.762481i \(-0.276019\pi\)
0.647010 + 0.762481i \(0.276019\pi\)
\(480\) 0 0
\(481\) 5.24369 0.239092
\(482\) 0 0
\(483\) −22.4001 −1.01924
\(484\) 0 0
\(485\) 8.77719 0.398551
\(486\) 0 0
\(487\) −34.4523 −1.56118 −0.780592 0.625041i \(-0.785082\pi\)
−0.780592 + 0.625041i \(0.785082\pi\)
\(488\) 0 0
\(489\) −9.91587 −0.448411
\(490\) 0 0
\(491\) 28.0569 1.26619 0.633096 0.774074i \(-0.281784\pi\)
0.633096 + 0.774074i \(0.281784\pi\)
\(492\) 0 0
\(493\) 0.407781 0.0183655
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −27.4276 −1.23029
\(498\) 0 0
\(499\) 40.6597 1.82018 0.910089 0.414413i \(-0.136013\pi\)
0.910089 + 0.414413i \(0.136013\pi\)
\(500\) 0 0
\(501\) −10.1420 −0.453111
\(502\) 0 0
\(503\) 35.5947 1.58709 0.793544 0.608513i \(-0.208234\pi\)
0.793544 + 0.608513i \(0.208234\pi\)
\(504\) 0 0
\(505\) 16.1579 0.719015
\(506\) 0 0
\(507\) 10.3603 0.460117
\(508\) 0 0
\(509\) −20.1064 −0.891202 −0.445601 0.895232i \(-0.647010\pi\)
−0.445601 + 0.895232i \(0.647010\pi\)
\(510\) 0 0
\(511\) −59.8565 −2.64790
\(512\) 0 0
\(513\) 14.0326 0.619554
\(514\) 0 0
\(515\) 4.58972 0.202247
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 12.7164 0.558186
\(520\) 0 0
\(521\) 23.0122 1.00818 0.504091 0.863651i \(-0.331828\pi\)
0.504091 + 0.863651i \(0.331828\pi\)
\(522\) 0 0
\(523\) 1.59213 0.0696191 0.0348096 0.999394i \(-0.488918\pi\)
0.0348096 + 0.999394i \(0.488918\pi\)
\(524\) 0 0
\(525\) 3.18003 0.138788
\(526\) 0 0
\(527\) −0.199586 −0.00869409
\(528\) 0 0
\(529\) 26.6178 1.15730
\(530\) 0 0
\(531\) 12.2634 0.532184
\(532\) 0 0
\(533\) −6.82721 −0.295719
\(534\) 0 0
\(535\) 0.823893 0.0356200
\(536\) 0 0
\(537\) 13.4102 0.578694
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 16.4539 0.707410 0.353705 0.935357i \(-0.384922\pi\)
0.353705 + 0.935357i \(0.384922\pi\)
\(542\) 0 0
\(543\) −17.0126 −0.730081
\(544\) 0 0
\(545\) −6.43124 −0.275484
\(546\) 0 0
\(547\) −36.0646 −1.54201 −0.771005 0.636829i \(-0.780246\pi\)
−0.771005 + 0.636829i \(0.780246\pi\)
\(548\) 0 0
\(549\) 15.2149 0.649358
\(550\) 0 0
\(551\) 4.94488 0.210659
\(552\) 0 0
\(553\) 11.0159 0.468445
\(554\) 0 0
\(555\) 4.29079 0.182134
\(556\) 0 0
\(557\) −24.9630 −1.05772 −0.528859 0.848710i \(-0.677380\pi\)
−0.528859 + 0.848710i \(0.677380\pi\)
\(558\) 0 0
\(559\) 2.80825 0.118776
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 32.6458 1.37586 0.687929 0.725778i \(-0.258520\pi\)
0.687929 + 0.725778i \(0.258520\pi\)
\(564\) 0 0
\(565\) −5.96800 −0.251076
\(566\) 0 0
\(567\) −9.88358 −0.415072
\(568\) 0 0
\(569\) −17.2126 −0.721590 −0.360795 0.932645i \(-0.617495\pi\)
−0.360795 + 0.932645i \(0.617495\pi\)
\(570\) 0 0
\(571\) −38.8975 −1.62781 −0.813905 0.580997i \(-0.802663\pi\)
−0.813905 + 0.580997i \(0.802663\pi\)
\(572\) 0 0
\(573\) 20.2867 0.847489
\(574\) 0 0
\(575\) −7.04399 −0.293755
\(576\) 0 0
\(577\) 13.8651 0.577212 0.288606 0.957448i \(-0.406808\pi\)
0.288606 + 0.957448i \(0.406808\pi\)
\(578\) 0 0
\(579\) −16.0194 −0.665743
\(580\) 0 0
\(581\) −47.9619 −1.98979
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 2.38798 0.0987310
\(586\) 0 0
\(587\) −6.62340 −0.273377 −0.136688 0.990614i \(-0.543646\pi\)
−0.136688 + 0.990614i \(0.543646\pi\)
\(588\) 0 0
\(589\) −2.42024 −0.0997241
\(590\) 0 0
\(591\) 9.25043 0.380512
\(592\) 0 0
\(593\) 11.2348 0.461357 0.230679 0.973030i \(-0.425905\pi\)
0.230679 + 0.973030i \(0.425905\pi\)
\(594\) 0 0
\(595\) −0.920908 −0.0377535
\(596\) 0 0
\(597\) 12.9058 0.528198
\(598\) 0 0
\(599\) 11.6028 0.474077 0.237038 0.971500i \(-0.423823\pi\)
0.237038 + 0.971500i \(0.423823\pi\)
\(600\) 0 0
\(601\) 47.2759 1.92843 0.964213 0.265129i \(-0.0854147\pi\)
0.964213 + 0.265129i \(0.0854147\pi\)
\(602\) 0 0
\(603\) 21.2480 0.865287
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −38.1056 −1.54666 −0.773330 0.634004i \(-0.781410\pi\)
−0.773330 + 0.634004i \(0.781410\pi\)
\(608\) 0 0
\(609\) 5.12625 0.207726
\(610\) 0 0
\(611\) 8.31120 0.336235
\(612\) 0 0
\(613\) −17.0761 −0.689698 −0.344849 0.938658i \(-0.612070\pi\)
−0.344849 + 0.938658i \(0.612070\pi\)
\(614\) 0 0
\(615\) −5.58654 −0.225271
\(616\) 0 0
\(617\) 12.5333 0.504572 0.252286 0.967653i \(-0.418818\pi\)
0.252286 + 0.967653i \(0.418818\pi\)
\(618\) 0 0
\(619\) −15.8379 −0.636577 −0.318289 0.947994i \(-0.603108\pi\)
−0.318289 + 0.947994i \(0.603108\pi\)
\(620\) 0 0
\(621\) −32.2234 −1.29308
\(622\) 0 0
\(623\) −45.3092 −1.81528
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.24258 −0.0495447
\(630\) 0 0
\(631\) 34.7403 1.38299 0.691494 0.722382i \(-0.256953\pi\)
0.691494 + 0.722382i \(0.256953\pi\)
\(632\) 0 0
\(633\) −20.2575 −0.805164
\(634\) 0 0
\(635\) −9.41320 −0.373552
\(636\) 0 0
\(637\) −6.67524 −0.264483
\(638\) 0 0
\(639\) −16.8534 −0.666711
\(640\) 0 0
\(641\) 3.72059 0.146954 0.0734772 0.997297i \(-0.476590\pi\)
0.0734772 + 0.997297i \(0.476590\pi\)
\(642\) 0 0
\(643\) −9.61585 −0.379212 −0.189606 0.981860i \(-0.560721\pi\)
−0.189606 + 0.981860i \(0.560721\pi\)
\(644\) 0 0
\(645\) 2.29792 0.0904806
\(646\) 0 0
\(647\) 18.1587 0.713892 0.356946 0.934125i \(-0.383818\pi\)
0.356946 + 0.934125i \(0.383818\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −2.50901 −0.0983359
\(652\) 0 0
\(653\) 24.9382 0.975907 0.487954 0.872870i \(-0.337744\pi\)
0.487954 + 0.872870i \(0.337744\pi\)
\(654\) 0 0
\(655\) −14.2507 −0.556822
\(656\) 0 0
\(657\) −36.7800 −1.43493
\(658\) 0 0
\(659\) −17.2527 −0.672071 −0.336035 0.941849i \(-0.609086\pi\)
−0.336035 + 0.941849i \(0.609086\pi\)
\(660\) 0 0
\(661\) 17.4811 0.679936 0.339968 0.940437i \(-0.389584\pi\)
0.339968 + 0.940437i \(0.389584\pi\)
\(662\) 0 0
\(663\) 0.235886 0.00916104
\(664\) 0 0
\(665\) −11.1672 −0.433046
\(666\) 0 0
\(667\) −11.3550 −0.439669
\(668\) 0 0
\(669\) 1.32020 0.0510420
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 24.7907 0.955611 0.477806 0.878466i \(-0.341432\pi\)
0.477806 + 0.878466i \(0.341432\pi\)
\(674\) 0 0
\(675\) 4.57459 0.176076
\(676\) 0 0
\(677\) 42.4611 1.63191 0.815957 0.578112i \(-0.196210\pi\)
0.815957 + 0.578112i \(0.196210\pi\)
\(678\) 0 0
\(679\) −31.9532 −1.22625
\(680\) 0 0
\(681\) −11.3309 −0.434202
\(682\) 0 0
\(683\) 48.3820 1.85128 0.925642 0.378401i \(-0.123526\pi\)
0.925642 + 0.378401i \(0.123526\pi\)
\(684\) 0 0
\(685\) 21.6893 0.828704
\(686\) 0 0
\(687\) −21.6864 −0.827386
\(688\) 0 0
\(689\) 6.74977 0.257146
\(690\) 0 0
\(691\) 11.3397 0.431382 0.215691 0.976462i \(-0.430800\pi\)
0.215691 + 0.976462i \(0.430800\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.26252 0.313415
\(696\) 0 0
\(697\) 1.61781 0.0612791
\(698\) 0 0
\(699\) 14.3437 0.542527
\(700\) 0 0
\(701\) 4.06603 0.153572 0.0767859 0.997048i \(-0.475534\pi\)
0.0767859 + 0.997048i \(0.475534\pi\)
\(702\) 0 0
\(703\) −15.0678 −0.568295
\(704\) 0 0
\(705\) 6.80086 0.256135
\(706\) 0 0
\(707\) −58.8224 −2.21224
\(708\) 0 0
\(709\) 39.9682 1.50104 0.750518 0.660850i \(-0.229804\pi\)
0.750518 + 0.660850i \(0.229804\pi\)
\(710\) 0 0
\(711\) 6.76896 0.253856
\(712\) 0 0
\(713\) 5.55765 0.208135
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −19.4646 −0.726918
\(718\) 0 0
\(719\) 27.8865 1.03999 0.519996 0.854169i \(-0.325934\pi\)
0.519996 + 0.854169i \(0.325934\pi\)
\(720\) 0 0
\(721\) −16.7088 −0.622268
\(722\) 0 0
\(723\) −19.7317 −0.733829
\(724\) 0 0
\(725\) 1.61202 0.0598688
\(726\) 0 0
\(727\) −51.5024 −1.91012 −0.955059 0.296417i \(-0.904208\pi\)
−0.955059 + 0.296417i \(0.904208\pi\)
\(728\) 0 0
\(729\) 5.91479 0.219066
\(730\) 0 0
\(731\) −0.665458 −0.0246128
\(732\) 0 0
\(733\) −36.3991 −1.34443 −0.672215 0.740356i \(-0.734657\pi\)
−0.672215 + 0.740356i \(0.734657\pi\)
\(734\) 0 0
\(735\) −5.46219 −0.201476
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −10.1048 −0.371710 −0.185855 0.982577i \(-0.559505\pi\)
−0.185855 + 0.982577i \(0.559505\pi\)
\(740\) 0 0
\(741\) 2.86042 0.105080
\(742\) 0 0
\(743\) −33.0572 −1.21275 −0.606375 0.795179i \(-0.707377\pi\)
−0.606375 + 0.795179i \(0.707377\pi\)
\(744\) 0 0
\(745\) 6.82953 0.250215
\(746\) 0 0
\(747\) −29.4711 −1.07829
\(748\) 0 0
\(749\) −2.99937 −0.109594
\(750\) 0 0
\(751\) 13.1368 0.479369 0.239684 0.970851i \(-0.422956\pi\)
0.239684 + 0.970851i \(0.422956\pi\)
\(752\) 0 0
\(753\) 11.9325 0.434843
\(754\) 0 0
\(755\) 17.0217 0.619482
\(756\) 0 0
\(757\) −14.2945 −0.519541 −0.259770 0.965670i \(-0.583647\pi\)
−0.259770 + 0.965670i \(0.583647\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −15.8121 −0.573187 −0.286594 0.958052i \(-0.592523\pi\)
−0.286594 + 0.958052i \(0.592523\pi\)
\(762\) 0 0
\(763\) 23.4128 0.847600
\(764\) 0 0
\(765\) −0.565870 −0.0204591
\(766\) 0 0
\(767\) 5.85224 0.211312
\(768\) 0 0
\(769\) 12.3972 0.447053 0.223527 0.974698i \(-0.428243\pi\)
0.223527 + 0.974698i \(0.428243\pi\)
\(770\) 0 0
\(771\) 12.4383 0.447953
\(772\) 0 0
\(773\) −17.2672 −0.621060 −0.310530 0.950564i \(-0.600506\pi\)
−0.310530 + 0.950564i \(0.600506\pi\)
\(774\) 0 0
\(775\) −0.788991 −0.0283414
\(776\) 0 0
\(777\) −15.6205 −0.560383
\(778\) 0 0
\(779\) 19.6181 0.702892
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 7.37431 0.263536
\(784\) 0 0
\(785\) 20.4419 0.729604
\(786\) 0 0
\(787\) −22.0893 −0.787398 −0.393699 0.919240i \(-0.628805\pi\)
−0.393699 + 0.919240i \(0.628805\pi\)
\(788\) 0 0
\(789\) −5.49401 −0.195592
\(790\) 0 0
\(791\) 21.7264 0.772502
\(792\) 0 0
\(793\) 7.26078 0.257838
\(794\) 0 0
\(795\) 5.52318 0.195887
\(796\) 0 0
\(797\) 12.7250 0.450742 0.225371 0.974273i \(-0.427641\pi\)
0.225371 + 0.974273i \(0.427641\pi\)
\(798\) 0 0
\(799\) −1.96947 −0.0696748
\(800\) 0 0
\(801\) −27.8412 −0.983719
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 25.6435 0.903815
\(806\) 0 0
\(807\) −18.9356 −0.666565
\(808\) 0 0
\(809\) −39.0381 −1.37251 −0.686254 0.727362i \(-0.740746\pi\)
−0.686254 + 0.727362i \(0.740746\pi\)
\(810\) 0 0
\(811\) −22.3975 −0.786484 −0.393242 0.919435i \(-0.628647\pi\)
−0.393242 + 0.919435i \(0.628647\pi\)
\(812\) 0 0
\(813\) 2.97361 0.104289
\(814\) 0 0
\(815\) 11.3516 0.397631
\(816\) 0 0
\(817\) −8.06955 −0.282318
\(818\) 0 0
\(819\) −8.69341 −0.303772
\(820\) 0 0
\(821\) 6.19209 0.216105 0.108053 0.994145i \(-0.465538\pi\)
0.108053 + 0.994145i \(0.465538\pi\)
\(822\) 0 0
\(823\) 13.0630 0.455349 0.227674 0.973737i \(-0.426888\pi\)
0.227674 + 0.973737i \(0.426888\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −28.4130 −0.988017 −0.494009 0.869457i \(-0.664469\pi\)
−0.494009 + 0.869457i \(0.664469\pi\)
\(828\) 0 0
\(829\) −15.2772 −0.530599 −0.265300 0.964166i \(-0.585471\pi\)
−0.265300 + 0.964166i \(0.585471\pi\)
\(830\) 0 0
\(831\) 16.4090 0.569221
\(832\) 0 0
\(833\) 1.58180 0.0548062
\(834\) 0 0
\(835\) 11.6105 0.401799
\(836\) 0 0
\(837\) −3.60931 −0.124756
\(838\) 0 0
\(839\) −26.0887 −0.900684 −0.450342 0.892856i \(-0.648698\pi\)
−0.450342 + 0.892856i \(0.648698\pi\)
\(840\) 0 0
\(841\) −26.4014 −0.910393
\(842\) 0 0
\(843\) −13.2628 −0.456795
\(844\) 0 0
\(845\) −11.8604 −0.408011
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −8.91169 −0.305848
\(850\) 0 0
\(851\) 34.6006 1.18609
\(852\) 0 0
\(853\) −54.5956 −1.86932 −0.934659 0.355545i \(-0.884295\pi\)
−0.934659 + 0.355545i \(0.884295\pi\)
\(854\) 0 0
\(855\) −6.86192 −0.234673
\(856\) 0 0
\(857\) −6.40938 −0.218940 −0.109470 0.993990i \(-0.534915\pi\)
−0.109470 + 0.993990i \(0.534915\pi\)
\(858\) 0 0
\(859\) −4.08632 −0.139423 −0.0697117 0.997567i \(-0.522208\pi\)
−0.0697117 + 0.997567i \(0.522208\pi\)
\(860\) 0 0
\(861\) 20.3377 0.693107
\(862\) 0 0
\(863\) 27.9434 0.951205 0.475602 0.879660i \(-0.342230\pi\)
0.475602 + 0.879660i \(0.342230\pi\)
\(864\) 0 0
\(865\) −14.5576 −0.494974
\(866\) 0 0
\(867\) 14.7939 0.502428
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 10.1398 0.343576
\(872\) 0 0
\(873\) −19.6343 −0.664519
\(874\) 0 0
\(875\) −3.64048 −0.123071
\(876\) 0 0
\(877\) 15.0039 0.506645 0.253323 0.967382i \(-0.418477\pi\)
0.253323 + 0.967382i \(0.418477\pi\)
\(878\) 0 0
\(879\) 2.48878 0.0839443
\(880\) 0 0
\(881\) −6.47635 −0.218194 −0.109097 0.994031i \(-0.534796\pi\)
−0.109097 + 0.994031i \(0.534796\pi\)
\(882\) 0 0
\(883\) −7.24979 −0.243975 −0.121987 0.992532i \(-0.538927\pi\)
−0.121987 + 0.992532i \(0.538927\pi\)
\(884\) 0 0
\(885\) 4.78875 0.160972
\(886\) 0 0
\(887\) −19.4233 −0.652170 −0.326085 0.945340i \(-0.605730\pi\)
−0.326085 + 0.945340i \(0.605730\pi\)
\(888\) 0 0
\(889\) 34.2686 1.14933
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −23.8824 −0.799194
\(894\) 0 0
\(895\) −15.3520 −0.513160
\(896\) 0 0
\(897\) −6.56845 −0.219314
\(898\) 0 0
\(899\) −1.27187 −0.0424191
\(900\) 0 0
\(901\) −1.59946 −0.0532859
\(902\) 0 0
\(903\) −8.36553 −0.278388
\(904\) 0 0
\(905\) 19.4760 0.647403
\(906\) 0 0
\(907\) −25.1730 −0.835854 −0.417927 0.908481i \(-0.637243\pi\)
−0.417927 + 0.908481i \(0.637243\pi\)
\(908\) 0 0
\(909\) −36.1446 −1.19884
\(910\) 0 0
\(911\) −52.5919 −1.74245 −0.871224 0.490885i \(-0.836674\pi\)
−0.871224 + 0.490885i \(0.836674\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 5.94132 0.196414
\(916\) 0 0
\(917\) 51.8794 1.71321
\(918\) 0 0
\(919\) 13.9381 0.459774 0.229887 0.973217i \(-0.426164\pi\)
0.229887 + 0.973217i \(0.426164\pi\)
\(920\) 0 0
\(921\) −7.96577 −0.262481
\(922\) 0 0
\(923\) −8.04268 −0.264728
\(924\) 0 0
\(925\) −4.91208 −0.161508
\(926\) 0 0
\(927\) −10.2671 −0.337214
\(928\) 0 0
\(929\) −41.5841 −1.36433 −0.682164 0.731199i \(-0.738961\pi\)
−0.682164 + 0.731199i \(0.738961\pi\)
\(930\) 0 0
\(931\) 19.1814 0.628646
\(932\) 0 0
\(933\) −13.0446 −0.427059
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 23.0636 0.753455 0.376727 0.926324i \(-0.377049\pi\)
0.376727 + 0.926324i \(0.377049\pi\)
\(938\) 0 0
\(939\) −12.1935 −0.397919
\(940\) 0 0
\(941\) 27.2409 0.888027 0.444014 0.896020i \(-0.353554\pi\)
0.444014 + 0.896020i \(0.353554\pi\)
\(942\) 0 0
\(943\) −45.0495 −1.46701
\(944\) 0 0
\(945\) −16.6537 −0.541744
\(946\) 0 0
\(947\) 44.2870 1.43913 0.719566 0.694424i \(-0.244341\pi\)
0.719566 + 0.694424i \(0.244341\pi\)
\(948\) 0 0
\(949\) −17.5519 −0.569760
\(950\) 0 0
\(951\) −13.3213 −0.431972
\(952\) 0 0
\(953\) −1.93291 −0.0626132 −0.0313066 0.999510i \(-0.509967\pi\)
−0.0313066 + 0.999510i \(0.509967\pi\)
\(954\) 0 0
\(955\) −23.2241 −0.751515
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −78.9593 −2.54973
\(960\) 0 0
\(961\) −30.3775 −0.979919
\(962\) 0 0
\(963\) −1.84302 −0.0593905
\(964\) 0 0
\(965\) 18.3389 0.590351
\(966\) 0 0
\(967\) 43.0790 1.38533 0.692663 0.721261i \(-0.256437\pi\)
0.692663 + 0.721261i \(0.256437\pi\)
\(968\) 0 0
\(969\) −0.677822 −0.0217748
\(970\) 0 0
\(971\) 27.2811 0.875493 0.437746 0.899099i \(-0.355777\pi\)
0.437746 + 0.899099i \(0.355777\pi\)
\(972\) 0 0
\(973\) −30.0795 −0.964306
\(974\) 0 0
\(975\) 0.932490 0.0298636
\(976\) 0 0
\(977\) 5.21603 0.166876 0.0834378 0.996513i \(-0.473410\pi\)
0.0834378 + 0.996513i \(0.473410\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 14.3865 0.459324
\(982\) 0 0
\(983\) 17.7199 0.565176 0.282588 0.959241i \(-0.408807\pi\)
0.282588 + 0.959241i \(0.408807\pi\)
\(984\) 0 0
\(985\) −10.5899 −0.337421
\(986\) 0 0
\(987\) −24.7584 −0.788068
\(988\) 0 0
\(989\) 18.5303 0.589229
\(990\) 0 0
\(991\) −39.3005 −1.24842 −0.624211 0.781256i \(-0.714579\pi\)
−0.624211 + 0.781256i \(0.714579\pi\)
\(992\) 0 0
\(993\) −15.0211 −0.476680
\(994\) 0 0
\(995\) −14.7745 −0.468382
\(996\) 0 0
\(997\) −59.0721 −1.87083 −0.935416 0.353550i \(-0.884974\pi\)
−0.935416 + 0.353550i \(0.884974\pi\)
\(998\) 0 0
\(999\) −22.4707 −0.710942
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.bd.1.3 yes 6
4.3 odd 2 9680.2.a.da.1.4 6
11.10 odd 2 4840.2.a.bc.1.3 6
44.43 even 2 9680.2.a.db.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.bc.1.3 6 11.10 odd 2
4840.2.a.bd.1.3 yes 6 1.1 even 1 trivial
9680.2.a.da.1.4 6 4.3 odd 2
9680.2.a.db.1.4 6 44.43 even 2