Properties

Label 4304.2.a.e.1.3
Level $4304$
Weight $2$
Character 4304.1
Self dual yes
Analytic conductor $34.368$
Analytic rank $1$
Dimension $4$
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] = [4304,2,Mod(1,4304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4304.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4304 = 2^{4} \cdot 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4304.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: \(34.3676130300\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4913.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 538)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.90570\) of defining polynomial
Character \(\chi\) \(=\) 4304.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.512072 q^{3} -1.04948 q^{5} +2.90570 q^{7} -2.73778 q^{9} -2.34415 q^{11} +4.24985 q^{13} +0.537410 q^{15} +7.66052 q^{17} -5.81141 q^{19} -1.48793 q^{21} -6.53741 q^{23} -3.89859 q^{25} +2.93816 q^{27} +0.119634 q^{29} -3.75481 q^{31} +1.20037 q^{33} -3.04948 q^{35} -3.27400 q^{37} -2.17623 q^{39} +4.46726 q^{41} +3.34051 q^{43} +2.87325 q^{45} -2.61468 q^{47} +1.44311 q^{49} -3.92273 q^{51} +2.87689 q^{53} +2.46014 q^{55} +2.97586 q^{57} -0.549773 q^{59} -8.24985 q^{61} -7.95518 q^{63} -4.46014 q^{65} -1.09430 q^{67} +3.34762 q^{69} +8.10363 q^{71} +9.15191 q^{73} +1.99636 q^{75} -6.81141 q^{77} -0.426084 q^{79} +6.70880 q^{81} -7.84547 q^{83} -8.03957 q^{85} -0.0612614 q^{87} +15.7424 q^{89} +12.3488 q^{91} +1.92273 q^{93} +6.09896 q^{95} -14.9051 q^{97} +6.41778 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{3} + 5 q^{5} + q^{7} + 3 q^{9} - 7 q^{11} + 4 q^{13} - 8 q^{15} + 4 q^{17} - 2 q^{19} - 5 q^{21} - 16 q^{23} - q^{25} - 6 q^{27} + q^{31} + q^{33} - 3 q^{35} - 2 q^{37} - 3 q^{39} - q^{41}+ \cdots + 16 q^{99}+O(q^{100}) \) 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
\(3\) −0.512072 −0.295645 −0.147822 0.989014i \(-0.547226\pi\)
−0.147822 + 0.989014i \(0.547226\pi\)
\(4\) 0 0
\(5\) −1.04948 −0.469342 −0.234671 0.972075i \(-0.575401\pi\)
−0.234671 + 0.972075i \(0.575401\pi\)
\(6\) 0 0
\(7\) 2.90570 1.09825 0.549126 0.835739i \(-0.314961\pi\)
0.549126 + 0.835739i \(0.314961\pi\)
\(8\) 0 0
\(9\) −2.73778 −0.912594
\(10\) 0 0
\(11\) −2.34415 −0.706788 −0.353394 0.935475i \(-0.614972\pi\)
−0.353394 + 0.935475i \(0.614972\pi\)
\(12\) 0 0
\(13\) 4.24985 1.17870 0.589349 0.807879i \(-0.299384\pi\)
0.589349 + 0.807879i \(0.299384\pi\)
\(14\) 0 0
\(15\) 0.537410 0.138759
\(16\) 0 0
\(17\) 7.66052 1.85795 0.928974 0.370145i \(-0.120692\pi\)
0.928974 + 0.370145i \(0.120692\pi\)
\(18\) 0 0
\(19\) −5.81141 −1.33323 −0.666614 0.745403i \(-0.732257\pi\)
−0.666614 + 0.745403i \(0.732257\pi\)
\(20\) 0 0
\(21\) −1.48793 −0.324693
\(22\) 0 0
\(23\) −6.53741 −1.36314 −0.681572 0.731751i \(-0.738703\pi\)
−0.681572 + 0.731751i \(0.738703\pi\)
\(24\) 0 0
\(25\) −3.89859 −0.779718
\(26\) 0 0
\(27\) 2.93816 0.565448
\(28\) 0 0
\(29\) 0.119634 0.0222155 0.0111078 0.999938i \(-0.496464\pi\)
0.0111078 + 0.999938i \(0.496464\pi\)
\(30\) 0 0
\(31\) −3.75481 −0.674384 −0.337192 0.941436i \(-0.609477\pi\)
−0.337192 + 0.941436i \(0.609477\pi\)
\(32\) 0 0
\(33\) 1.20037 0.208958
\(34\) 0 0
\(35\) −3.04948 −0.515456
\(36\) 0 0
\(37\) −3.27400 −0.538242 −0.269121 0.963106i \(-0.586733\pi\)
−0.269121 + 0.963106i \(0.586733\pi\)
\(38\) 0 0
\(39\) −2.17623 −0.348476
\(40\) 0 0
\(41\) 4.46726 0.697668 0.348834 0.937184i \(-0.386578\pi\)
0.348834 + 0.937184i \(0.386578\pi\)
\(42\) 0 0
\(43\) 3.34051 0.509423 0.254711 0.967017i \(-0.418020\pi\)
0.254711 + 0.967017i \(0.418020\pi\)
\(44\) 0 0
\(45\) 2.87325 0.428319
\(46\) 0 0
\(47\) −2.61468 −0.381390 −0.190695 0.981649i \(-0.561074\pi\)
−0.190695 + 0.981649i \(0.561074\pi\)
\(48\) 0 0
\(49\) 1.44311 0.206159
\(50\) 0 0
\(51\) −3.92273 −0.549292
\(52\) 0 0
\(53\) 2.87689 0.395172 0.197586 0.980286i \(-0.436690\pi\)
0.197586 + 0.980286i \(0.436690\pi\)
\(54\) 0 0
\(55\) 2.46014 0.331725
\(56\) 0 0
\(57\) 2.97586 0.394162
\(58\) 0 0
\(59\) −0.549773 −0.0715743 −0.0357871 0.999359i \(-0.511394\pi\)
−0.0357871 + 0.999359i \(0.511394\pi\)
\(60\) 0 0
\(61\) −8.24985 −1.05629 −0.528143 0.849156i \(-0.677111\pi\)
−0.528143 + 0.849156i \(0.677111\pi\)
\(62\) 0 0
\(63\) −7.95518 −1.00226
\(64\) 0 0
\(65\) −4.46014 −0.553213
\(66\) 0 0
\(67\) −1.09430 −0.133690 −0.0668448 0.997763i \(-0.521293\pi\)
−0.0668448 + 0.997763i \(0.521293\pi\)
\(68\) 0 0
\(69\) 3.34762 0.403006
\(70\) 0 0
\(71\) 8.10363 0.961724 0.480862 0.876796i \(-0.340324\pi\)
0.480862 + 0.876796i \(0.340324\pi\)
\(72\) 0 0
\(73\) 9.15191 1.07115 0.535575 0.844487i \(-0.320095\pi\)
0.535575 + 0.844487i \(0.320095\pi\)
\(74\) 0 0
\(75\) 1.99636 0.230519
\(76\) 0 0
\(77\) −6.81141 −0.776232
\(78\) 0 0
\(79\) −0.426084 −0.0479382 −0.0239691 0.999713i \(-0.507630\pi\)
−0.0239691 + 0.999713i \(0.507630\pi\)
\(80\) 0 0
\(81\) 6.70880 0.745422
\(82\) 0 0
\(83\) −7.84547 −0.861152 −0.430576 0.902554i \(-0.641689\pi\)
−0.430576 + 0.902554i \(0.641689\pi\)
\(84\) 0 0
\(85\) −8.03957 −0.872013
\(86\) 0 0
\(87\) −0.0612614 −0.00656791
\(88\) 0 0
\(89\) 15.7424 1.66870 0.834348 0.551238i \(-0.185844\pi\)
0.834348 + 0.551238i \(0.185844\pi\)
\(90\) 0 0
\(91\) 12.3488 1.29451
\(92\) 0 0
\(93\) 1.92273 0.199378
\(94\) 0 0
\(95\) 6.09896 0.625740
\(96\) 0 0
\(97\) −14.9051 −1.51339 −0.756693 0.653771i \(-0.773186\pi\)
−0.756693 + 0.653771i \(0.773186\pi\)
\(98\) 0 0
\(99\) 6.41778 0.645011
\(100\) 0 0
\(101\) −13.0300 −1.29653 −0.648267 0.761413i \(-0.724506\pi\)
−0.648267 + 0.761413i \(0.724506\pi\)
\(102\) 0 0
\(103\) −8.52505 −0.839998 −0.419999 0.907525i \(-0.637970\pi\)
−0.419999 + 0.907525i \(0.637970\pi\)
\(104\) 0 0
\(105\) 1.56155 0.152392
\(106\) 0 0
\(107\) −11.0371 −1.06700 −0.533499 0.845801i \(-0.679123\pi\)
−0.533499 + 0.845801i \(0.679123\pi\)
\(108\) 0 0
\(109\) −4.51919 −0.432860 −0.216430 0.976298i \(-0.569441\pi\)
−0.216430 + 0.976298i \(0.569441\pi\)
\(110\) 0 0
\(111\) 1.67652 0.159128
\(112\) 0 0
\(113\) −5.82616 −0.548079 −0.274039 0.961718i \(-0.588360\pi\)
−0.274039 + 0.961718i \(0.588360\pi\)
\(114\) 0 0
\(115\) 6.86089 0.639781
\(116\) 0 0
\(117\) −11.6352 −1.07567
\(118\) 0 0
\(119\) 22.2592 2.04050
\(120\) 0 0
\(121\) −5.50496 −0.500451
\(122\) 0 0
\(123\) −2.28756 −0.206262
\(124\) 0 0
\(125\) 9.33890 0.835297
\(126\) 0 0
\(127\) 5.97102 0.529842 0.264921 0.964270i \(-0.414654\pi\)
0.264921 + 0.964270i \(0.414654\pi\)
\(128\) 0 0
\(129\) −1.71058 −0.150608
\(130\) 0 0
\(131\) −11.3535 −0.991958 −0.495979 0.868334i \(-0.665191\pi\)
−0.495979 + 0.868334i \(0.665191\pi\)
\(132\) 0 0
\(133\) −16.8862 −1.46422
\(134\) 0 0
\(135\) −3.08354 −0.265389
\(136\) 0 0
\(137\) 10.6240 0.907670 0.453835 0.891086i \(-0.350055\pi\)
0.453835 + 0.891086i \(0.350055\pi\)
\(138\) 0 0
\(139\) −8.41186 −0.713484 −0.356742 0.934203i \(-0.616113\pi\)
−0.356742 + 0.934203i \(0.616113\pi\)
\(140\) 0 0
\(141\) 1.33890 0.112756
\(142\) 0 0
\(143\) −9.96230 −0.833089
\(144\) 0 0
\(145\) −0.125554 −0.0104267
\(146\) 0 0
\(147\) −0.738977 −0.0609498
\(148\) 0 0
\(149\) −17.9321 −1.46905 −0.734526 0.678581i \(-0.762595\pi\)
−0.734526 + 0.678581i \(0.762595\pi\)
\(150\) 0 0
\(151\) 0.938156 0.0763460 0.0381730 0.999271i \(-0.487846\pi\)
0.0381730 + 0.999271i \(0.487846\pi\)
\(152\) 0 0
\(153\) −20.9728 −1.69555
\(154\) 0 0
\(155\) 3.94060 0.316517
\(156\) 0 0
\(157\) −1.01948 −0.0813632 −0.0406816 0.999172i \(-0.512953\pi\)
−0.0406816 + 0.999172i \(0.512953\pi\)
\(158\) 0 0
\(159\) −1.47318 −0.116830
\(160\) 0 0
\(161\) −18.9958 −1.49708
\(162\) 0 0
\(163\) −13.3440 −1.04518 −0.522591 0.852584i \(-0.675034\pi\)
−0.522591 + 0.852584i \(0.675034\pi\)
\(164\) 0 0
\(165\) −1.25977 −0.0980729
\(166\) 0 0
\(167\) −19.8414 −1.53537 −0.767687 0.640825i \(-0.778593\pi\)
−0.767687 + 0.640825i \(0.778593\pi\)
\(168\) 0 0
\(169\) 5.06126 0.389328
\(170\) 0 0
\(171\) 15.9104 1.21670
\(172\) 0 0
\(173\) 14.7860 1.12416 0.562080 0.827083i \(-0.310001\pi\)
0.562080 + 0.827083i \(0.310001\pi\)
\(174\) 0 0
\(175\) −11.3281 −0.856327
\(176\) 0 0
\(177\) 0.281523 0.0211606
\(178\) 0 0
\(179\) 13.1868 0.985624 0.492812 0.870136i \(-0.335969\pi\)
0.492812 + 0.870136i \(0.335969\pi\)
\(180\) 0 0
\(181\) −21.7542 −1.61698 −0.808490 0.588511i \(-0.799715\pi\)
−0.808490 + 0.588511i \(0.799715\pi\)
\(182\) 0 0
\(183\) 4.22452 0.312285
\(184\) 0 0
\(185\) 3.43600 0.252620
\(186\) 0 0
\(187\) −17.9574 −1.31318
\(188\) 0 0
\(189\) 8.53741 0.621005
\(190\) 0 0
\(191\) 1.23630 0.0894553 0.0447276 0.998999i \(-0.485758\pi\)
0.0447276 + 0.998999i \(0.485758\pi\)
\(192\) 0 0
\(193\) 1.52749 0.109951 0.0549757 0.998488i \(-0.482492\pi\)
0.0549757 + 0.998488i \(0.482492\pi\)
\(194\) 0 0
\(195\) 2.28391 0.163554
\(196\) 0 0
\(197\) −22.3689 −1.59372 −0.796859 0.604165i \(-0.793507\pi\)
−0.796859 + 0.604165i \(0.793507\pi\)
\(198\) 0 0
\(199\) 16.7266 1.18572 0.592859 0.805306i \(-0.297999\pi\)
0.592859 + 0.805306i \(0.297999\pi\)
\(200\) 0 0
\(201\) 0.560358 0.0395246
\(202\) 0 0
\(203\) 0.347622 0.0243983
\(204\) 0 0
\(205\) −4.68830 −0.327445
\(206\) 0 0
\(207\) 17.8980 1.24400
\(208\) 0 0
\(209\) 13.6228 0.942310
\(210\) 0 0
\(211\) 2.00644 0.138129 0.0690646 0.997612i \(-0.477999\pi\)
0.0690646 + 0.997612i \(0.477999\pi\)
\(212\) 0 0
\(213\) −4.14964 −0.284329
\(214\) 0 0
\(215\) −3.50580 −0.239094
\(216\) 0 0
\(217\) −10.9104 −0.740644
\(218\) 0 0
\(219\) −4.68644 −0.316680
\(220\) 0 0
\(221\) 32.5561 2.18996
\(222\) 0 0
\(223\) −17.0201 −1.13975 −0.569875 0.821732i \(-0.693008\pi\)
−0.569875 + 0.821732i \(0.693008\pi\)
\(224\) 0 0
\(225\) 10.6735 0.711566
\(226\) 0 0
\(227\) −17.9605 −1.19208 −0.596041 0.802954i \(-0.703260\pi\)
−0.596041 + 0.802954i \(0.703260\pi\)
\(228\) 0 0
\(229\) 3.85377 0.254665 0.127332 0.991860i \(-0.459359\pi\)
0.127332 + 0.991860i \(0.459359\pi\)
\(230\) 0 0
\(231\) 3.48793 0.229489
\(232\) 0 0
\(233\) 19.8538 1.30066 0.650332 0.759650i \(-0.274630\pi\)
0.650332 + 0.759650i \(0.274630\pi\)
\(234\) 0 0
\(235\) 2.74405 0.179002
\(236\) 0 0
\(237\) 0.218186 0.0141727
\(238\) 0 0
\(239\) 11.1250 0.719615 0.359807 0.933027i \(-0.382842\pi\)
0.359807 + 0.933027i \(0.382842\pi\)
\(240\) 0 0
\(241\) −17.2740 −1.11272 −0.556358 0.830943i \(-0.687802\pi\)
−0.556358 + 0.830943i \(0.687802\pi\)
\(242\) 0 0
\(243\) −12.2499 −0.785829
\(244\) 0 0
\(245\) −1.51452 −0.0967591
\(246\) 0 0
\(247\) −24.6976 −1.57147
\(248\) 0 0
\(249\) 4.01744 0.254595
\(250\) 0 0
\(251\) −2.07362 −0.130886 −0.0654430 0.997856i \(-0.520846\pi\)
−0.0654430 + 0.997856i \(0.520846\pi\)
\(252\) 0 0
\(253\) 15.3247 0.963454
\(254\) 0 0
\(255\) 4.11683 0.257806
\(256\) 0 0
\(257\) −26.0799 −1.62682 −0.813411 0.581690i \(-0.802392\pi\)
−0.813411 + 0.581690i \(0.802392\pi\)
\(258\) 0 0
\(259\) −9.51327 −0.591126
\(260\) 0 0
\(261\) −0.327533 −0.0202738
\(262\) 0 0
\(263\) 9.93935 0.612887 0.306443 0.951889i \(-0.400861\pi\)
0.306443 + 0.951889i \(0.400861\pi\)
\(264\) 0 0
\(265\) −3.01925 −0.185471
\(266\) 0 0
\(267\) −8.06126 −0.493341
\(268\) 0 0
\(269\) 1.00000 0.0609711
\(270\) 0 0
\(271\) −9.40702 −0.571436 −0.285718 0.958314i \(-0.592232\pi\)
−0.285718 + 0.958314i \(0.592232\pi\)
\(272\) 0 0
\(273\) −6.32348 −0.382714
\(274\) 0 0
\(275\) 9.13888 0.551095
\(276\) 0 0
\(277\) 1.10380 0.0663209 0.0331605 0.999450i \(-0.489443\pi\)
0.0331605 + 0.999450i \(0.489443\pi\)
\(278\) 0 0
\(279\) 10.2799 0.615439
\(280\) 0 0
\(281\) −16.2356 −0.968536 −0.484268 0.874920i \(-0.660914\pi\)
−0.484268 + 0.874920i \(0.660914\pi\)
\(282\) 0 0
\(283\) −22.0635 −1.31154 −0.655771 0.754960i \(-0.727656\pi\)
−0.655771 + 0.754960i \(0.727656\pi\)
\(284\) 0 0
\(285\) −3.12311 −0.184997
\(286\) 0 0
\(287\) 12.9805 0.766216
\(288\) 0 0
\(289\) 41.6835 2.45197
\(290\) 0 0
\(291\) 7.63249 0.447424
\(292\) 0 0
\(293\) 12.1496 0.709789 0.354895 0.934906i \(-0.384517\pi\)
0.354895 + 0.934906i \(0.384517\pi\)
\(294\) 0 0
\(295\) 0.576976 0.0335928
\(296\) 0 0
\(297\) −6.88748 −0.399652
\(298\) 0 0
\(299\) −27.7830 −1.60673
\(300\) 0 0
\(301\) 9.70653 0.559475
\(302\) 0 0
\(303\) 6.67230 0.383313
\(304\) 0 0
\(305\) 8.65807 0.495759
\(306\) 0 0
\(307\) 22.7400 1.29784 0.648920 0.760856i \(-0.275221\pi\)
0.648920 + 0.760856i \(0.275221\pi\)
\(308\) 0 0
\(309\) 4.36543 0.248341
\(310\) 0 0
\(311\) 10.9941 0.623417 0.311709 0.950178i \(-0.399099\pi\)
0.311709 + 0.950178i \(0.399099\pi\)
\(312\) 0 0
\(313\) −25.0357 −1.41510 −0.707551 0.706663i \(-0.750200\pi\)
−0.707551 + 0.706663i \(0.750200\pi\)
\(314\) 0 0
\(315\) 8.34882 0.470403
\(316\) 0 0
\(317\) 29.2168 1.64098 0.820490 0.571661i \(-0.193701\pi\)
0.820490 + 0.571661i \(0.193701\pi\)
\(318\) 0 0
\(319\) −0.280441 −0.0157017
\(320\) 0 0
\(321\) 5.65180 0.315452
\(322\) 0 0
\(323\) −44.5184 −2.47707
\(324\) 0 0
\(325\) −16.5684 −0.919051
\(326\) 0 0
\(327\) 2.31415 0.127973
\(328\) 0 0
\(329\) −7.59748 −0.418862
\(330\) 0 0
\(331\) −15.9156 −0.874801 −0.437401 0.899267i \(-0.644101\pi\)
−0.437401 + 0.899267i \(0.644101\pi\)
\(332\) 0 0
\(333\) 8.96349 0.491196
\(334\) 0 0
\(335\) 1.14844 0.0627462
\(336\) 0 0
\(337\) −35.1956 −1.91722 −0.958612 0.284715i \(-0.908101\pi\)
−0.958612 + 0.284715i \(0.908101\pi\)
\(338\) 0 0
\(339\) 2.98341 0.162037
\(340\) 0 0
\(341\) 8.80184 0.476647
\(342\) 0 0
\(343\) −16.1467 −0.871838
\(344\) 0 0
\(345\) −3.51327 −0.189148
\(346\) 0 0
\(347\) 24.8903 1.33618 0.668091 0.744080i \(-0.267112\pi\)
0.668091 + 0.744080i \(0.267112\pi\)
\(348\) 0 0
\(349\) 1.73175 0.0926985 0.0463492 0.998925i \(-0.485241\pi\)
0.0463492 + 0.998925i \(0.485241\pi\)
\(350\) 0 0
\(351\) 12.4867 0.666493
\(352\) 0 0
\(353\) −21.3682 −1.13732 −0.568658 0.822574i \(-0.692537\pi\)
−0.568658 + 0.822574i \(0.692537\pi\)
\(354\) 0 0
\(355\) −8.50461 −0.451378
\(356\) 0 0
\(357\) −11.3983 −0.603262
\(358\) 0 0
\(359\) 19.8025 1.04514 0.522568 0.852597i \(-0.324974\pi\)
0.522568 + 0.852597i \(0.324974\pi\)
\(360\) 0 0
\(361\) 14.7725 0.777497
\(362\) 0 0
\(363\) 2.81893 0.147956
\(364\) 0 0
\(365\) −9.60476 −0.502736
\(366\) 0 0
\(367\) 1.72720 0.0901590 0.0450795 0.998983i \(-0.485646\pi\)
0.0450795 + 0.998983i \(0.485646\pi\)
\(368\) 0 0
\(369\) −12.2304 −0.636688
\(370\) 0 0
\(371\) 8.35940 0.433999
\(372\) 0 0
\(373\) −31.3423 −1.62284 −0.811422 0.584461i \(-0.801306\pi\)
−0.811422 + 0.584461i \(0.801306\pi\)
\(374\) 0 0
\(375\) −4.78219 −0.246951
\(376\) 0 0
\(377\) 0.508429 0.0261854
\(378\) 0 0
\(379\) −21.2263 −1.09032 −0.545161 0.838331i \(-0.683532\pi\)
−0.545161 + 0.838331i \(0.683532\pi\)
\(380\) 0 0
\(381\) −3.05759 −0.156645
\(382\) 0 0
\(383\) 34.5798 1.76695 0.883473 0.468483i \(-0.155199\pi\)
0.883473 + 0.468483i \(0.155199\pi\)
\(384\) 0 0
\(385\) 7.14844 0.364318
\(386\) 0 0
\(387\) −9.14558 −0.464896
\(388\) 0 0
\(389\) 2.21454 0.112282 0.0561409 0.998423i \(-0.482120\pi\)
0.0561409 + 0.998423i \(0.482120\pi\)
\(390\) 0 0
\(391\) −50.0799 −2.53265
\(392\) 0 0
\(393\) 5.81380 0.293267
\(394\) 0 0
\(395\) 0.447167 0.0224994
\(396\) 0 0
\(397\) −28.7782 −1.44434 −0.722168 0.691717i \(-0.756854\pi\)
−0.722168 + 0.691717i \(0.756854\pi\)
\(398\) 0 0
\(399\) 8.64696 0.432889
\(400\) 0 0
\(401\) −9.00119 −0.449498 −0.224749 0.974417i \(-0.572156\pi\)
−0.224749 + 0.974417i \(0.572156\pi\)
\(402\) 0 0
\(403\) −15.9574 −0.794895
\(404\) 0 0
\(405\) −7.04076 −0.349858
\(406\) 0 0
\(407\) 7.67474 0.380423
\(408\) 0 0
\(409\) 36.6526 1.81236 0.906178 0.422896i \(-0.138986\pi\)
0.906178 + 0.422896i \(0.138986\pi\)
\(410\) 0 0
\(411\) −5.44025 −0.268348
\(412\) 0 0
\(413\) −1.59748 −0.0786067
\(414\) 0 0
\(415\) 8.23367 0.404175
\(416\) 0 0
\(417\) 4.30747 0.210938
\(418\) 0 0
\(419\) 1.67822 0.0819862 0.0409931 0.999159i \(-0.486948\pi\)
0.0409931 + 0.999159i \(0.486948\pi\)
\(420\) 0 0
\(421\) −8.60926 −0.419589 −0.209795 0.977745i \(-0.567280\pi\)
−0.209795 + 0.977745i \(0.567280\pi\)
\(422\) 0 0
\(423\) 7.15842 0.348054
\(424\) 0 0
\(425\) −29.8652 −1.44868
\(426\) 0 0
\(427\) −23.9716 −1.16007
\(428\) 0 0
\(429\) 5.10141 0.246298
\(430\) 0 0
\(431\) −20.5339 −0.989081 −0.494540 0.869155i \(-0.664664\pi\)
−0.494540 + 0.869155i \(0.664664\pi\)
\(432\) 0 0
\(433\) 0.678127 0.0325887 0.0162944 0.999867i \(-0.494813\pi\)
0.0162944 + 0.999867i \(0.494813\pi\)
\(434\) 0 0
\(435\) 0.0642926 0.00308260
\(436\) 0 0
\(437\) 37.9915 1.81738
\(438\) 0 0
\(439\) 33.4206 1.59508 0.797539 0.603267i \(-0.206135\pi\)
0.797539 + 0.603267i \(0.206135\pi\)
\(440\) 0 0
\(441\) −3.95093 −0.188140
\(442\) 0 0
\(443\) −21.7843 −1.03500 −0.517501 0.855682i \(-0.673138\pi\)
−0.517501 + 0.855682i \(0.673138\pi\)
\(444\) 0 0
\(445\) −16.5214 −0.783190
\(446\) 0 0
\(447\) 9.18250 0.434317
\(448\) 0 0
\(449\) −15.6603 −0.739057 −0.369529 0.929219i \(-0.620481\pi\)
−0.369529 + 0.929219i \(0.620481\pi\)
\(450\) 0 0
\(451\) −10.4719 −0.493104
\(452\) 0 0
\(453\) −0.480403 −0.0225713
\(454\) 0 0
\(455\) −12.9599 −0.607567
\(456\) 0 0
\(457\) −4.59453 −0.214923 −0.107461 0.994209i \(-0.534272\pi\)
−0.107461 + 0.994209i \(0.534272\pi\)
\(458\) 0 0
\(459\) 22.5078 1.05057
\(460\) 0 0
\(461\) −32.2651 −1.50274 −0.751368 0.659884i \(-0.770606\pi\)
−0.751368 + 0.659884i \(0.770606\pi\)
\(462\) 0 0
\(463\) −20.4853 −0.952033 −0.476017 0.879436i \(-0.657920\pi\)
−0.476017 + 0.879436i \(0.657920\pi\)
\(464\) 0 0
\(465\) −2.01787 −0.0935766
\(466\) 0 0
\(467\) −36.5500 −1.69133 −0.845667 0.533711i \(-0.820797\pi\)
−0.845667 + 0.533711i \(0.820797\pi\)
\(468\) 0 0
\(469\) −3.17970 −0.146825
\(470\) 0 0
\(471\) 0.522045 0.0240546
\(472\) 0 0
\(473\) −7.83065 −0.360054
\(474\) 0 0
\(475\) 22.6563 1.03954
\(476\) 0 0
\(477\) −7.87631 −0.360632
\(478\) 0 0
\(479\) 36.6751 1.67573 0.837864 0.545879i \(-0.183804\pi\)
0.837864 + 0.545879i \(0.183804\pi\)
\(480\) 0 0
\(481\) −13.9140 −0.634424
\(482\) 0 0
\(483\) 9.72720 0.442603
\(484\) 0 0
\(485\) 15.6426 0.710296
\(486\) 0 0
\(487\) −34.8684 −1.58004 −0.790020 0.613081i \(-0.789930\pi\)
−0.790020 + 0.613081i \(0.789930\pi\)
\(488\) 0 0
\(489\) 6.83307 0.309002
\(490\) 0 0
\(491\) 2.32389 0.104876 0.0524378 0.998624i \(-0.483301\pi\)
0.0524378 + 0.998624i \(0.483301\pi\)
\(492\) 0 0
\(493\) 0.916461 0.0412753
\(494\) 0 0
\(495\) −6.73533 −0.302731
\(496\) 0 0
\(497\) 23.5467 1.05622
\(498\) 0 0
\(499\) −25.6747 −1.14936 −0.574680 0.818378i \(-0.694873\pi\)
−0.574680 + 0.818378i \(0.694873\pi\)
\(500\) 0 0
\(501\) 10.1602 0.453925
\(502\) 0 0
\(503\) 35.8649 1.59914 0.799569 0.600575i \(-0.205061\pi\)
0.799569 + 0.600575i \(0.205061\pi\)
\(504\) 0 0
\(505\) 13.6747 0.608518
\(506\) 0 0
\(507\) −2.59173 −0.115103
\(508\) 0 0
\(509\) 26.1779 1.16032 0.580158 0.814504i \(-0.302991\pi\)
0.580158 + 0.814504i \(0.302991\pi\)
\(510\) 0 0
\(511\) 26.5928 1.17639
\(512\) 0 0
\(513\) −17.0748 −0.753872
\(514\) 0 0
\(515\) 8.94688 0.394246
\(516\) 0 0
\(517\) 6.12920 0.269562
\(518\) 0 0
\(519\) −7.57150 −0.332352
\(520\) 0 0
\(521\) −30.9190 −1.35459 −0.677294 0.735713i \(-0.736847\pi\)
−0.677294 + 0.735713i \(0.736847\pi\)
\(522\) 0 0
\(523\) 17.6068 0.769892 0.384946 0.922939i \(-0.374220\pi\)
0.384946 + 0.922939i \(0.374220\pi\)
\(524\) 0 0
\(525\) 5.80082 0.253169
\(526\) 0 0
\(527\) −28.7638 −1.25297
\(528\) 0 0
\(529\) 19.7377 0.858162
\(530\) 0 0
\(531\) 1.50516 0.0653183
\(532\) 0 0
\(533\) 18.9852 0.822340
\(534\) 0 0
\(535\) 11.5832 0.500787
\(536\) 0 0
\(537\) −6.75256 −0.291395
\(538\) 0 0
\(539\) −3.38287 −0.145711
\(540\) 0 0
\(541\) 4.03939 0.173667 0.0868336 0.996223i \(-0.472325\pi\)
0.0868336 + 0.996223i \(0.472325\pi\)
\(542\) 0 0
\(543\) 11.1397 0.478051
\(544\) 0 0
\(545\) 4.74280 0.203159
\(546\) 0 0
\(547\) 40.7182 1.74098 0.870491 0.492184i \(-0.163801\pi\)
0.870491 + 0.492184i \(0.163801\pi\)
\(548\) 0 0
\(549\) 22.5863 0.963960
\(550\) 0 0
\(551\) −0.695244 −0.0296184
\(552\) 0 0
\(553\) −1.23807 −0.0526483
\(554\) 0 0
\(555\) −1.75948 −0.0746857
\(556\) 0 0
\(557\) 19.3259 0.818866 0.409433 0.912340i \(-0.365727\pi\)
0.409433 + 0.912340i \(0.365727\pi\)
\(558\) 0 0
\(559\) 14.1967 0.600455
\(560\) 0 0
\(561\) 9.19548 0.388233
\(562\) 0 0
\(563\) 3.59240 0.151402 0.0757008 0.997131i \(-0.475881\pi\)
0.0757008 + 0.997131i \(0.475881\pi\)
\(564\) 0 0
\(565\) 6.11444 0.257237
\(566\) 0 0
\(567\) 19.4938 0.818662
\(568\) 0 0
\(569\) 15.0425 0.630616 0.315308 0.948989i \(-0.397892\pi\)
0.315308 + 0.948989i \(0.397892\pi\)
\(570\) 0 0
\(571\) 27.0643 1.13261 0.566303 0.824197i \(-0.308373\pi\)
0.566303 + 0.824197i \(0.308373\pi\)
\(572\) 0 0
\(573\) −0.633072 −0.0264470
\(574\) 0 0
\(575\) 25.4867 1.06287
\(576\) 0 0
\(577\) 16.5926 0.690761 0.345380 0.938463i \(-0.387750\pi\)
0.345380 + 0.938463i \(0.387750\pi\)
\(578\) 0 0
\(579\) −0.782187 −0.0325066
\(580\) 0 0
\(581\) −22.7966 −0.945762
\(582\) 0 0
\(583\) −6.74387 −0.279303
\(584\) 0 0
\(585\) 12.2109 0.504859
\(586\) 0 0
\(587\) 4.01362 0.165660 0.0828298 0.996564i \(-0.473604\pi\)
0.0828298 + 0.996564i \(0.473604\pi\)
\(588\) 0 0
\(589\) 21.8207 0.899108
\(590\) 0 0
\(591\) 11.4545 0.471174
\(592\) 0 0
\(593\) 13.1761 0.541076 0.270538 0.962709i \(-0.412798\pi\)
0.270538 + 0.962709i \(0.412798\pi\)
\(594\) 0 0
\(595\) −23.3606 −0.957691
\(596\) 0 0
\(597\) −8.56522 −0.350551
\(598\) 0 0
\(599\) 6.44903 0.263500 0.131750 0.991283i \(-0.457940\pi\)
0.131750 + 0.991283i \(0.457940\pi\)
\(600\) 0 0
\(601\) 25.6859 1.04775 0.523876 0.851795i \(-0.324486\pi\)
0.523876 + 0.851795i \(0.324486\pi\)
\(602\) 0 0
\(603\) 2.99595 0.122004
\(604\) 0 0
\(605\) 5.77735 0.234883
\(606\) 0 0
\(607\) 31.9037 1.29493 0.647466 0.762095i \(-0.275829\pi\)
0.647466 + 0.762095i \(0.275829\pi\)
\(608\) 0 0
\(609\) −0.178007 −0.00721322
\(610\) 0 0
\(611\) −11.1120 −0.449543
\(612\) 0 0
\(613\) 20.5761 0.831062 0.415531 0.909579i \(-0.363596\pi\)
0.415531 + 0.909579i \(0.363596\pi\)
\(614\) 0 0
\(615\) 2.40075 0.0968074
\(616\) 0 0
\(617\) 35.3614 1.42360 0.711798 0.702384i \(-0.247881\pi\)
0.711798 + 0.702384i \(0.247881\pi\)
\(618\) 0 0
\(619\) −44.9921 −1.80839 −0.904193 0.427124i \(-0.859527\pi\)
−0.904193 + 0.427124i \(0.859527\pi\)
\(620\) 0 0
\(621\) −19.2079 −0.770788
\(622\) 0 0
\(623\) 45.7429 1.83265
\(624\) 0 0
\(625\) 9.69194 0.387678
\(626\) 0 0
\(627\) −6.97586 −0.278589
\(628\) 0 0
\(629\) −25.0805 −1.00003
\(630\) 0 0
\(631\) 39.3156 1.56513 0.782565 0.622569i \(-0.213911\pi\)
0.782565 + 0.622569i \(0.213911\pi\)
\(632\) 0 0
\(633\) −1.02744 −0.0408372
\(634\) 0 0
\(635\) −6.26647 −0.248677
\(636\) 0 0
\(637\) 6.13302 0.242999
\(638\) 0 0
\(639\) −22.1860 −0.877664
\(640\) 0 0
\(641\) 28.3628 1.12026 0.560132 0.828403i \(-0.310750\pi\)
0.560132 + 0.828403i \(0.310750\pi\)
\(642\) 0 0
\(643\) −46.4066 −1.83010 −0.915049 0.403343i \(-0.867848\pi\)
−0.915049 + 0.403343i \(0.867848\pi\)
\(644\) 0 0
\(645\) 1.79522 0.0706867
\(646\) 0 0
\(647\) −9.47203 −0.372384 −0.186192 0.982513i \(-0.559615\pi\)
−0.186192 + 0.982513i \(0.559615\pi\)
\(648\) 0 0
\(649\) 1.28875 0.0505879
\(650\) 0 0
\(651\) 5.58689 0.218968
\(652\) 0 0
\(653\) 25.5129 0.998398 0.499199 0.866487i \(-0.333628\pi\)
0.499199 + 0.866487i \(0.333628\pi\)
\(654\) 0 0
\(655\) 11.9153 0.465568
\(656\) 0 0
\(657\) −25.0560 −0.977526
\(658\) 0 0
\(659\) −11.7305 −0.456955 −0.228478 0.973549i \(-0.573375\pi\)
−0.228478 + 0.973549i \(0.573375\pi\)
\(660\) 0 0
\(661\) −20.7855 −0.808462 −0.404231 0.914657i \(-0.632461\pi\)
−0.404231 + 0.914657i \(0.632461\pi\)
\(662\) 0 0
\(663\) −16.6710 −0.647450
\(664\) 0 0
\(665\) 17.7218 0.687221
\(666\) 0 0
\(667\) −0.782099 −0.0302830
\(668\) 0 0
\(669\) 8.71550 0.336961
\(670\) 0 0
\(671\) 19.3389 0.746570
\(672\) 0 0
\(673\) −5.05829 −0.194983 −0.0974914 0.995236i \(-0.531082\pi\)
−0.0974914 + 0.995236i \(0.531082\pi\)
\(674\) 0 0
\(675\) −11.4547 −0.440890
\(676\) 0 0
\(677\) 14.5131 0.557784 0.278892 0.960323i \(-0.410033\pi\)
0.278892 + 0.960323i \(0.410033\pi\)
\(678\) 0 0
\(679\) −43.3099 −1.66208
\(680\) 0 0
\(681\) 9.19707 0.352433
\(682\) 0 0
\(683\) 48.9508 1.87305 0.936525 0.350601i \(-0.114023\pi\)
0.936525 + 0.350601i \(0.114023\pi\)
\(684\) 0 0
\(685\) −11.1497 −0.426008
\(686\) 0 0
\(687\) −1.97341 −0.0752902
\(688\) 0 0
\(689\) 12.2264 0.465788
\(690\) 0 0
\(691\) 35.2573 1.34125 0.670626 0.741795i \(-0.266025\pi\)
0.670626 + 0.741795i \(0.266025\pi\)
\(692\) 0 0
\(693\) 18.6482 0.708385
\(694\) 0 0
\(695\) 8.82808 0.334868
\(696\) 0 0
\(697\) 34.2215 1.29623
\(698\) 0 0
\(699\) −10.1666 −0.384534
\(700\) 0 0
\(701\) 21.9096 0.827515 0.413757 0.910387i \(-0.364216\pi\)
0.413757 + 0.910387i \(0.364216\pi\)
\(702\) 0 0
\(703\) 19.0265 0.717599
\(704\) 0 0
\(705\) −1.40515 −0.0529211
\(706\) 0 0
\(707\) −37.8613 −1.42392
\(708\) 0 0
\(709\) 7.16159 0.268959 0.134480 0.990916i \(-0.457064\pi\)
0.134480 + 0.990916i \(0.457064\pi\)
\(710\) 0 0
\(711\) 1.16653 0.0437481
\(712\) 0 0
\(713\) 24.5467 0.919283
\(714\) 0 0
\(715\) 10.4552 0.391004
\(716\) 0 0
\(717\) −5.69678 −0.212750
\(718\) 0 0
\(719\) −13.0574 −0.486958 −0.243479 0.969906i \(-0.578289\pi\)
−0.243479 + 0.969906i \(0.578289\pi\)
\(720\) 0 0
\(721\) −24.7713 −0.922530
\(722\) 0 0
\(723\) 8.84552 0.328969
\(724\) 0 0
\(725\) −0.466405 −0.0173219
\(726\) 0 0
\(727\) −39.8163 −1.47670 −0.738352 0.674416i \(-0.764396\pi\)
−0.738352 + 0.674416i \(0.764396\pi\)
\(728\) 0 0
\(729\) −13.8536 −0.513096
\(730\) 0 0
\(731\) 25.5900 0.946481
\(732\) 0 0
\(733\) −10.9054 −0.402799 −0.201399 0.979509i \(-0.564549\pi\)
−0.201399 + 0.979509i \(0.564549\pi\)
\(734\) 0 0
\(735\) 0.775543 0.0286063
\(736\) 0 0
\(737\) 2.56520 0.0944902
\(738\) 0 0
\(739\) 51.0337 1.87731 0.938653 0.344862i \(-0.112074\pi\)
0.938653 + 0.344862i \(0.112074\pi\)
\(740\) 0 0
\(741\) 12.6470 0.464598
\(742\) 0 0
\(743\) 20.1783 0.740269 0.370135 0.928978i \(-0.379312\pi\)
0.370135 + 0.928978i \(0.379312\pi\)
\(744\) 0 0
\(745\) 18.8194 0.689488
\(746\) 0 0
\(747\) 21.4792 0.785882
\(748\) 0 0
\(749\) −32.0706 −1.17183
\(750\) 0 0
\(751\) −5.83301 −0.212850 −0.106425 0.994321i \(-0.533940\pi\)
−0.106425 + 0.994321i \(0.533940\pi\)
\(752\) 0 0
\(753\) 1.06184 0.0386958
\(754\) 0 0
\(755\) −0.984577 −0.0358324
\(756\) 0 0
\(757\) 37.8086 1.37418 0.687089 0.726573i \(-0.258888\pi\)
0.687089 + 0.726573i \(0.258888\pi\)
\(758\) 0 0
\(759\) −7.84733 −0.284840
\(760\) 0 0
\(761\) −22.2483 −0.806499 −0.403249 0.915090i \(-0.632119\pi\)
−0.403249 + 0.915090i \(0.632119\pi\)
\(762\) 0 0
\(763\) −13.1314 −0.475389
\(764\) 0 0
\(765\) 22.0106 0.795794
\(766\) 0 0
\(767\) −2.33645 −0.0843644
\(768\) 0 0
\(769\) 7.40743 0.267119 0.133559 0.991041i \(-0.457359\pi\)
0.133559 + 0.991041i \(0.457359\pi\)
\(770\) 0 0
\(771\) 13.3548 0.480961
\(772\) 0 0
\(773\) −28.3513 −1.01973 −0.509863 0.860256i \(-0.670304\pi\)
−0.509863 + 0.860256i \(0.670304\pi\)
\(774\) 0 0
\(775\) 14.6385 0.525829
\(776\) 0 0
\(777\) 4.87147 0.174763
\(778\) 0 0
\(779\) −25.9610 −0.930151
\(780\) 0 0
\(781\) −18.9961 −0.679735
\(782\) 0 0
\(783\) 0.351504 0.0125617
\(784\) 0 0
\(785\) 1.06992 0.0381872
\(786\) 0 0
\(787\) 4.71084 0.167923 0.0839616 0.996469i \(-0.473243\pi\)
0.0839616 + 0.996469i \(0.473243\pi\)
\(788\) 0 0
\(789\) −5.08966 −0.181197
\(790\) 0 0
\(791\) −16.9291 −0.601929
\(792\) 0 0
\(793\) −35.0607 −1.24504
\(794\) 0 0
\(795\) 1.54607 0.0548335
\(796\) 0 0
\(797\) −17.8537 −0.632411 −0.316206 0.948691i \(-0.602409\pi\)
−0.316206 + 0.948691i \(0.602409\pi\)
\(798\) 0 0
\(799\) −20.0298 −0.708602
\(800\) 0 0
\(801\) −43.0994 −1.52284
\(802\) 0 0
\(803\) −21.4535 −0.757076
\(804\) 0 0
\(805\) 19.9357 0.702641
\(806\) 0 0
\(807\) −0.512072 −0.0180258
\(808\) 0 0
\(809\) −34.3781 −1.20867 −0.604334 0.796731i \(-0.706561\pi\)
−0.604334 + 0.796731i \(0.706561\pi\)
\(810\) 0 0
\(811\) 29.4277 1.03335 0.516673 0.856183i \(-0.327170\pi\)
0.516673 + 0.856183i \(0.327170\pi\)
\(812\) 0 0
\(813\) 4.81707 0.168942
\(814\) 0 0
\(815\) 14.0043 0.490548
\(816\) 0 0
\(817\) −19.4131 −0.679177
\(818\) 0 0
\(819\) −33.8084 −1.18136
\(820\) 0 0
\(821\) −29.8950 −1.04334 −0.521671 0.853147i \(-0.674691\pi\)
−0.521671 + 0.853147i \(0.674691\pi\)
\(822\) 0 0
\(823\) −25.6212 −0.893099 −0.446550 0.894759i \(-0.647347\pi\)
−0.446550 + 0.894759i \(0.647347\pi\)
\(824\) 0 0
\(825\) −4.67976 −0.162928
\(826\) 0 0
\(827\) −33.6271 −1.16933 −0.584664 0.811276i \(-0.698774\pi\)
−0.584664 + 0.811276i \(0.698774\pi\)
\(828\) 0 0
\(829\) −9.56047 −0.332049 −0.166024 0.986122i \(-0.553093\pi\)
−0.166024 + 0.986122i \(0.553093\pi\)
\(830\) 0 0
\(831\) −0.565225 −0.0196074
\(832\) 0 0
\(833\) 11.0550 0.383033
\(834\) 0 0
\(835\) 20.8232 0.720616
\(836\) 0 0
\(837\) −11.0322 −0.381329
\(838\) 0 0
\(839\) 3.75123 0.129507 0.0647534 0.997901i \(-0.479374\pi\)
0.0647534 + 0.997901i \(0.479374\pi\)
\(840\) 0 0
\(841\) −28.9857 −0.999506
\(842\) 0 0
\(843\) 8.31380 0.286343
\(844\) 0 0
\(845\) −5.31170 −0.182728
\(846\) 0 0
\(847\) −15.9958 −0.549621
\(848\) 0 0
\(849\) 11.2981 0.387750
\(850\) 0 0
\(851\) 21.4035 0.733701
\(852\) 0 0
\(853\) 23.1483 0.792583 0.396291 0.918125i \(-0.370297\pi\)
0.396291 + 0.918125i \(0.370297\pi\)
\(854\) 0 0
\(855\) −16.6976 −0.571047
\(856\) 0 0
\(857\) −47.4439 −1.62065 −0.810326 0.585979i \(-0.800710\pi\)
−0.810326 + 0.585979i \(0.800710\pi\)
\(858\) 0 0
\(859\) 10.3635 0.353597 0.176798 0.984247i \(-0.443426\pi\)
0.176798 + 0.984247i \(0.443426\pi\)
\(860\) 0 0
\(861\) −6.64696 −0.226528
\(862\) 0 0
\(863\) −19.6044 −0.667342 −0.333671 0.942690i \(-0.608287\pi\)
−0.333671 + 0.942690i \(0.608287\pi\)
\(864\) 0 0
\(865\) −15.5176 −0.527616
\(866\) 0 0
\(867\) −21.3449 −0.724912
\(868\) 0 0
\(869\) 0.998805 0.0338822
\(870\) 0 0
\(871\) −4.65060 −0.157580
\(872\) 0 0
\(873\) 40.8070 1.38111
\(874\) 0 0
\(875\) 27.1361 0.917367
\(876\) 0 0
\(877\) 47.4632 1.60272 0.801359 0.598183i \(-0.204111\pi\)
0.801359 + 0.598183i \(0.204111\pi\)
\(878\) 0 0
\(879\) −6.22149 −0.209845
\(880\) 0 0
\(881\) 56.7144 1.91076 0.955379 0.295383i \(-0.0954473\pi\)
0.955379 + 0.295383i \(0.0954473\pi\)
\(882\) 0 0
\(883\) 45.3481 1.52609 0.763043 0.646348i \(-0.223705\pi\)
0.763043 + 0.646348i \(0.223705\pi\)
\(884\) 0 0
\(885\) −0.295453 −0.00993155
\(886\) 0 0
\(887\) 4.41133 0.148118 0.0740590 0.997254i \(-0.476405\pi\)
0.0740590 + 0.997254i \(0.476405\pi\)
\(888\) 0 0
\(889\) 17.3500 0.581901
\(890\) 0 0
\(891\) −15.7264 −0.526856
\(892\) 0 0
\(893\) 15.1950 0.508480
\(894\) 0 0
\(895\) −13.8393 −0.462595
\(896\) 0 0
\(897\) 14.2269 0.475023
\(898\) 0 0
\(899\) −0.449204 −0.0149818
\(900\) 0 0
\(901\) 22.0385 0.734209
\(902\) 0 0
\(903\) −4.97044 −0.165406
\(904\) 0 0
\(905\) 22.8307 0.758917
\(906\) 0 0
\(907\) −48.2246 −1.60127 −0.800635 0.599152i \(-0.795504\pi\)
−0.800635 + 0.599152i \(0.795504\pi\)
\(908\) 0 0
\(909\) 35.6733 1.18321
\(910\) 0 0
\(911\) 9.66308 0.320152 0.160076 0.987105i \(-0.448826\pi\)
0.160076 + 0.987105i \(0.448826\pi\)
\(912\) 0 0
\(913\) 18.3910 0.608652
\(914\) 0 0
\(915\) −4.43355 −0.146569
\(916\) 0 0
\(917\) −32.9899 −1.08942
\(918\) 0 0
\(919\) 2.61634 0.0863052 0.0431526 0.999068i \(-0.486260\pi\)
0.0431526 + 0.999068i \(0.486260\pi\)
\(920\) 0 0
\(921\) −11.6445 −0.383700
\(922\) 0 0
\(923\) 34.4392 1.13358
\(924\) 0 0
\(925\) 12.7640 0.419677
\(926\) 0 0
\(927\) 23.3397 0.766577
\(928\) 0 0
\(929\) 34.1534 1.12054 0.560268 0.828311i \(-0.310698\pi\)
0.560268 + 0.828311i \(0.310698\pi\)
\(930\) 0 0
\(931\) −8.38652 −0.274857
\(932\) 0 0
\(933\) −5.62976 −0.184310
\(934\) 0 0
\(935\) 18.8460 0.616329
\(936\) 0 0
\(937\) −0.840971 −0.0274733 −0.0137367 0.999906i \(-0.504373\pi\)
−0.0137367 + 0.999906i \(0.504373\pi\)
\(938\) 0 0
\(939\) 12.8201 0.418367
\(940\) 0 0
\(941\) −16.3930 −0.534398 −0.267199 0.963641i \(-0.586098\pi\)
−0.267199 + 0.963641i \(0.586098\pi\)
\(942\) 0 0
\(943\) −29.2043 −0.951022
\(944\) 0 0
\(945\) −8.95985 −0.291464
\(946\) 0 0
\(947\) 46.3550 1.50634 0.753168 0.657828i \(-0.228525\pi\)
0.753168 + 0.657828i \(0.228525\pi\)
\(948\) 0 0
\(949\) 38.8943 1.26256
\(950\) 0 0
\(951\) −14.9611 −0.485147
\(952\) 0 0
\(953\) −56.3635 −1.82579 −0.912896 0.408191i \(-0.866160\pi\)
−0.912896 + 0.408191i \(0.866160\pi\)
\(954\) 0 0
\(955\) −1.29747 −0.0419851
\(956\) 0 0
\(957\) 0.143606 0.00464212
\(958\) 0 0
\(959\) 30.8702 0.996851
\(960\) 0 0
\(961\) −16.9014 −0.545206
\(962\) 0 0
\(963\) 30.2172 0.973737
\(964\) 0 0
\(965\) −1.60308 −0.0516049
\(966\) 0 0
\(967\) 50.1806 1.61370 0.806849 0.590758i \(-0.201171\pi\)
0.806849 + 0.590758i \(0.201171\pi\)
\(968\) 0 0
\(969\) 22.7966 0.732332
\(970\) 0 0
\(971\) −20.7585 −0.666173 −0.333087 0.942896i \(-0.608090\pi\)
−0.333087 + 0.942896i \(0.608090\pi\)
\(972\) 0 0
\(973\) −24.4424 −0.783586
\(974\) 0 0
\(975\) 8.48423 0.271713
\(976\) 0 0
\(977\) 5.01009 0.160287 0.0801434 0.996783i \(-0.474462\pi\)
0.0801434 + 0.996783i \(0.474462\pi\)
\(978\) 0 0
\(979\) −36.9027 −1.17941
\(980\) 0 0
\(981\) 12.3725 0.395025
\(982\) 0 0
\(983\) 20.6726 0.659355 0.329677 0.944094i \(-0.393060\pi\)
0.329677 + 0.944094i \(0.393060\pi\)
\(984\) 0 0
\(985\) 23.4757 0.747999
\(986\) 0 0
\(987\) 3.89045 0.123834
\(988\) 0 0
\(989\) −21.8383 −0.694416
\(990\) 0 0
\(991\) −43.4299 −1.37960 −0.689799 0.724001i \(-0.742301\pi\)
−0.689799 + 0.724001i \(0.742301\pi\)
\(992\) 0 0
\(993\) 8.14994 0.258630
\(994\) 0 0
\(995\) −17.5543 −0.556508
\(996\) 0 0
\(997\) −4.39494 −0.139189 −0.0695946 0.997575i \(-0.522171\pi\)
−0.0695946 + 0.997575i \(0.522171\pi\)
\(998\) 0 0
\(999\) −9.61951 −0.304348
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 4304.2.a.e.1.3 4
4.3 odd 2 538.2.a.c.1.2 4
12.11 even 2 4842.2.a.j.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
538.2.a.c.1.2 4 4.3 odd 2
4304.2.a.e.1.3 4 1.1 even 1 trivial
4842.2.a.j.1.4 4 12.11 even 2