Properties

Label 8016.2.a.r.1.3
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $5$
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] = [8016,2,Mod(1,8016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, 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("8016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.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: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.11256624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 16x^{3} + 20x^{2} + 31x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1002)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.50259\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.368511 q^{5} -4.85901 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.368511 q^{5} -4.85901 q^{7} +1.00000 q^{9} +4.29593 q^{11} +3.50259 q^{13} +0.368511 q^{15} -1.50259 q^{17} -6.29593 q^{19} +4.85901 q^{21} -4.29593 q^{23} -4.86420 q^{25} -1.00000 q^{27} +7.30112 q^{29} +5.15494 q^{31} -4.29593 q^{33} +1.79060 q^{35} -0.424828 q^{37} -3.50259 q^{39} -5.06151 q^{41} +11.0191 q^{43} -0.368511 q^{45} -0.146177 q^{47} +16.6100 q^{49} +1.50259 q^{51} +11.1195 q^{53} -1.58310 q^{55} +6.29593 q^{57} -5.87468 q^{59} +4.71283 q^{61} -4.85901 q^{63} -1.29075 q^{65} +11.9414 q^{67} +4.29593 q^{69} -12.2821 q^{71} -8.75098 q^{73} +4.86420 q^{75} -20.8740 q^{77} -7.06151 q^{79} +1.00000 q^{81} -10.3020 q^{83} +0.553723 q^{85} -7.30112 q^{87} -5.57185 q^{89} -17.0191 q^{91} -5.15494 q^{93} +2.32012 q^{95} +4.41434 q^{97} +4.29593 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{3} - q^{5} - 9 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{3} - q^{5} - 9 q^{7} + 5 q^{9} + 2 q^{11} + 6 q^{13} + q^{15} + 4 q^{17} - 12 q^{19} + 9 q^{21} - 2 q^{23} + 14 q^{25} - 5 q^{27} - 6 q^{29} - 9 q^{31} - 2 q^{33} - 3 q^{35} + 5 q^{37} - 6 q^{39} + 4 q^{41} - 18 q^{43} - q^{45} + 7 q^{47} + 16 q^{49} - 4 q^{51} + 3 q^{53} + 4 q^{55} + 12 q^{57} - 13 q^{59} + 16 q^{61} - 9 q^{63} - 10 q^{65} - 9 q^{67} + 2 q^{69} + 10 q^{71} + 8 q^{73} - 14 q^{75} + 6 q^{77} - 6 q^{79} + 5 q^{81} - q^{83} + 8 q^{85} + 6 q^{87} - 5 q^{89} - 12 q^{91} + 9 q^{93} - 2 q^{95} - 7 q^{97} + 2 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −0.368511 −0.164803 −0.0824016 0.996599i \(-0.526259\pi\)
−0.0824016 + 0.996599i \(0.526259\pi\)
\(6\) 0 0
\(7\) −4.85901 −1.83653 −0.918267 0.395962i \(-0.870411\pi\)
−0.918267 + 0.395962i \(0.870411\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.29593 1.29527 0.647636 0.761950i \(-0.275758\pi\)
0.647636 + 0.761950i \(0.275758\pi\)
\(12\) 0 0
\(13\) 3.50259 0.971445 0.485722 0.874113i \(-0.338557\pi\)
0.485722 + 0.874113i \(0.338557\pi\)
\(14\) 0 0
\(15\) 0.368511 0.0951492
\(16\) 0 0
\(17\) −1.50259 −0.364433 −0.182216 0.983258i \(-0.558327\pi\)
−0.182216 + 0.983258i \(0.558327\pi\)
\(18\) 0 0
\(19\) −6.29593 −1.44439 −0.722193 0.691691i \(-0.756866\pi\)
−0.722193 + 0.691691i \(0.756866\pi\)
\(20\) 0 0
\(21\) 4.85901 1.06032
\(22\) 0 0
\(23\) −4.29593 −0.895764 −0.447882 0.894093i \(-0.647822\pi\)
−0.447882 + 0.894093i \(0.647822\pi\)
\(24\) 0 0
\(25\) −4.86420 −0.972840
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 7.30112 1.35578 0.677892 0.735161i \(-0.262894\pi\)
0.677892 + 0.735161i \(0.262894\pi\)
\(30\) 0 0
\(31\) 5.15494 0.925855 0.462928 0.886396i \(-0.346799\pi\)
0.462928 + 0.886396i \(0.346799\pi\)
\(32\) 0 0
\(33\) −4.29593 −0.747826
\(34\) 0 0
\(35\) 1.79060 0.302667
\(36\) 0 0
\(37\) −0.424828 −0.0698413 −0.0349207 0.999390i \(-0.511118\pi\)
−0.0349207 + 0.999390i \(0.511118\pi\)
\(38\) 0 0
\(39\) −3.50259 −0.560864
\(40\) 0 0
\(41\) −5.06151 −0.790474 −0.395237 0.918579i \(-0.629338\pi\)
−0.395237 + 0.918579i \(0.629338\pi\)
\(42\) 0 0
\(43\) 11.0191 1.68040 0.840202 0.542274i \(-0.182436\pi\)
0.840202 + 0.542274i \(0.182436\pi\)
\(44\) 0 0
\(45\) −0.368511 −0.0549344
\(46\) 0 0
\(47\) −0.146177 −0.0213221 −0.0106611 0.999943i \(-0.503394\pi\)
−0.0106611 + 0.999943i \(0.503394\pi\)
\(48\) 0 0
\(49\) 16.6100 2.37286
\(50\) 0 0
\(51\) 1.50259 0.210405
\(52\) 0 0
\(53\) 11.1195 1.52738 0.763690 0.645584i \(-0.223386\pi\)
0.763690 + 0.645584i \(0.223386\pi\)
\(54\) 0 0
\(55\) −1.58310 −0.213465
\(56\) 0 0
\(57\) 6.29593 0.833917
\(58\) 0 0
\(59\) −5.87468 −0.764819 −0.382409 0.923993i \(-0.624906\pi\)
−0.382409 + 0.923993i \(0.624906\pi\)
\(60\) 0 0
\(61\) 4.71283 0.603417 0.301708 0.953400i \(-0.402443\pi\)
0.301708 + 0.953400i \(0.402443\pi\)
\(62\) 0 0
\(63\) −4.85901 −0.612178
\(64\) 0 0
\(65\) −1.29075 −0.160097
\(66\) 0 0
\(67\) 11.9414 1.45887 0.729436 0.684049i \(-0.239783\pi\)
0.729436 + 0.684049i \(0.239783\pi\)
\(68\) 0 0
\(69\) 4.29593 0.517170
\(70\) 0 0
\(71\) −12.2821 −1.45762 −0.728810 0.684716i \(-0.759926\pi\)
−0.728810 + 0.684716i \(0.759926\pi\)
\(72\) 0 0
\(73\) −8.75098 −1.02422 −0.512112 0.858919i \(-0.671137\pi\)
−0.512112 + 0.858919i \(0.671137\pi\)
\(74\) 0 0
\(75\) 4.86420 0.561669
\(76\) 0 0
\(77\) −20.8740 −2.37881
\(78\) 0 0
\(79\) −7.06151 −0.794481 −0.397241 0.917714i \(-0.630032\pi\)
−0.397241 + 0.917714i \(0.630032\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.3020 −1.13079 −0.565394 0.824821i \(-0.691276\pi\)
−0.565394 + 0.824821i \(0.691276\pi\)
\(84\) 0 0
\(85\) 0.553723 0.0600597
\(86\) 0 0
\(87\) −7.30112 −0.782762
\(88\) 0 0
\(89\) −5.57185 −0.590614 −0.295307 0.955402i \(-0.595422\pi\)
−0.295307 + 0.955402i \(0.595422\pi\)
\(90\) 0 0
\(91\) −17.0191 −1.78409
\(92\) 0 0
\(93\) −5.15494 −0.534543
\(94\) 0 0
\(95\) 2.32012 0.238039
\(96\) 0 0
\(97\) 4.41434 0.448209 0.224104 0.974565i \(-0.428054\pi\)
0.224104 + 0.974565i \(0.428054\pi\)
\(98\) 0 0
\(99\) 4.29593 0.431758
\(100\) 0 0
\(101\) −12.5193 −1.24572 −0.622858 0.782335i \(-0.714029\pi\)
−0.622858 + 0.782335i \(0.714029\pi\)
\(102\) 0 0
\(103\) −10.8037 −1.06452 −0.532261 0.846580i \(-0.678657\pi\)
−0.532261 + 0.846580i \(0.678657\pi\)
\(104\) 0 0
\(105\) −1.79060 −0.174745
\(106\) 0 0
\(107\) 12.1731 1.17681 0.588407 0.808565i \(-0.299755\pi\)
0.588407 + 0.808565i \(0.299755\pi\)
\(108\) 0 0
\(109\) 9.08927 0.870594 0.435297 0.900287i \(-0.356643\pi\)
0.435297 + 0.900287i \(0.356643\pi\)
\(110\) 0 0
\(111\) 0.424828 0.0403229
\(112\) 0 0
\(113\) 9.24838 0.870015 0.435007 0.900427i \(-0.356746\pi\)
0.435007 + 0.900427i \(0.356746\pi\)
\(114\) 0 0
\(115\) 1.58310 0.147625
\(116\) 0 0
\(117\) 3.50259 0.323815
\(118\) 0 0
\(119\) 7.30112 0.669293
\(120\) 0 0
\(121\) 7.45504 0.677731
\(122\) 0 0
\(123\) 5.06151 0.456381
\(124\) 0 0
\(125\) 3.63507 0.325130
\(126\) 0 0
\(127\) 14.6152 1.29689 0.648444 0.761263i \(-0.275420\pi\)
0.648444 + 0.761263i \(0.275420\pi\)
\(128\) 0 0
\(129\) −11.0191 −0.970182
\(130\) 0 0
\(131\) −5.11346 −0.446765 −0.223383 0.974731i \(-0.571710\pi\)
−0.223383 + 0.974731i \(0.571710\pi\)
\(132\) 0 0
\(133\) 30.5920 2.65266
\(134\) 0 0
\(135\) 0.368511 0.0317164
\(136\) 0 0
\(137\) 6.46988 0.552759 0.276380 0.961049i \(-0.410865\pi\)
0.276380 + 0.961049i \(0.410865\pi\)
\(138\) 0 0
\(139\) 6.79937 0.576715 0.288357 0.957523i \(-0.406891\pi\)
0.288357 + 0.957523i \(0.406891\pi\)
\(140\) 0 0
\(141\) 0.146177 0.0123103
\(142\) 0 0
\(143\) 15.0469 1.25829
\(144\) 0 0
\(145\) −2.69055 −0.223438
\(146\) 0 0
\(147\) −16.6100 −1.36997
\(148\) 0 0
\(149\) −9.51410 −0.779426 −0.389713 0.920936i \(-0.627426\pi\)
−0.389713 + 0.920936i \(0.627426\pi\)
\(150\) 0 0
\(151\) −4.92468 −0.400765 −0.200383 0.979718i \(-0.564218\pi\)
−0.200383 + 0.979718i \(0.564218\pi\)
\(152\) 0 0
\(153\) −1.50259 −0.121478
\(154\) 0 0
\(155\) −1.89965 −0.152584
\(156\) 0 0
\(157\) 11.0365 0.880811 0.440406 0.897799i \(-0.354835\pi\)
0.440406 + 0.897799i \(0.354835\pi\)
\(158\) 0 0
\(159\) −11.1195 −0.881833
\(160\) 0 0
\(161\) 20.8740 1.64510
\(162\) 0 0
\(163\) 8.67482 0.679464 0.339732 0.940522i \(-0.389664\pi\)
0.339732 + 0.940522i \(0.389664\pi\)
\(164\) 0 0
\(165\) 1.58310 0.123244
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) −0.731834 −0.0562949
\(170\) 0 0
\(171\) −6.29593 −0.481462
\(172\) 0 0
\(173\) −2.68149 −0.203870 −0.101935 0.994791i \(-0.532503\pi\)
−0.101935 + 0.994791i \(0.532503\pi\)
\(174\) 0 0
\(175\) 23.6352 1.78665
\(176\) 0 0
\(177\) 5.87468 0.441568
\(178\) 0 0
\(179\) 5.82708 0.435536 0.217768 0.976001i \(-0.430122\pi\)
0.217768 + 0.976001i \(0.430122\pi\)
\(180\) 0 0
\(181\) −18.1695 −1.35053 −0.675264 0.737576i \(-0.735970\pi\)
−0.675264 + 0.737576i \(0.735970\pi\)
\(182\) 0 0
\(183\) −4.71283 −0.348383
\(184\) 0 0
\(185\) 0.156554 0.0115101
\(186\) 0 0
\(187\) −6.45504 −0.472040
\(188\) 0 0
\(189\) 4.85901 0.353441
\(190\) 0 0
\(191\) 21.7468 1.57354 0.786772 0.617244i \(-0.211751\pi\)
0.786772 + 0.617244i \(0.211751\pi\)
\(192\) 0 0
\(193\) −24.9950 −1.79918 −0.899588 0.436739i \(-0.856133\pi\)
−0.899588 + 0.436739i \(0.856133\pi\)
\(194\) 0 0
\(195\) 1.29075 0.0924322
\(196\) 0 0
\(197\) 6.56410 0.467673 0.233836 0.972276i \(-0.424872\pi\)
0.233836 + 0.972276i \(0.424872\pi\)
\(198\) 0 0
\(199\) −5.19755 −0.368444 −0.184222 0.982885i \(-0.558977\pi\)
−0.184222 + 0.982885i \(0.558977\pi\)
\(200\) 0 0
\(201\) −11.9414 −0.842280
\(202\) 0 0
\(203\) −35.4762 −2.48994
\(204\) 0 0
\(205\) 1.86522 0.130273
\(206\) 0 0
\(207\) −4.29593 −0.298588
\(208\) 0 0
\(209\) −27.0469 −1.87087
\(210\) 0 0
\(211\) −25.0331 −1.72335 −0.861675 0.507461i \(-0.830584\pi\)
−0.861675 + 0.507461i \(0.830584\pi\)
\(212\) 0 0
\(213\) 12.2821 0.841557
\(214\) 0 0
\(215\) −4.06068 −0.276936
\(216\) 0 0
\(217\) −25.0479 −1.70036
\(218\) 0 0
\(219\) 8.75098 0.591336
\(220\) 0 0
\(221\) −5.26298 −0.354026
\(222\) 0 0
\(223\) −5.03208 −0.336973 −0.168486 0.985704i \(-0.553888\pi\)
−0.168486 + 0.985704i \(0.553888\pi\)
\(224\) 0 0
\(225\) −4.86420 −0.324280
\(226\) 0 0
\(227\) 15.3349 1.01781 0.508907 0.860821i \(-0.330050\pi\)
0.508907 + 0.860821i \(0.330050\pi\)
\(228\) 0 0
\(229\) −24.3099 −1.60644 −0.803221 0.595681i \(-0.796882\pi\)
−0.803221 + 0.595681i \(0.796882\pi\)
\(230\) 0 0
\(231\) 20.8740 1.37341
\(232\) 0 0
\(233\) −7.89197 −0.517020 −0.258510 0.966009i \(-0.583231\pi\)
−0.258510 + 0.966009i \(0.583231\pi\)
\(234\) 0 0
\(235\) 0.0538679 0.00351396
\(236\) 0 0
\(237\) 7.06151 0.458694
\(238\) 0 0
\(239\) −6.17307 −0.399303 −0.199651 0.979867i \(-0.563981\pi\)
−0.199651 + 0.979867i \(0.563981\pi\)
\(240\) 0 0
\(241\) −15.9013 −1.02429 −0.512147 0.858898i \(-0.671150\pi\)
−0.512147 + 0.858898i \(0.671150\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −6.12097 −0.391054
\(246\) 0 0
\(247\) −22.0521 −1.40314
\(248\) 0 0
\(249\) 10.3020 0.652860
\(250\) 0 0
\(251\) 0.977421 0.0616943 0.0308471 0.999524i \(-0.490179\pi\)
0.0308471 + 0.999524i \(0.490179\pi\)
\(252\) 0 0
\(253\) −18.4550 −1.16026
\(254\) 0 0
\(255\) −0.553723 −0.0346755
\(256\) 0 0
\(257\) −12.3626 −0.771159 −0.385580 0.922675i \(-0.625999\pi\)
−0.385580 + 0.922675i \(0.625999\pi\)
\(258\) 0 0
\(259\) 2.06424 0.128266
\(260\) 0 0
\(261\) 7.30112 0.451928
\(262\) 0 0
\(263\) 29.0183 1.78934 0.894671 0.446725i \(-0.147410\pi\)
0.894671 + 0.446725i \(0.147410\pi\)
\(264\) 0 0
\(265\) −4.09766 −0.251717
\(266\) 0 0
\(267\) 5.57185 0.340991
\(268\) 0 0
\(269\) 5.03619 0.307062 0.153531 0.988144i \(-0.450936\pi\)
0.153531 + 0.988144i \(0.450936\pi\)
\(270\) 0 0
\(271\) 21.4646 1.30388 0.651941 0.758270i \(-0.273955\pi\)
0.651941 + 0.758270i \(0.273955\pi\)
\(272\) 0 0
\(273\) 17.0191 1.03005
\(274\) 0 0
\(275\) −20.8963 −1.26009
\(276\) 0 0
\(277\) 22.4250 1.34739 0.673693 0.739011i \(-0.264707\pi\)
0.673693 + 0.739011i \(0.264707\pi\)
\(278\) 0 0
\(279\) 5.15494 0.308618
\(280\) 0 0
\(281\) −28.7465 −1.71487 −0.857437 0.514589i \(-0.827944\pi\)
−0.857437 + 0.514589i \(0.827944\pi\)
\(282\) 0 0
\(283\) 2.84966 0.169394 0.0846972 0.996407i \(-0.473008\pi\)
0.0846972 + 0.996407i \(0.473008\pi\)
\(284\) 0 0
\(285\) −2.32012 −0.137432
\(286\) 0 0
\(287\) 24.5939 1.45173
\(288\) 0 0
\(289\) −14.7422 −0.867189
\(290\) 0 0
\(291\) −4.41434 −0.258773
\(292\) 0 0
\(293\) −20.8546 −1.21834 −0.609168 0.793041i \(-0.708497\pi\)
−0.609168 + 0.793041i \(0.708497\pi\)
\(294\) 0 0
\(295\) 2.16489 0.126045
\(296\) 0 0
\(297\) −4.29593 −0.249275
\(298\) 0 0
\(299\) −15.0469 −0.870185
\(300\) 0 0
\(301\) −53.5421 −3.08612
\(302\) 0 0
\(303\) 12.5193 0.719214
\(304\) 0 0
\(305\) −1.73673 −0.0994450
\(306\) 0 0
\(307\) −13.2508 −0.756265 −0.378132 0.925751i \(-0.623434\pi\)
−0.378132 + 0.925751i \(0.623434\pi\)
\(308\) 0 0
\(309\) 10.8037 0.614602
\(310\) 0 0
\(311\) 2.17394 0.123273 0.0616365 0.998099i \(-0.480368\pi\)
0.0616365 + 0.998099i \(0.480368\pi\)
\(312\) 0 0
\(313\) −19.2769 −1.08960 −0.544798 0.838567i \(-0.683394\pi\)
−0.544798 + 0.838567i \(0.683394\pi\)
\(314\) 0 0
\(315\) 1.79060 0.100889
\(316\) 0 0
\(317\) 5.64901 0.317280 0.158640 0.987336i \(-0.449289\pi\)
0.158640 + 0.987336i \(0.449289\pi\)
\(318\) 0 0
\(319\) 31.3651 1.75611
\(320\) 0 0
\(321\) −12.1731 −0.679434
\(322\) 0 0
\(323\) 9.46023 0.526381
\(324\) 0 0
\(325\) −17.0373 −0.945060
\(326\) 0 0
\(327\) −9.08927 −0.502638
\(328\) 0 0
\(329\) 0.710277 0.0391588
\(330\) 0 0
\(331\) −13.6490 −0.750218 −0.375109 0.926981i \(-0.622395\pi\)
−0.375109 + 0.926981i \(0.622395\pi\)
\(332\) 0 0
\(333\) −0.424828 −0.0232804
\(334\) 0 0
\(335\) −4.40053 −0.240427
\(336\) 0 0
\(337\) 6.27072 0.341588 0.170794 0.985307i \(-0.445367\pi\)
0.170794 + 0.985307i \(0.445367\pi\)
\(338\) 0 0
\(339\) −9.24838 −0.502303
\(340\) 0 0
\(341\) 22.1453 1.19924
\(342\) 0 0
\(343\) −46.6950 −2.52130
\(344\) 0 0
\(345\) −1.58310 −0.0852312
\(346\) 0 0
\(347\) −18.0357 −0.968207 −0.484103 0.875011i \(-0.660854\pi\)
−0.484103 + 0.875011i \(0.660854\pi\)
\(348\) 0 0
\(349\) −31.1722 −1.66861 −0.834306 0.551302i \(-0.814131\pi\)
−0.834306 + 0.551302i \(0.814131\pi\)
\(350\) 0 0
\(351\) −3.50259 −0.186955
\(352\) 0 0
\(353\) −34.6969 −1.84673 −0.923366 0.383921i \(-0.874574\pi\)
−0.923366 + 0.383921i \(0.874574\pi\)
\(354\) 0 0
\(355\) 4.52610 0.240220
\(356\) 0 0
\(357\) −7.30112 −0.386416
\(358\) 0 0
\(359\) −28.4571 −1.50191 −0.750954 0.660355i \(-0.770406\pi\)
−0.750954 + 0.660355i \(0.770406\pi\)
\(360\) 0 0
\(361\) 20.6388 1.08625
\(362\) 0 0
\(363\) −7.45504 −0.391288
\(364\) 0 0
\(365\) 3.22483 0.168795
\(366\) 0 0
\(367\) 24.4726 1.27746 0.638729 0.769432i \(-0.279460\pi\)
0.638729 + 0.769432i \(0.279460\pi\)
\(368\) 0 0
\(369\) −5.06151 −0.263491
\(370\) 0 0
\(371\) −54.0297 −2.80508
\(372\) 0 0
\(373\) 27.1966 1.40818 0.704092 0.710109i \(-0.251354\pi\)
0.704092 + 0.710109i \(0.251354\pi\)
\(374\) 0 0
\(375\) −3.63507 −0.187714
\(376\) 0 0
\(377\) 25.5729 1.31707
\(378\) 0 0
\(379\) −22.5055 −1.15603 −0.578014 0.816027i \(-0.696172\pi\)
−0.578014 + 0.816027i \(0.696172\pi\)
\(380\) 0 0
\(381\) −14.6152 −0.748758
\(382\) 0 0
\(383\) −13.0294 −0.665770 −0.332885 0.942967i \(-0.608022\pi\)
−0.332885 + 0.942967i \(0.608022\pi\)
\(384\) 0 0
\(385\) 7.69230 0.392036
\(386\) 0 0
\(387\) 11.0191 0.560135
\(388\) 0 0
\(389\) −5.04833 −0.255961 −0.127980 0.991777i \(-0.540849\pi\)
−0.127980 + 0.991777i \(0.540849\pi\)
\(390\) 0 0
\(391\) 6.45504 0.326446
\(392\) 0 0
\(393\) 5.11346 0.257940
\(394\) 0 0
\(395\) 2.60224 0.130933
\(396\) 0 0
\(397\) 20.8268 1.04527 0.522633 0.852558i \(-0.324950\pi\)
0.522633 + 0.852558i \(0.324950\pi\)
\(398\) 0 0
\(399\) −30.5920 −1.53152
\(400\) 0 0
\(401\) −20.4008 −1.01877 −0.509383 0.860540i \(-0.670126\pi\)
−0.509383 + 0.860540i \(0.670126\pi\)
\(402\) 0 0
\(403\) 18.0557 0.899417
\(404\) 0 0
\(405\) −0.368511 −0.0183115
\(406\) 0 0
\(407\) −1.82503 −0.0904635
\(408\) 0 0
\(409\) 3.47404 0.171780 0.0858902 0.996305i \(-0.472627\pi\)
0.0858902 + 0.996305i \(0.472627\pi\)
\(410\) 0 0
\(411\) −6.46988 −0.319136
\(412\) 0 0
\(413\) 28.5452 1.40462
\(414\) 0 0
\(415\) 3.79639 0.186357
\(416\) 0 0
\(417\) −6.79937 −0.332966
\(418\) 0 0
\(419\) 27.1457 1.32616 0.663078 0.748550i \(-0.269250\pi\)
0.663078 + 0.748550i \(0.269250\pi\)
\(420\) 0 0
\(421\) −23.9446 −1.16699 −0.583494 0.812117i \(-0.698315\pi\)
−0.583494 + 0.812117i \(0.698315\pi\)
\(422\) 0 0
\(423\) −0.146177 −0.00710738
\(424\) 0 0
\(425\) 7.30892 0.354535
\(426\) 0 0
\(427\) −22.8997 −1.10819
\(428\) 0 0
\(429\) −15.0469 −0.726472
\(430\) 0 0
\(431\) −13.3330 −0.642229 −0.321114 0.947040i \(-0.604057\pi\)
−0.321114 + 0.947040i \(0.604057\pi\)
\(432\) 0 0
\(433\) −37.4335 −1.79894 −0.899469 0.436984i \(-0.856047\pi\)
−0.899469 + 0.436984i \(0.856047\pi\)
\(434\) 0 0
\(435\) 2.69055 0.129002
\(436\) 0 0
\(437\) 27.0469 1.29383
\(438\) 0 0
\(439\) −35.0185 −1.67134 −0.835671 0.549230i \(-0.814921\pi\)
−0.835671 + 0.549230i \(0.814921\pi\)
\(440\) 0 0
\(441\) 16.6100 0.790952
\(442\) 0 0
\(443\) −41.8058 −1.98625 −0.993127 0.117040i \(-0.962659\pi\)
−0.993127 + 0.117040i \(0.962659\pi\)
\(444\) 0 0
\(445\) 2.05329 0.0973352
\(446\) 0 0
\(447\) 9.51410 0.450002
\(448\) 0 0
\(449\) −25.4981 −1.20333 −0.601664 0.798749i \(-0.705496\pi\)
−0.601664 + 0.798749i \(0.705496\pi\)
\(450\) 0 0
\(451\) −21.7439 −1.02388
\(452\) 0 0
\(453\) 4.92468 0.231382
\(454\) 0 0
\(455\) 6.27175 0.294024
\(456\) 0 0
\(457\) 16.6022 0.776620 0.388310 0.921529i \(-0.373059\pi\)
0.388310 + 0.921529i \(0.373059\pi\)
\(458\) 0 0
\(459\) 1.50259 0.0701351
\(460\) 0 0
\(461\) 16.0557 0.747788 0.373894 0.927472i \(-0.378022\pi\)
0.373894 + 0.927472i \(0.378022\pi\)
\(462\) 0 0
\(463\) −18.3662 −0.853550 −0.426775 0.904358i \(-0.640350\pi\)
−0.426775 + 0.904358i \(0.640350\pi\)
\(464\) 0 0
\(465\) 1.89965 0.0880944
\(466\) 0 0
\(467\) −31.4102 −1.45349 −0.726745 0.686908i \(-0.758968\pi\)
−0.726745 + 0.686908i \(0.758968\pi\)
\(468\) 0 0
\(469\) −58.0233 −2.67927
\(470\) 0 0
\(471\) −11.0365 −0.508537
\(472\) 0 0
\(473\) 47.3375 2.17658
\(474\) 0 0
\(475\) 30.6247 1.40516
\(476\) 0 0
\(477\) 11.1195 0.509126
\(478\) 0 0
\(479\) 30.0521 1.37312 0.686558 0.727075i \(-0.259121\pi\)
0.686558 + 0.727075i \(0.259121\pi\)
\(480\) 0 0
\(481\) −1.48800 −0.0678470
\(482\) 0 0
\(483\) −20.8740 −0.949799
\(484\) 0 0
\(485\) −1.62673 −0.0738662
\(486\) 0 0
\(487\) −5.34706 −0.242299 −0.121149 0.992634i \(-0.538658\pi\)
−0.121149 + 0.992634i \(0.538658\pi\)
\(488\) 0 0
\(489\) −8.67482 −0.392289
\(490\) 0 0
\(491\) −8.10548 −0.365795 −0.182897 0.983132i \(-0.558548\pi\)
−0.182897 + 0.983132i \(0.558548\pi\)
\(492\) 0 0
\(493\) −10.9706 −0.494092
\(494\) 0 0
\(495\) −1.58310 −0.0711550
\(496\) 0 0
\(497\) 59.6790 2.67697
\(498\) 0 0
\(499\) −13.2824 −0.594602 −0.297301 0.954784i \(-0.596087\pi\)
−0.297301 + 0.954784i \(0.596087\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) 2.68031 0.119509 0.0597546 0.998213i \(-0.480968\pi\)
0.0597546 + 0.998213i \(0.480968\pi\)
\(504\) 0 0
\(505\) 4.61350 0.205298
\(506\) 0 0
\(507\) 0.731834 0.0325019
\(508\) 0 0
\(509\) −12.1627 −0.539102 −0.269551 0.962986i \(-0.586875\pi\)
−0.269551 + 0.962986i \(0.586875\pi\)
\(510\) 0 0
\(511\) 42.5211 1.88102
\(512\) 0 0
\(513\) 6.29593 0.277972
\(514\) 0 0
\(515\) 3.98129 0.175437
\(516\) 0 0
\(517\) −0.627968 −0.0276180
\(518\) 0 0
\(519\) 2.68149 0.117704
\(520\) 0 0
\(521\) −6.19124 −0.271243 −0.135622 0.990761i \(-0.543303\pi\)
−0.135622 + 0.990761i \(0.543303\pi\)
\(522\) 0 0
\(523\) −21.6144 −0.945134 −0.472567 0.881295i \(-0.656672\pi\)
−0.472567 + 0.881295i \(0.656672\pi\)
\(524\) 0 0
\(525\) −23.6352 −1.03152
\(526\) 0 0
\(527\) −7.74579 −0.337412
\(528\) 0 0
\(529\) −4.54496 −0.197607
\(530\) 0 0
\(531\) −5.87468 −0.254940
\(532\) 0 0
\(533\) −17.7284 −0.767902
\(534\) 0 0
\(535\) −4.48591 −0.193943
\(536\) 0 0
\(537\) −5.82708 −0.251457
\(538\) 0 0
\(539\) 71.3554 3.07350
\(540\) 0 0
\(541\) 18.8438 0.810157 0.405079 0.914282i \(-0.367244\pi\)
0.405079 + 0.914282i \(0.367244\pi\)
\(542\) 0 0
\(543\) 18.1695 0.779727
\(544\) 0 0
\(545\) −3.34950 −0.143477
\(546\) 0 0
\(547\) 29.8050 1.27437 0.637186 0.770710i \(-0.280098\pi\)
0.637186 + 0.770710i \(0.280098\pi\)
\(548\) 0 0
\(549\) 4.71283 0.201139
\(550\) 0 0
\(551\) −45.9674 −1.95828
\(552\) 0 0
\(553\) 34.3119 1.45909
\(554\) 0 0
\(555\) −0.156554 −0.00664534
\(556\) 0 0
\(557\) 12.0194 0.509280 0.254640 0.967036i \(-0.418043\pi\)
0.254640 + 0.967036i \(0.418043\pi\)
\(558\) 0 0
\(559\) 38.5956 1.63242
\(560\) 0 0
\(561\) 6.45504 0.272532
\(562\) 0 0
\(563\) −33.2081 −1.39955 −0.699777 0.714362i \(-0.746717\pi\)
−0.699777 + 0.714362i \(0.746717\pi\)
\(564\) 0 0
\(565\) −3.40813 −0.143381
\(566\) 0 0
\(567\) −4.85901 −0.204059
\(568\) 0 0
\(569\) −14.8156 −0.621103 −0.310552 0.950556i \(-0.600514\pi\)
−0.310552 + 0.950556i \(0.600514\pi\)
\(570\) 0 0
\(571\) −17.7454 −0.742624 −0.371312 0.928508i \(-0.621092\pi\)
−0.371312 + 0.928508i \(0.621092\pi\)
\(572\) 0 0
\(573\) −21.7468 −0.908486
\(574\) 0 0
\(575\) 20.8963 0.871435
\(576\) 0 0
\(577\) 9.51631 0.396169 0.198085 0.980185i \(-0.436528\pi\)
0.198085 + 0.980185i \(0.436528\pi\)
\(578\) 0 0
\(579\) 24.9950 1.03875
\(580\) 0 0
\(581\) 50.0573 2.07673
\(582\) 0 0
\(583\) 47.7686 1.97837
\(584\) 0 0
\(585\) −1.29075 −0.0533657
\(586\) 0 0
\(587\) 37.6116 1.55240 0.776198 0.630490i \(-0.217146\pi\)
0.776198 + 0.630490i \(0.217146\pi\)
\(588\) 0 0
\(589\) −32.4552 −1.33729
\(590\) 0 0
\(591\) −6.56410 −0.270011
\(592\) 0 0
\(593\) 13.3295 0.547378 0.273689 0.961818i \(-0.411756\pi\)
0.273689 + 0.961818i \(0.411756\pi\)
\(594\) 0 0
\(595\) −2.69055 −0.110302
\(596\) 0 0
\(597\) 5.19755 0.212721
\(598\) 0 0
\(599\) 43.5217 1.77825 0.889124 0.457666i \(-0.151315\pi\)
0.889124 + 0.457666i \(0.151315\pi\)
\(600\) 0 0
\(601\) −23.6773 −0.965818 −0.482909 0.875671i \(-0.660420\pi\)
−0.482909 + 0.875671i \(0.660420\pi\)
\(602\) 0 0
\(603\) 11.9414 0.486291
\(604\) 0 0
\(605\) −2.74727 −0.111692
\(606\) 0 0
\(607\) −0.173561 −0.00704461 −0.00352231 0.999994i \(-0.501121\pi\)
−0.00352231 + 0.999994i \(0.501121\pi\)
\(608\) 0 0
\(609\) 35.4762 1.43757
\(610\) 0 0
\(611\) −0.511999 −0.0207133
\(612\) 0 0
\(613\) −45.9055 −1.85411 −0.927053 0.374929i \(-0.877667\pi\)
−0.927053 + 0.374929i \(0.877667\pi\)
\(614\) 0 0
\(615\) −1.86522 −0.0752130
\(616\) 0 0
\(617\) 12.2940 0.494939 0.247470 0.968896i \(-0.420401\pi\)
0.247470 + 0.968896i \(0.420401\pi\)
\(618\) 0 0
\(619\) −22.4206 −0.901161 −0.450580 0.892736i \(-0.648783\pi\)
−0.450580 + 0.892736i \(0.648783\pi\)
\(620\) 0 0
\(621\) 4.29593 0.172390
\(622\) 0 0
\(623\) 27.0737 1.08468
\(624\) 0 0
\(625\) 22.9814 0.919257
\(626\) 0 0
\(627\) 27.0469 1.08015
\(628\) 0 0
\(629\) 0.638344 0.0254525
\(630\) 0 0
\(631\) −28.7953 −1.14632 −0.573161 0.819443i \(-0.694283\pi\)
−0.573161 + 0.819443i \(0.694283\pi\)
\(632\) 0 0
\(633\) 25.0331 0.994976
\(634\) 0 0
\(635\) −5.38586 −0.213731
\(636\) 0 0
\(637\) 58.1781 2.30510
\(638\) 0 0
\(639\) −12.2821 −0.485873
\(640\) 0 0
\(641\) −13.3314 −0.526557 −0.263278 0.964720i \(-0.584804\pi\)
−0.263278 + 0.964720i \(0.584804\pi\)
\(642\) 0 0
\(643\) −19.3737 −0.764025 −0.382012 0.924157i \(-0.624769\pi\)
−0.382012 + 0.924157i \(0.624769\pi\)
\(644\) 0 0
\(645\) 4.06068 0.159889
\(646\) 0 0
\(647\) −11.9846 −0.471164 −0.235582 0.971854i \(-0.575700\pi\)
−0.235582 + 0.971854i \(0.575700\pi\)
\(648\) 0 0
\(649\) −25.2373 −0.990649
\(650\) 0 0
\(651\) 25.0479 0.981706
\(652\) 0 0
\(653\) −49.8602 −1.95118 −0.975591 0.219597i \(-0.929526\pi\)
−0.975591 + 0.219597i \(0.929526\pi\)
\(654\) 0 0
\(655\) 1.88437 0.0736283
\(656\) 0 0
\(657\) −8.75098 −0.341408
\(658\) 0 0
\(659\) 13.6776 0.532803 0.266401 0.963862i \(-0.414165\pi\)
0.266401 + 0.963862i \(0.414165\pi\)
\(660\) 0 0
\(661\) −16.7673 −0.652170 −0.326085 0.945340i \(-0.605730\pi\)
−0.326085 + 0.945340i \(0.605730\pi\)
\(662\) 0 0
\(663\) 5.26298 0.204397
\(664\) 0 0
\(665\) −11.2735 −0.437168
\(666\) 0 0
\(667\) −31.3651 −1.21446
\(668\) 0 0
\(669\) 5.03208 0.194551
\(670\) 0 0
\(671\) 20.2460 0.781589
\(672\) 0 0
\(673\) −4.06233 −0.156591 −0.0782957 0.996930i \(-0.524948\pi\)
−0.0782957 + 0.996930i \(0.524948\pi\)
\(674\) 0 0
\(675\) 4.86420 0.187223
\(676\) 0 0
\(677\) 31.7458 1.22009 0.610045 0.792367i \(-0.291151\pi\)
0.610045 + 0.792367i \(0.291151\pi\)
\(678\) 0 0
\(679\) −21.4493 −0.823150
\(680\) 0 0
\(681\) −15.3349 −0.587635
\(682\) 0 0
\(683\) −2.63079 −0.100665 −0.0503323 0.998733i \(-0.516028\pi\)
−0.0503323 + 0.998733i \(0.516028\pi\)
\(684\) 0 0
\(685\) −2.38422 −0.0910965
\(686\) 0 0
\(687\) 24.3099 0.927480
\(688\) 0 0
\(689\) 38.9471 1.48376
\(690\) 0 0
\(691\) 30.6234 1.16497 0.582484 0.812842i \(-0.302081\pi\)
0.582484 + 0.812842i \(0.302081\pi\)
\(692\) 0 0
\(693\) −20.8740 −0.792937
\(694\) 0 0
\(695\) −2.50564 −0.0950445
\(696\) 0 0
\(697\) 7.60539 0.288075
\(698\) 0 0
\(699\) 7.89197 0.298502
\(700\) 0 0
\(701\) 13.4102 0.506495 0.253248 0.967401i \(-0.418501\pi\)
0.253248 + 0.967401i \(0.418501\pi\)
\(702\) 0 0
\(703\) 2.67469 0.100878
\(704\) 0 0
\(705\) −0.0538679 −0.00202878
\(706\) 0 0
\(707\) 60.8314 2.28780
\(708\) 0 0
\(709\) −16.2761 −0.611262 −0.305631 0.952150i \(-0.598867\pi\)
−0.305631 + 0.952150i \(0.598867\pi\)
\(710\) 0 0
\(711\) −7.06151 −0.264827
\(712\) 0 0
\(713\) −22.1453 −0.829348
\(714\) 0 0
\(715\) −5.54496 −0.207370
\(716\) 0 0
\(717\) 6.17307 0.230537
\(718\) 0 0
\(719\) −33.8035 −1.26066 −0.630329 0.776328i \(-0.717080\pi\)
−0.630329 + 0.776328i \(0.717080\pi\)
\(720\) 0 0
\(721\) 52.4954 1.95503
\(722\) 0 0
\(723\) 15.9013 0.591377
\(724\) 0 0
\(725\) −35.5141 −1.31896
\(726\) 0 0
\(727\) 7.96312 0.295336 0.147668 0.989037i \(-0.452823\pi\)
0.147668 + 0.989037i \(0.452823\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −16.5573 −0.612394
\(732\) 0 0
\(733\) 34.5816 1.27730 0.638651 0.769497i \(-0.279493\pi\)
0.638651 + 0.769497i \(0.279493\pi\)
\(734\) 0 0
\(735\) 6.12097 0.225775
\(736\) 0 0
\(737\) 51.2994 1.88964
\(738\) 0 0
\(739\) 9.91361 0.364678 0.182339 0.983236i \(-0.441633\pi\)
0.182339 + 0.983236i \(0.441633\pi\)
\(740\) 0 0
\(741\) 22.0521 0.810104
\(742\) 0 0
\(743\) −35.1326 −1.28889 −0.644444 0.764651i \(-0.722911\pi\)
−0.644444 + 0.764651i \(0.722911\pi\)
\(744\) 0 0
\(745\) 3.50605 0.128452
\(746\) 0 0
\(747\) −10.3020 −0.376929
\(748\) 0 0
\(749\) −59.1491 −2.16126
\(750\) 0 0
\(751\) −0.810730 −0.0295840 −0.0147920 0.999891i \(-0.504709\pi\)
−0.0147920 + 0.999891i \(0.504709\pi\)
\(752\) 0 0
\(753\) −0.977421 −0.0356192
\(754\) 0 0
\(755\) 1.81480 0.0660474
\(756\) 0 0
\(757\) 8.08330 0.293793 0.146896 0.989152i \(-0.453072\pi\)
0.146896 + 0.989152i \(0.453072\pi\)
\(758\) 0 0
\(759\) 18.4550 0.669876
\(760\) 0 0
\(761\) 47.6549 1.72749 0.863745 0.503929i \(-0.168113\pi\)
0.863745 + 0.503929i \(0.168113\pi\)
\(762\) 0 0
\(763\) −44.1649 −1.59888
\(764\) 0 0
\(765\) 0.553723 0.0200199
\(766\) 0 0
\(767\) −20.5766 −0.742979
\(768\) 0 0
\(769\) −22.1365 −0.798262 −0.399131 0.916894i \(-0.630688\pi\)
−0.399131 + 0.916894i \(0.630688\pi\)
\(770\) 0 0
\(771\) 12.3626 0.445229
\(772\) 0 0
\(773\) 32.7111 1.17654 0.588268 0.808666i \(-0.299810\pi\)
0.588268 + 0.808666i \(0.299810\pi\)
\(774\) 0 0
\(775\) −25.0747 −0.900709
\(776\) 0 0
\(777\) −2.06424 −0.0740544
\(778\) 0 0
\(779\) 31.8669 1.14175
\(780\) 0 0
\(781\) −52.7632 −1.88801
\(782\) 0 0
\(783\) −7.30112 −0.260921
\(784\) 0 0
\(785\) −4.06709 −0.145161
\(786\) 0 0
\(787\) −38.1925 −1.36142 −0.680708 0.732555i \(-0.738328\pi\)
−0.680708 + 0.732555i \(0.738328\pi\)
\(788\) 0 0
\(789\) −29.0183 −1.03308
\(790\) 0 0
\(791\) −44.9380 −1.59781
\(792\) 0 0
\(793\) 16.5071 0.586186
\(794\) 0 0
\(795\) 4.09766 0.145329
\(796\) 0 0
\(797\) −0.496624 −0.0175913 −0.00879566 0.999961i \(-0.502800\pi\)
−0.00879566 + 0.999961i \(0.502800\pi\)
\(798\) 0 0
\(799\) 0.219645 0.00777048
\(800\) 0 0
\(801\) −5.57185 −0.196871
\(802\) 0 0
\(803\) −37.5936 −1.32665
\(804\) 0 0
\(805\) −7.69230 −0.271118
\(806\) 0 0
\(807\) −5.03619 −0.177282
\(808\) 0 0
\(809\) 30.1686 1.06067 0.530336 0.847788i \(-0.322066\pi\)
0.530336 + 0.847788i \(0.322066\pi\)
\(810\) 0 0
\(811\) −3.34079 −0.117311 −0.0586555 0.998278i \(-0.518681\pi\)
−0.0586555 + 0.998278i \(0.518681\pi\)
\(812\) 0 0
\(813\) −21.4646 −0.752796
\(814\) 0 0
\(815\) −3.19677 −0.111978
\(816\) 0 0
\(817\) −69.3758 −2.42715
\(818\) 0 0
\(819\) −17.0191 −0.594697
\(820\) 0 0
\(821\) −26.6959 −0.931694 −0.465847 0.884865i \(-0.654250\pi\)
−0.465847 + 0.884865i \(0.654250\pi\)
\(822\) 0 0
\(823\) −19.7998 −0.690179 −0.345090 0.938570i \(-0.612151\pi\)
−0.345090 + 0.938570i \(0.612151\pi\)
\(824\) 0 0
\(825\) 20.8963 0.727515
\(826\) 0 0
\(827\) 52.9527 1.84135 0.920674 0.390333i \(-0.127640\pi\)
0.920674 + 0.390333i \(0.127640\pi\)
\(828\) 0 0
\(829\) 17.4115 0.604725 0.302362 0.953193i \(-0.402225\pi\)
0.302362 + 0.953193i \(0.402225\pi\)
\(830\) 0 0
\(831\) −22.4250 −0.777914
\(832\) 0 0
\(833\) −24.9581 −0.864746
\(834\) 0 0
\(835\) −0.368511 −0.0127529
\(836\) 0 0
\(837\) −5.15494 −0.178181
\(838\) 0 0
\(839\) 34.5735 1.19361 0.596804 0.802387i \(-0.296437\pi\)
0.596804 + 0.802387i \(0.296437\pi\)
\(840\) 0 0
\(841\) 24.3064 0.838151
\(842\) 0 0
\(843\) 28.7465 0.990083
\(844\) 0 0
\(845\) 0.269689 0.00927758
\(846\) 0 0
\(847\) −36.2241 −1.24468
\(848\) 0 0
\(849\) −2.84966 −0.0978000
\(850\) 0 0
\(851\) 1.82503 0.0625613
\(852\) 0 0
\(853\) −37.8981 −1.29761 −0.648804 0.760956i \(-0.724730\pi\)
−0.648804 + 0.760956i \(0.724730\pi\)
\(854\) 0 0
\(855\) 2.32012 0.0793465
\(856\) 0 0
\(857\) −14.7888 −0.505177 −0.252588 0.967574i \(-0.581282\pi\)
−0.252588 + 0.967574i \(0.581282\pi\)
\(858\) 0 0
\(859\) −15.6848 −0.535158 −0.267579 0.963536i \(-0.586224\pi\)
−0.267579 + 0.963536i \(0.586224\pi\)
\(860\) 0 0
\(861\) −24.5939 −0.838158
\(862\) 0 0
\(863\) 25.4309 0.865679 0.432839 0.901471i \(-0.357512\pi\)
0.432839 + 0.901471i \(0.357512\pi\)
\(864\) 0 0
\(865\) 0.988158 0.0335984
\(866\) 0 0
\(867\) 14.7422 0.500672
\(868\) 0 0
\(869\) −30.3358 −1.02907
\(870\) 0 0
\(871\) 41.8258 1.41721
\(872\) 0 0
\(873\) 4.41434 0.149403
\(874\) 0 0
\(875\) −17.6628 −0.597113
\(876\) 0 0
\(877\) −10.8789 −0.367354 −0.183677 0.982987i \(-0.558800\pi\)
−0.183677 + 0.982987i \(0.558800\pi\)
\(878\) 0 0
\(879\) 20.8546 0.703407
\(880\) 0 0
\(881\) 35.0839 1.18201 0.591004 0.806669i \(-0.298732\pi\)
0.591004 + 0.806669i \(0.298732\pi\)
\(882\) 0 0
\(883\) −31.8391 −1.07147 −0.535736 0.844385i \(-0.679966\pi\)
−0.535736 + 0.844385i \(0.679966\pi\)
\(884\) 0 0
\(885\) −2.16489 −0.0727719
\(886\) 0 0
\(887\) 47.8743 1.60746 0.803731 0.594993i \(-0.202845\pi\)
0.803731 + 0.594993i \(0.202845\pi\)
\(888\) 0 0
\(889\) −71.0153 −2.38178
\(890\) 0 0
\(891\) 4.29593 0.143919
\(892\) 0 0
\(893\) 0.920322 0.0307974
\(894\) 0 0
\(895\) −2.14734 −0.0717778
\(896\) 0 0
\(897\) 15.0469 0.502402
\(898\) 0 0
\(899\) 37.6369 1.25526
\(900\) 0 0
\(901\) −16.7081 −0.556627
\(902\) 0 0
\(903\) 53.5421 1.78177
\(904\) 0 0
\(905\) 6.69566 0.222571
\(906\) 0 0
\(907\) −16.1837 −0.537371 −0.268685 0.963228i \(-0.586589\pi\)
−0.268685 + 0.963228i \(0.586589\pi\)
\(908\) 0 0
\(909\) −12.5193 −0.415239
\(910\) 0 0
\(911\) 8.65866 0.286874 0.143437 0.989659i \(-0.454185\pi\)
0.143437 + 0.989659i \(0.454185\pi\)
\(912\) 0 0
\(913\) −44.2565 −1.46468
\(914\) 0 0
\(915\) 1.73673 0.0574146
\(916\) 0 0
\(917\) 24.8464 0.820499
\(918\) 0 0
\(919\) 38.7780 1.27917 0.639584 0.768721i \(-0.279107\pi\)
0.639584 + 0.768721i \(0.279107\pi\)
\(920\) 0 0
\(921\) 13.2508 0.436630
\(922\) 0 0
\(923\) −43.0193 −1.41600
\(924\) 0 0
\(925\) 2.06645 0.0679444
\(926\) 0 0
\(927\) −10.8037 −0.354841
\(928\) 0 0
\(929\) 38.8787 1.27557 0.637785 0.770214i \(-0.279851\pi\)
0.637785 + 0.770214i \(0.279851\pi\)
\(930\) 0 0
\(931\) −104.575 −3.42732
\(932\) 0 0
\(933\) −2.17394 −0.0711718
\(934\) 0 0
\(935\) 2.37876 0.0777936
\(936\) 0 0
\(937\) 18.5796 0.606969 0.303484 0.952836i \(-0.401850\pi\)
0.303484 + 0.952836i \(0.401850\pi\)
\(938\) 0 0
\(939\) 19.2769 0.629079
\(940\) 0 0
\(941\) 48.2550 1.57307 0.786534 0.617547i \(-0.211874\pi\)
0.786534 + 0.617547i \(0.211874\pi\)
\(942\) 0 0
\(943\) 21.7439 0.708078
\(944\) 0 0
\(945\) −1.79060 −0.0582482
\(946\) 0 0
\(947\) −54.1601 −1.75997 −0.879984 0.475004i \(-0.842447\pi\)
−0.879984 + 0.475004i \(0.842447\pi\)
\(948\) 0 0
\(949\) −30.6511 −0.994978
\(950\) 0 0
\(951\) −5.64901 −0.183182
\(952\) 0 0
\(953\) 42.2537 1.36873 0.684366 0.729139i \(-0.260079\pi\)
0.684366 + 0.729139i \(0.260079\pi\)
\(954\) 0 0
\(955\) −8.01394 −0.259325
\(956\) 0 0
\(957\) −31.3651 −1.01389
\(958\) 0 0
\(959\) −31.4372 −1.01516
\(960\) 0 0
\(961\) −4.42655 −0.142792
\(962\) 0 0
\(963\) 12.1731 0.392272
\(964\) 0 0
\(965\) 9.21092 0.296510
\(966\) 0 0
\(967\) 7.54868 0.242749 0.121375 0.992607i \(-0.461270\pi\)
0.121375 + 0.992607i \(0.461270\pi\)
\(968\) 0 0
\(969\) −9.46023 −0.303906
\(970\) 0 0
\(971\) −25.8294 −0.828904 −0.414452 0.910071i \(-0.636027\pi\)
−0.414452 + 0.910071i \(0.636027\pi\)
\(972\) 0 0
\(973\) −33.0382 −1.05916
\(974\) 0 0
\(975\) 17.0373 0.545631
\(976\) 0 0
\(977\) −5.80544 −0.185733 −0.0928663 0.995679i \(-0.529603\pi\)
−0.0928663 + 0.995679i \(0.529603\pi\)
\(978\) 0 0
\(979\) −23.9363 −0.765007
\(980\) 0 0
\(981\) 9.08927 0.290198
\(982\) 0 0
\(983\) 50.5608 1.61264 0.806320 0.591480i \(-0.201456\pi\)
0.806320 + 0.591480i \(0.201456\pi\)
\(984\) 0 0
\(985\) −2.41894 −0.0770740
\(986\) 0 0
\(987\) −0.710277 −0.0226084
\(988\) 0 0
\(989\) −47.3375 −1.50525
\(990\) 0 0
\(991\) 19.6618 0.624579 0.312290 0.949987i \(-0.398904\pi\)
0.312290 + 0.949987i \(0.398904\pi\)
\(992\) 0 0
\(993\) 13.6490 0.433138
\(994\) 0 0
\(995\) 1.91535 0.0607208
\(996\) 0 0
\(997\) 19.2060 0.608261 0.304130 0.952630i \(-0.401634\pi\)
0.304130 + 0.952630i \(0.401634\pi\)
\(998\) 0 0
\(999\) 0.424828 0.0134410
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 8016.2.a.r.1.3 5
4.3 odd 2 1002.2.a.j.1.3 5
12.11 even 2 3006.2.a.t.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1002.2.a.j.1.3 5 4.3 odd 2
3006.2.a.t.1.3 5 12.11 even 2
8016.2.a.r.1.3 5 1.1 even 1 trivial