Properties

Label 124.4.a.a.1.4
Level $124$
Weight $4$
Character 124.1
Self dual yes
Analytic conductor $7.316$
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] = [124,4,Mod(1,124)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(124, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("124.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 124.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: \(7.31623684071\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.841724.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 16x^{2} + 11x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-3.81863\) of defining polynomial
Character \(\chi\) \(=\) 124.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.12869 q^{3} -10.8920 q^{5} -29.9908 q^{7} -9.95389 q^{9} +O(q^{10})\) \(q+4.12869 q^{3} -10.8920 q^{5} -29.9908 q^{7} -9.95389 q^{9} +4.75411 q^{11} -75.3498 q^{13} -44.9697 q^{15} +111.675 q^{17} +90.8213 q^{19} -123.823 q^{21} -42.3266 q^{23} -6.36432 q^{25} -152.571 q^{27} -202.535 q^{29} -31.0000 q^{31} +19.6283 q^{33} +326.660 q^{35} +334.906 q^{37} -311.096 q^{39} -367.314 q^{41} +82.6879 q^{43} +108.418 q^{45} -276.578 q^{47} +556.448 q^{49} +461.073 q^{51} +581.851 q^{53} -51.7818 q^{55} +374.973 q^{57} -630.494 q^{59} +527.474 q^{61} +298.525 q^{63} +820.710 q^{65} -684.749 q^{67} -174.754 q^{69} -230.564 q^{71} +802.656 q^{73} -26.2763 q^{75} -142.580 q^{77} -455.003 q^{79} -361.165 q^{81} -214.005 q^{83} -1216.37 q^{85} -836.206 q^{87} -340.002 q^{89} +2259.80 q^{91} -127.989 q^{93} -989.226 q^{95} -995.113 q^{97} -47.3219 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 14 q^{5} - 12 q^{7} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 14 q^{5} - 12 q^{7} - 24 q^{9} - 98 q^{11} - 128 q^{13} - 162 q^{15} - 86 q^{17} - 116 q^{19} - 118 q^{21} - 214 q^{23} + 102 q^{25} - 244 q^{27} - 168 q^{29} - 124 q^{31} + 324 q^{33} - 272 q^{35} + 598 q^{37} + 152 q^{39} + 218 q^{41} - 192 q^{43} + 990 q^{45} + 32 q^{47} + 1110 q^{49} + 240 q^{51} + 290 q^{53} + 280 q^{55} + 1358 q^{57} - 376 q^{59} + 196 q^{61} + 144 q^{63} + 822 q^{65} + 40 q^{67} + 740 q^{69} - 536 q^{71} + 1436 q^{73} + 22 q^{75} - 1708 q^{77} - 1010 q^{79} + 696 q^{81} - 1174 q^{83} + 1122 q^{85} - 332 q^{87} - 518 q^{89} - 438 q^{91} + 124 q^{93} - 3148 q^{95} - 466 q^{97} - 14 q^{99}+O(q^{100}) \) 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
\(3\) 4.12869 0.794567 0.397284 0.917696i \(-0.369953\pi\)
0.397284 + 0.917696i \(0.369953\pi\)
\(4\) 0 0
\(5\) −10.8920 −0.974210 −0.487105 0.873343i \(-0.661947\pi\)
−0.487105 + 0.873343i \(0.661947\pi\)
\(6\) 0 0
\(7\) −29.9908 −1.61935 −0.809676 0.586878i \(-0.800357\pi\)
−0.809676 + 0.586878i \(0.800357\pi\)
\(8\) 0 0
\(9\) −9.95389 −0.368663
\(10\) 0 0
\(11\) 4.75411 0.130311 0.0651554 0.997875i \(-0.479246\pi\)
0.0651554 + 0.997875i \(0.479246\pi\)
\(12\) 0 0
\(13\) −75.3498 −1.60756 −0.803780 0.594927i \(-0.797181\pi\)
−0.803780 + 0.594927i \(0.797181\pi\)
\(14\) 0 0
\(15\) −44.9697 −0.774076
\(16\) 0 0
\(17\) 111.675 1.59325 0.796625 0.604473i \(-0.206616\pi\)
0.796625 + 0.604473i \(0.206616\pi\)
\(18\) 0 0
\(19\) 90.8213 1.09662 0.548311 0.836274i \(-0.315271\pi\)
0.548311 + 0.836274i \(0.315271\pi\)
\(20\) 0 0
\(21\) −123.823 −1.28668
\(22\) 0 0
\(23\) −42.3266 −0.383727 −0.191863 0.981422i \(-0.561453\pi\)
−0.191863 + 0.981422i \(0.561453\pi\)
\(24\) 0 0
\(25\) −6.36432 −0.0509145
\(26\) 0 0
\(27\) −152.571 −1.08749
\(28\) 0 0
\(29\) −202.535 −1.29689 −0.648446 0.761261i \(-0.724581\pi\)
−0.648446 + 0.761261i \(0.724581\pi\)
\(30\) 0 0
\(31\) −31.0000 −0.179605
\(32\) 0 0
\(33\) 19.6283 0.103541
\(34\) 0 0
\(35\) 326.660 1.57759
\(36\) 0 0
\(37\) 334.906 1.48806 0.744030 0.668147i \(-0.232912\pi\)
0.744030 + 0.668147i \(0.232912\pi\)
\(38\) 0 0
\(39\) −311.096 −1.27731
\(40\) 0 0
\(41\) −367.314 −1.39914 −0.699571 0.714563i \(-0.746626\pi\)
−0.699571 + 0.714563i \(0.746626\pi\)
\(42\) 0 0
\(43\) 82.6879 0.293251 0.146625 0.989192i \(-0.453159\pi\)
0.146625 + 0.989192i \(0.453159\pi\)
\(44\) 0 0
\(45\) 108.418 0.359155
\(46\) 0 0
\(47\) −276.578 −0.858363 −0.429182 0.903218i \(-0.641198\pi\)
−0.429182 + 0.903218i \(0.641198\pi\)
\(48\) 0 0
\(49\) 556.448 1.62230
\(50\) 0 0
\(51\) 461.073 1.26594
\(52\) 0 0
\(53\) 581.851 1.50799 0.753994 0.656882i \(-0.228125\pi\)
0.753994 + 0.656882i \(0.228125\pi\)
\(54\) 0 0
\(55\) −51.7818 −0.126950
\(56\) 0 0
\(57\) 374.973 0.871341
\(58\) 0 0
\(59\) −630.494 −1.39124 −0.695621 0.718409i \(-0.744871\pi\)
−0.695621 + 0.718409i \(0.744871\pi\)
\(60\) 0 0
\(61\) 527.474 1.10715 0.553574 0.832800i \(-0.313263\pi\)
0.553574 + 0.832800i \(0.313263\pi\)
\(62\) 0 0
\(63\) 298.525 0.596994
\(64\) 0 0
\(65\) 820.710 1.56610
\(66\) 0 0
\(67\) −684.749 −1.24859 −0.624294 0.781189i \(-0.714613\pi\)
−0.624294 + 0.781189i \(0.714613\pi\)
\(68\) 0 0
\(69\) −174.754 −0.304897
\(70\) 0 0
\(71\) −230.564 −0.385393 −0.192696 0.981258i \(-0.561723\pi\)
−0.192696 + 0.981258i \(0.561723\pi\)
\(72\) 0 0
\(73\) 802.656 1.28690 0.643451 0.765488i \(-0.277502\pi\)
0.643451 + 0.765488i \(0.277502\pi\)
\(74\) 0 0
\(75\) −26.2763 −0.0404550
\(76\) 0 0
\(77\) −142.580 −0.211019
\(78\) 0 0
\(79\) −455.003 −0.647998 −0.323999 0.946057i \(-0.605027\pi\)
−0.323999 + 0.946057i \(0.605027\pi\)
\(80\) 0 0
\(81\) −361.165 −0.495425
\(82\) 0 0
\(83\) −214.005 −0.283013 −0.141507 0.989937i \(-0.545195\pi\)
−0.141507 + 0.989937i \(0.545195\pi\)
\(84\) 0 0
\(85\) −1216.37 −1.55216
\(86\) 0 0
\(87\) −836.206 −1.03047
\(88\) 0 0
\(89\) −340.002 −0.404946 −0.202473 0.979288i \(-0.564898\pi\)
−0.202473 + 0.979288i \(0.564898\pi\)
\(90\) 0 0
\(91\) 2259.80 2.60320
\(92\) 0 0
\(93\) −127.989 −0.142709
\(94\) 0 0
\(95\) −989.226 −1.06834
\(96\) 0 0
\(97\) −995.113 −1.04163 −0.520817 0.853669i \(-0.674373\pi\)
−0.520817 + 0.853669i \(0.674373\pi\)
\(98\) 0 0
\(99\) −47.3219 −0.0480407
\(100\) 0 0
\(101\) −585.825 −0.577146 −0.288573 0.957458i \(-0.593181\pi\)
−0.288573 + 0.957458i \(0.593181\pi\)
\(102\) 0 0
\(103\) 492.440 0.471083 0.235542 0.971864i \(-0.424314\pi\)
0.235542 + 0.971864i \(0.424314\pi\)
\(104\) 0 0
\(105\) 1348.68 1.25350
\(106\) 0 0
\(107\) −1883.39 −1.70163 −0.850813 0.525469i \(-0.823890\pi\)
−0.850813 + 0.525469i \(0.823890\pi\)
\(108\) 0 0
\(109\) −1176.96 −1.03424 −0.517122 0.855912i \(-0.672997\pi\)
−0.517122 + 0.855912i \(0.672997\pi\)
\(110\) 0 0
\(111\) 1382.72 1.18236
\(112\) 0 0
\(113\) 1946.32 1.62031 0.810154 0.586218i \(-0.199384\pi\)
0.810154 + 0.586218i \(0.199384\pi\)
\(114\) 0 0
\(115\) 461.022 0.373831
\(116\) 0 0
\(117\) 750.024 0.592647
\(118\) 0 0
\(119\) −3349.23 −2.58003
\(120\) 0 0
\(121\) −1308.40 −0.983019
\(122\) 0 0
\(123\) −1516.53 −1.11171
\(124\) 0 0
\(125\) 1430.82 1.02381
\(126\) 0 0
\(127\) 487.162 0.340383 0.170191 0.985411i \(-0.445561\pi\)
0.170191 + 0.985411i \(0.445561\pi\)
\(128\) 0 0
\(129\) 341.393 0.233008
\(130\) 0 0
\(131\) −453.362 −0.302370 −0.151185 0.988506i \(-0.548309\pi\)
−0.151185 + 0.988506i \(0.548309\pi\)
\(132\) 0 0
\(133\) −2723.80 −1.77582
\(134\) 0 0
\(135\) 1661.81 1.05945
\(136\) 0 0
\(137\) −869.263 −0.542089 −0.271044 0.962567i \(-0.587369\pi\)
−0.271044 + 0.962567i \(0.587369\pi\)
\(138\) 0 0
\(139\) −291.016 −0.177580 −0.0887900 0.996050i \(-0.528300\pi\)
−0.0887900 + 0.996050i \(0.528300\pi\)
\(140\) 0 0
\(141\) −1141.91 −0.682027
\(142\) 0 0
\(143\) −358.221 −0.209482
\(144\) 0 0
\(145\) 2206.01 1.26344
\(146\) 0 0
\(147\) 2297.41 1.28903
\(148\) 0 0
\(149\) 815.345 0.448293 0.224147 0.974555i \(-0.428041\pi\)
0.224147 + 0.974555i \(0.428041\pi\)
\(150\) 0 0
\(151\) 2288.65 1.23343 0.616715 0.787187i \(-0.288463\pi\)
0.616715 + 0.787187i \(0.288463\pi\)
\(152\) 0 0
\(153\) −1111.60 −0.587372
\(154\) 0 0
\(155\) 337.652 0.174973
\(156\) 0 0
\(157\) 851.521 0.432858 0.216429 0.976298i \(-0.430559\pi\)
0.216429 + 0.976298i \(0.430559\pi\)
\(158\) 0 0
\(159\) 2402.28 1.19820
\(160\) 0 0
\(161\) 1269.41 0.621388
\(162\) 0 0
\(163\) −2400.93 −1.15371 −0.576857 0.816845i \(-0.695721\pi\)
−0.576857 + 0.816845i \(0.695721\pi\)
\(164\) 0 0
\(165\) −213.791 −0.100870
\(166\) 0 0
\(167\) −820.836 −0.380349 −0.190174 0.981750i \(-0.560905\pi\)
−0.190174 + 0.981750i \(0.560905\pi\)
\(168\) 0 0
\(169\) 3480.59 1.58425
\(170\) 0 0
\(171\) −904.025 −0.404284
\(172\) 0 0
\(173\) −744.786 −0.327312 −0.163656 0.986517i \(-0.552329\pi\)
−0.163656 + 0.986517i \(0.552329\pi\)
\(174\) 0 0
\(175\) 190.871 0.0824485
\(176\) 0 0
\(177\) −2603.12 −1.10544
\(178\) 0 0
\(179\) 12.4598 0.00520275 0.00260137 0.999997i \(-0.499172\pi\)
0.00260137 + 0.999997i \(0.499172\pi\)
\(180\) 0 0
\(181\) −806.271 −0.331103 −0.165552 0.986201i \(-0.552940\pi\)
−0.165552 + 0.986201i \(0.552940\pi\)
\(182\) 0 0
\(183\) 2177.78 0.879704
\(184\) 0 0
\(185\) −3647.80 −1.44968
\(186\) 0 0
\(187\) 530.917 0.207618
\(188\) 0 0
\(189\) 4575.74 1.76104
\(190\) 0 0
\(191\) −2065.90 −0.782633 −0.391317 0.920256i \(-0.627980\pi\)
−0.391317 + 0.920256i \(0.627980\pi\)
\(192\) 0 0
\(193\) −32.4222 −0.0120922 −0.00604611 0.999982i \(-0.501925\pi\)
−0.00604611 + 0.999982i \(0.501925\pi\)
\(194\) 0 0
\(195\) 3388.46 1.24437
\(196\) 0 0
\(197\) −776.177 −0.280712 −0.140356 0.990101i \(-0.544825\pi\)
−0.140356 + 0.990101i \(0.544825\pi\)
\(198\) 0 0
\(199\) 1682.12 0.599208 0.299604 0.954064i \(-0.403146\pi\)
0.299604 + 0.954064i \(0.403146\pi\)
\(200\) 0 0
\(201\) −2827.12 −0.992088
\(202\) 0 0
\(203\) 6074.19 2.10012
\(204\) 0 0
\(205\) 4000.79 1.36306
\(206\) 0 0
\(207\) 421.315 0.141466
\(208\) 0 0
\(209\) 431.775 0.142902
\(210\) 0 0
\(211\) 1826.70 0.595996 0.297998 0.954566i \(-0.403681\pi\)
0.297998 + 0.954566i \(0.403681\pi\)
\(212\) 0 0
\(213\) −951.927 −0.306220
\(214\) 0 0
\(215\) −900.637 −0.285688
\(216\) 0 0
\(217\) 929.715 0.290844
\(218\) 0 0
\(219\) 3313.92 1.02253
\(220\) 0 0
\(221\) −8414.72 −2.56125
\(222\) 0 0
\(223\) 6174.56 1.85417 0.927083 0.374856i \(-0.122308\pi\)
0.927083 + 0.374856i \(0.122308\pi\)
\(224\) 0 0
\(225\) 63.3497 0.0187703
\(226\) 0 0
\(227\) −4255.39 −1.24423 −0.622115 0.782926i \(-0.713726\pi\)
−0.622115 + 0.782926i \(0.713726\pi\)
\(228\) 0 0
\(229\) −3459.92 −0.998420 −0.499210 0.866481i \(-0.666376\pi\)
−0.499210 + 0.866481i \(0.666376\pi\)
\(230\) 0 0
\(231\) −588.668 −0.167669
\(232\) 0 0
\(233\) −2248.39 −0.632177 −0.316088 0.948730i \(-0.602370\pi\)
−0.316088 + 0.948730i \(0.602370\pi\)
\(234\) 0 0
\(235\) 3012.49 0.836226
\(236\) 0 0
\(237\) −1878.57 −0.514878
\(238\) 0 0
\(239\) −3876.19 −1.04908 −0.524540 0.851386i \(-0.675763\pi\)
−0.524540 + 0.851386i \(0.675763\pi\)
\(240\) 0 0
\(241\) 2935.28 0.784556 0.392278 0.919847i \(-0.371687\pi\)
0.392278 + 0.919847i \(0.371687\pi\)
\(242\) 0 0
\(243\) 2628.28 0.693846
\(244\) 0 0
\(245\) −6060.84 −1.58046
\(246\) 0 0
\(247\) −6843.37 −1.76289
\(248\) 0 0
\(249\) −883.561 −0.224873
\(250\) 0 0
\(251\) −3810.36 −0.958199 −0.479099 0.877761i \(-0.659037\pi\)
−0.479099 + 0.877761i \(0.659037\pi\)
\(252\) 0 0
\(253\) −201.226 −0.0500038
\(254\) 0 0
\(255\) −5022.01 −1.23330
\(256\) 0 0
\(257\) 5898.77 1.43173 0.715866 0.698238i \(-0.246032\pi\)
0.715866 + 0.698238i \(0.246032\pi\)
\(258\) 0 0
\(259\) −10044.1 −2.40969
\(260\) 0 0
\(261\) 2016.01 0.478115
\(262\) 0 0
\(263\) −4054.69 −0.950657 −0.475328 0.879808i \(-0.657671\pi\)
−0.475328 + 0.879808i \(0.657671\pi\)
\(264\) 0 0
\(265\) −6337.52 −1.46910
\(266\) 0 0
\(267\) −1403.77 −0.321757
\(268\) 0 0
\(269\) −3047.31 −0.690697 −0.345349 0.938474i \(-0.612239\pi\)
−0.345349 + 0.938474i \(0.612239\pi\)
\(270\) 0 0
\(271\) 1252.70 0.280798 0.140399 0.990095i \(-0.455162\pi\)
0.140399 + 0.990095i \(0.455162\pi\)
\(272\) 0 0
\(273\) 9330.03 2.06842
\(274\) 0 0
\(275\) −30.2567 −0.00663472
\(276\) 0 0
\(277\) −813.610 −0.176480 −0.0882402 0.996099i \(-0.528124\pi\)
−0.0882402 + 0.996099i \(0.528124\pi\)
\(278\) 0 0
\(279\) 308.571 0.0662138
\(280\) 0 0
\(281\) 5237.25 1.11184 0.555922 0.831234i \(-0.312365\pi\)
0.555922 + 0.831234i \(0.312365\pi\)
\(282\) 0 0
\(283\) 1857.01 0.390062 0.195031 0.980797i \(-0.437519\pi\)
0.195031 + 0.980797i \(0.437519\pi\)
\(284\) 0 0
\(285\) −4084.21 −0.848869
\(286\) 0 0
\(287\) 11016.0 2.26570
\(288\) 0 0
\(289\) 7558.39 1.53845
\(290\) 0 0
\(291\) −4108.52 −0.827648
\(292\) 0 0
\(293\) 1879.24 0.374698 0.187349 0.982293i \(-0.440011\pi\)
0.187349 + 0.982293i \(0.440011\pi\)
\(294\) 0 0
\(295\) 6867.34 1.35536
\(296\) 0 0
\(297\) −725.341 −0.141712
\(298\) 0 0
\(299\) 3189.30 0.616864
\(300\) 0 0
\(301\) −2479.88 −0.474876
\(302\) 0 0
\(303\) −2418.69 −0.458582
\(304\) 0 0
\(305\) −5745.24 −1.07860
\(306\) 0 0
\(307\) −10205.3 −1.89723 −0.948613 0.316439i \(-0.897513\pi\)
−0.948613 + 0.316439i \(0.897513\pi\)
\(308\) 0 0
\(309\) 2033.13 0.374307
\(310\) 0 0
\(311\) 679.582 0.123909 0.0619543 0.998079i \(-0.480267\pi\)
0.0619543 + 0.998079i \(0.480267\pi\)
\(312\) 0 0
\(313\) −506.236 −0.0914190 −0.0457095 0.998955i \(-0.514555\pi\)
−0.0457095 + 0.998955i \(0.514555\pi\)
\(314\) 0 0
\(315\) −3251.54 −0.581598
\(316\) 0 0
\(317\) 104.454 0.0185070 0.00925352 0.999957i \(-0.497054\pi\)
0.00925352 + 0.999957i \(0.497054\pi\)
\(318\) 0 0
\(319\) −962.875 −0.168999
\(320\) 0 0
\(321\) −7775.93 −1.35206
\(322\) 0 0
\(323\) 10142.5 1.74720
\(324\) 0 0
\(325\) 479.550 0.0818482
\(326\) 0 0
\(327\) −4859.32 −0.821777
\(328\) 0 0
\(329\) 8294.80 1.38999
\(330\) 0 0
\(331\) 8147.05 1.35288 0.676439 0.736499i \(-0.263522\pi\)
0.676439 + 0.736499i \(0.263522\pi\)
\(332\) 0 0
\(333\) −3333.62 −0.548592
\(334\) 0 0
\(335\) 7458.29 1.21639
\(336\) 0 0
\(337\) 1397.48 0.225893 0.112946 0.993601i \(-0.463971\pi\)
0.112946 + 0.993601i \(0.463971\pi\)
\(338\) 0 0
\(339\) 8035.77 1.28744
\(340\) 0 0
\(341\) −147.378 −0.0234045
\(342\) 0 0
\(343\) −6401.49 −1.00772
\(344\) 0 0
\(345\) 1903.42 0.297034
\(346\) 0 0
\(347\) 7205.65 1.11475 0.557377 0.830260i \(-0.311808\pi\)
0.557377 + 0.830260i \(0.311808\pi\)
\(348\) 0 0
\(349\) −10772.5 −1.65227 −0.826134 0.563474i \(-0.809464\pi\)
−0.826134 + 0.563474i \(0.809464\pi\)
\(350\) 0 0
\(351\) 11496.2 1.74821
\(352\) 0 0
\(353\) 10072.5 1.51871 0.759356 0.650675i \(-0.225514\pi\)
0.759356 + 0.650675i \(0.225514\pi\)
\(354\) 0 0
\(355\) 2511.30 0.375453
\(356\) 0 0
\(357\) −13828.0 −2.05001
\(358\) 0 0
\(359\) −9359.95 −1.37604 −0.688021 0.725691i \(-0.741520\pi\)
−0.688021 + 0.725691i \(0.741520\pi\)
\(360\) 0 0
\(361\) 1389.51 0.202582
\(362\) 0 0
\(363\) −5401.98 −0.781075
\(364\) 0 0
\(365\) −8742.53 −1.25371
\(366\) 0 0
\(367\) 12168.9 1.73083 0.865413 0.501060i \(-0.167056\pi\)
0.865413 + 0.501060i \(0.167056\pi\)
\(368\) 0 0
\(369\) 3656.21 0.515812
\(370\) 0 0
\(371\) −17450.2 −2.44196
\(372\) 0 0
\(373\) 10713.1 1.48715 0.743573 0.668655i \(-0.233130\pi\)
0.743573 + 0.668655i \(0.233130\pi\)
\(374\) 0 0
\(375\) 5907.42 0.813487
\(376\) 0 0
\(377\) 15261.0 2.08483
\(378\) 0 0
\(379\) −11273.6 −1.52793 −0.763964 0.645259i \(-0.776749\pi\)
−0.763964 + 0.645259i \(0.776749\pi\)
\(380\) 0 0
\(381\) 2011.34 0.270457
\(382\) 0 0
\(383\) 13371.9 1.78400 0.891999 0.452038i \(-0.149303\pi\)
0.891999 + 0.452038i \(0.149303\pi\)
\(384\) 0 0
\(385\) 1552.98 0.205577
\(386\) 0 0
\(387\) −823.066 −0.108111
\(388\) 0 0
\(389\) 4121.34 0.537173 0.268586 0.963256i \(-0.413444\pi\)
0.268586 + 0.963256i \(0.413444\pi\)
\(390\) 0 0
\(391\) −4726.84 −0.611373
\(392\) 0 0
\(393\) −1871.79 −0.240253
\(394\) 0 0
\(395\) 4955.89 0.631286
\(396\) 0 0
\(397\) −7854.41 −0.992951 −0.496475 0.868051i \(-0.665373\pi\)
−0.496475 + 0.868051i \(0.665373\pi\)
\(398\) 0 0
\(399\) −11245.8 −1.41101
\(400\) 0 0
\(401\) 2013.90 0.250796 0.125398 0.992106i \(-0.459979\pi\)
0.125398 + 0.992106i \(0.459979\pi\)
\(402\) 0 0
\(403\) 2335.84 0.288726
\(404\) 0 0
\(405\) 3933.81 0.482648
\(406\) 0 0
\(407\) 1592.18 0.193910
\(408\) 0 0
\(409\) −3894.04 −0.470777 −0.235389 0.971901i \(-0.575636\pi\)
−0.235389 + 0.971901i \(0.575636\pi\)
\(410\) 0 0
\(411\) −3588.92 −0.430726
\(412\) 0 0
\(413\) 18909.0 2.25291
\(414\) 0 0
\(415\) 2330.94 0.275714
\(416\) 0 0
\(417\) −1201.51 −0.141099
\(418\) 0 0
\(419\) 1355.84 0.158084 0.0790418 0.996871i \(-0.474814\pi\)
0.0790418 + 0.996871i \(0.474814\pi\)
\(420\) 0 0
\(421\) −5451.46 −0.631088 −0.315544 0.948911i \(-0.602187\pi\)
−0.315544 + 0.948911i \(0.602187\pi\)
\(422\) 0 0
\(423\) 2753.03 0.316446
\(424\) 0 0
\(425\) −710.738 −0.0811196
\(426\) 0 0
\(427\) −15819.4 −1.79286
\(428\) 0 0
\(429\) −1478.99 −0.166448
\(430\) 0 0
\(431\) −13618.0 −1.52194 −0.760970 0.648787i \(-0.775276\pi\)
−0.760970 + 0.648787i \(0.775276\pi\)
\(432\) 0 0
\(433\) 17611.2 1.95460 0.977299 0.211867i \(-0.0679543\pi\)
0.977299 + 0.211867i \(0.0679543\pi\)
\(434\) 0 0
\(435\) 9107.95 1.00389
\(436\) 0 0
\(437\) −3844.16 −0.420804
\(438\) 0 0
\(439\) 11551.7 1.25588 0.627939 0.778262i \(-0.283899\pi\)
0.627939 + 0.778262i \(0.283899\pi\)
\(440\) 0 0
\(441\) −5538.83 −0.598081
\(442\) 0 0
\(443\) −10596.2 −1.13644 −0.568219 0.822878i \(-0.692367\pi\)
−0.568219 + 0.822878i \(0.692367\pi\)
\(444\) 0 0
\(445\) 3703.30 0.394502
\(446\) 0 0
\(447\) 3366.31 0.356199
\(448\) 0 0
\(449\) −6897.92 −0.725018 −0.362509 0.931980i \(-0.618080\pi\)
−0.362509 + 0.931980i \(0.618080\pi\)
\(450\) 0 0
\(451\) −1746.25 −0.182323
\(452\) 0 0
\(453\) 9449.14 0.980043
\(454\) 0 0
\(455\) −24613.8 −2.53607
\(456\) 0 0
\(457\) 11429.2 1.16989 0.584943 0.811075i \(-0.301117\pi\)
0.584943 + 0.811075i \(0.301117\pi\)
\(458\) 0 0
\(459\) −17038.5 −1.73265
\(460\) 0 0
\(461\) −7635.92 −0.771454 −0.385727 0.922613i \(-0.626049\pi\)
−0.385727 + 0.922613i \(0.626049\pi\)
\(462\) 0 0
\(463\) −8531.66 −0.856372 −0.428186 0.903691i \(-0.640847\pi\)
−0.428186 + 0.903691i \(0.640847\pi\)
\(464\) 0 0
\(465\) 1394.06 0.139028
\(466\) 0 0
\(467\) −8529.80 −0.845208 −0.422604 0.906314i \(-0.638884\pi\)
−0.422604 + 0.906314i \(0.638884\pi\)
\(468\) 0 0
\(469\) 20536.2 2.02190
\(470\) 0 0
\(471\) 3515.67 0.343935
\(472\) 0 0
\(473\) 393.108 0.0382138
\(474\) 0 0
\(475\) −578.016 −0.0558341
\(476\) 0 0
\(477\) −5791.68 −0.555938
\(478\) 0 0
\(479\) −2957.05 −0.282069 −0.141034 0.990005i \(-0.545043\pi\)
−0.141034 + 0.990005i \(0.545043\pi\)
\(480\) 0 0
\(481\) −25235.1 −2.39214
\(482\) 0 0
\(483\) 5241.01 0.493735
\(484\) 0 0
\(485\) 10838.8 1.01477
\(486\) 0 0
\(487\) −13582.8 −1.26385 −0.631927 0.775028i \(-0.717736\pi\)
−0.631927 + 0.775028i \(0.717736\pi\)
\(488\) 0 0
\(489\) −9912.71 −0.916704
\(490\) 0 0
\(491\) −9271.49 −0.852172 −0.426086 0.904683i \(-0.640108\pi\)
−0.426086 + 0.904683i \(0.640108\pi\)
\(492\) 0 0
\(493\) −22618.2 −2.06627
\(494\) 0 0
\(495\) 515.430 0.0468018
\(496\) 0 0
\(497\) 6914.79 0.624086
\(498\) 0 0
\(499\) 17511.3 1.57097 0.785483 0.618883i \(-0.212414\pi\)
0.785483 + 0.618883i \(0.212414\pi\)
\(500\) 0 0
\(501\) −3388.98 −0.302213
\(502\) 0 0
\(503\) 10035.8 0.889607 0.444804 0.895628i \(-0.353273\pi\)
0.444804 + 0.895628i \(0.353273\pi\)
\(504\) 0 0
\(505\) 6380.81 0.562262
\(506\) 0 0
\(507\) 14370.3 1.25879
\(508\) 0 0
\(509\) 5524.96 0.481119 0.240559 0.970634i \(-0.422669\pi\)
0.240559 + 0.970634i \(0.422669\pi\)
\(510\) 0 0
\(511\) −24072.3 −2.08394
\(512\) 0 0
\(513\) −13856.7 −1.19257
\(514\) 0 0
\(515\) −5363.66 −0.458934
\(516\) 0 0
\(517\) −1314.88 −0.111854
\(518\) 0 0
\(519\) −3074.99 −0.260072
\(520\) 0 0
\(521\) 9697.79 0.815485 0.407743 0.913097i \(-0.366316\pi\)
0.407743 + 0.913097i \(0.366316\pi\)
\(522\) 0 0
\(523\) 1395.38 0.116665 0.0583323 0.998297i \(-0.481422\pi\)
0.0583323 + 0.998297i \(0.481422\pi\)
\(524\) 0 0
\(525\) 788.048 0.0655109
\(526\) 0 0
\(527\) −3461.94 −0.286156
\(528\) 0 0
\(529\) −10375.5 −0.852754
\(530\) 0 0
\(531\) 6275.87 0.512899
\(532\) 0 0
\(533\) 27677.1 2.24920
\(534\) 0 0
\(535\) 20513.9 1.65774
\(536\) 0 0
\(537\) 51.4428 0.00413393
\(538\) 0 0
\(539\) 2645.42 0.211403
\(540\) 0 0
\(541\) −5935.90 −0.471727 −0.235863 0.971786i \(-0.575792\pi\)
−0.235863 + 0.971786i \(0.575792\pi\)
\(542\) 0 0
\(543\) −3328.85 −0.263084
\(544\) 0 0
\(545\) 12819.5 1.00757
\(546\) 0 0
\(547\) 2930.48 0.229065 0.114532 0.993420i \(-0.463463\pi\)
0.114532 + 0.993420i \(0.463463\pi\)
\(548\) 0 0
\(549\) −5250.41 −0.408164
\(550\) 0 0
\(551\) −18394.5 −1.42220
\(552\) 0 0
\(553\) 13645.9 1.04934
\(554\) 0 0
\(555\) −15060.6 −1.15187
\(556\) 0 0
\(557\) 12687.7 0.965162 0.482581 0.875851i \(-0.339699\pi\)
0.482581 + 0.875851i \(0.339699\pi\)
\(558\) 0 0
\(559\) −6230.52 −0.471418
\(560\) 0 0
\(561\) 2192.00 0.164966
\(562\) 0 0
\(563\) 23865.7 1.78654 0.893269 0.449522i \(-0.148406\pi\)
0.893269 + 0.449522i \(0.148406\pi\)
\(564\) 0 0
\(565\) −21199.4 −1.57852
\(566\) 0 0
\(567\) 10831.6 0.802268
\(568\) 0 0
\(569\) −12451.1 −0.917356 −0.458678 0.888603i \(-0.651677\pi\)
−0.458678 + 0.888603i \(0.651677\pi\)
\(570\) 0 0
\(571\) −8498.58 −0.622863 −0.311431 0.950269i \(-0.600808\pi\)
−0.311431 + 0.950269i \(0.600808\pi\)
\(572\) 0 0
\(573\) −8529.45 −0.621855
\(574\) 0 0
\(575\) 269.380 0.0195373
\(576\) 0 0
\(577\) 14527.9 1.04819 0.524094 0.851661i \(-0.324404\pi\)
0.524094 + 0.851661i \(0.324404\pi\)
\(578\) 0 0
\(579\) −133.861 −0.00960808
\(580\) 0 0
\(581\) 6418.18 0.458298
\(582\) 0 0
\(583\) 2766.18 0.196507
\(584\) 0 0
\(585\) −8169.26 −0.577363
\(586\) 0 0
\(587\) 24367.2 1.71336 0.856681 0.515847i \(-0.172523\pi\)
0.856681 + 0.515847i \(0.172523\pi\)
\(588\) 0 0
\(589\) −2815.46 −0.196959
\(590\) 0 0
\(591\) −3204.60 −0.223045
\(592\) 0 0
\(593\) −27008.9 −1.87036 −0.935179 0.354177i \(-0.884761\pi\)
−0.935179 + 0.354177i \(0.884761\pi\)
\(594\) 0 0
\(595\) 36479.9 2.51349
\(596\) 0 0
\(597\) 6944.96 0.476111
\(598\) 0 0
\(599\) 20539.1 1.40101 0.700504 0.713648i \(-0.252958\pi\)
0.700504 + 0.713648i \(0.252958\pi\)
\(600\) 0 0
\(601\) −9748.76 −0.661664 −0.330832 0.943690i \(-0.607329\pi\)
−0.330832 + 0.943690i \(0.607329\pi\)
\(602\) 0 0
\(603\) 6815.92 0.460308
\(604\) 0 0
\(605\) 14251.1 0.957667
\(606\) 0 0
\(607\) −15825.2 −1.05819 −0.529097 0.848561i \(-0.677469\pi\)
−0.529097 + 0.848561i \(0.677469\pi\)
\(608\) 0 0
\(609\) 25078.5 1.66869
\(610\) 0 0
\(611\) 20840.1 1.37987
\(612\) 0 0
\(613\) −8498.46 −0.559950 −0.279975 0.960007i \(-0.590326\pi\)
−0.279975 + 0.960007i \(0.590326\pi\)
\(614\) 0 0
\(615\) 16518.0 1.08304
\(616\) 0 0
\(617\) 6035.14 0.393785 0.196893 0.980425i \(-0.436915\pi\)
0.196893 + 0.980425i \(0.436915\pi\)
\(618\) 0 0
\(619\) −12286.8 −0.797815 −0.398908 0.916991i \(-0.630611\pi\)
−0.398908 + 0.916991i \(0.630611\pi\)
\(620\) 0 0
\(621\) 6457.83 0.417301
\(622\) 0 0
\(623\) 10196.9 0.655749
\(624\) 0 0
\(625\) −14789.0 −0.946493
\(626\) 0 0
\(627\) 1782.67 0.113545
\(628\) 0 0
\(629\) 37400.7 2.37085
\(630\) 0 0
\(631\) 4932.61 0.311195 0.155597 0.987821i \(-0.450270\pi\)
0.155597 + 0.987821i \(0.450270\pi\)
\(632\) 0 0
\(633\) 7541.88 0.473559
\(634\) 0 0
\(635\) −5306.17 −0.331604
\(636\) 0 0
\(637\) −41928.3 −2.60794
\(638\) 0 0
\(639\) 2295.01 0.142080
\(640\) 0 0
\(641\) −10027.0 −0.617852 −0.308926 0.951086i \(-0.599970\pi\)
−0.308926 + 0.951086i \(0.599970\pi\)
\(642\) 0 0
\(643\) 12963.5 0.795071 0.397535 0.917587i \(-0.369866\pi\)
0.397535 + 0.917587i \(0.369866\pi\)
\(644\) 0 0
\(645\) −3718.45 −0.226998
\(646\) 0 0
\(647\) −30692.4 −1.86498 −0.932490 0.361196i \(-0.882369\pi\)
−0.932490 + 0.361196i \(0.882369\pi\)
\(648\) 0 0
\(649\) −2997.44 −0.181294
\(650\) 0 0
\(651\) 3838.51 0.231095
\(652\) 0 0
\(653\) −23445.6 −1.40505 −0.702524 0.711660i \(-0.747944\pi\)
−0.702524 + 0.711660i \(0.747944\pi\)
\(654\) 0 0
\(655\) 4938.02 0.294572
\(656\) 0 0
\(657\) −7989.55 −0.474432
\(658\) 0 0
\(659\) −6039.31 −0.356993 −0.178496 0.983941i \(-0.557123\pi\)
−0.178496 + 0.983941i \(0.557123\pi\)
\(660\) 0 0
\(661\) 6549.05 0.385369 0.192684 0.981261i \(-0.438281\pi\)
0.192684 + 0.981261i \(0.438281\pi\)
\(662\) 0 0
\(663\) −34741.8 −2.03508
\(664\) 0 0
\(665\) 29667.7 1.73002
\(666\) 0 0
\(667\) 8572.64 0.497652
\(668\) 0 0
\(669\) 25492.9 1.47326
\(670\) 0 0
\(671\) 2507.67 0.144273
\(672\) 0 0
\(673\) 1144.93 0.0655778 0.0327889 0.999462i \(-0.489561\pi\)
0.0327889 + 0.999462i \(0.489561\pi\)
\(674\) 0 0
\(675\) 971.012 0.0553693
\(676\) 0 0
\(677\) −19501.4 −1.10709 −0.553544 0.832820i \(-0.686725\pi\)
−0.553544 + 0.832820i \(0.686725\pi\)
\(678\) 0 0
\(679\) 29844.2 1.68677
\(680\) 0 0
\(681\) −17569.2 −0.988624
\(682\) 0 0
\(683\) −10787.3 −0.604338 −0.302169 0.953254i \(-0.597711\pi\)
−0.302169 + 0.953254i \(0.597711\pi\)
\(684\) 0 0
\(685\) 9468.01 0.528108
\(686\) 0 0
\(687\) −14285.0 −0.793312
\(688\) 0 0
\(689\) −43842.3 −2.42418
\(690\) 0 0
\(691\) −16050.0 −0.883603 −0.441801 0.897113i \(-0.645660\pi\)
−0.441801 + 0.897113i \(0.645660\pi\)
\(692\) 0 0
\(693\) 1419.22 0.0777948
\(694\) 0 0
\(695\) 3169.74 0.173000
\(696\) 0 0
\(697\) −41020.0 −2.22918
\(698\) 0 0
\(699\) −9282.93 −0.502307
\(700\) 0 0
\(701\) 8234.20 0.443654 0.221827 0.975086i \(-0.428798\pi\)
0.221827 + 0.975086i \(0.428798\pi\)
\(702\) 0 0
\(703\) 30416.6 1.63184
\(704\) 0 0
\(705\) 12437.6 0.664438
\(706\) 0 0
\(707\) 17569.4 0.934603
\(708\) 0 0
\(709\) −18749.0 −0.993136 −0.496568 0.867998i \(-0.665407\pi\)
−0.496568 + 0.867998i \(0.665407\pi\)
\(710\) 0 0
\(711\) 4529.05 0.238892
\(712\) 0 0
\(713\) 1312.13 0.0689194
\(714\) 0 0
\(715\) 3901.75 0.204080
\(716\) 0 0
\(717\) −16003.6 −0.833564
\(718\) 0 0
\(719\) 2057.31 0.106710 0.0533551 0.998576i \(-0.483008\pi\)
0.0533551 + 0.998576i \(0.483008\pi\)
\(720\) 0 0
\(721\) −14768.7 −0.762849
\(722\) 0 0
\(723\) 12118.9 0.623382
\(724\) 0 0
\(725\) 1289.00 0.0660306
\(726\) 0 0
\(727\) 5762.61 0.293980 0.146990 0.989138i \(-0.453042\pi\)
0.146990 + 0.989138i \(0.453042\pi\)
\(728\) 0 0
\(729\) 20602.8 1.04673
\(730\) 0 0
\(731\) 9234.20 0.467222
\(732\) 0 0
\(733\) 28273.9 1.42472 0.712361 0.701813i \(-0.247626\pi\)
0.712361 + 0.701813i \(0.247626\pi\)
\(734\) 0 0
\(735\) −25023.3 −1.25578
\(736\) 0 0
\(737\) −3255.37 −0.162705
\(738\) 0 0
\(739\) −33535.1 −1.66929 −0.834647 0.550785i \(-0.814328\pi\)
−0.834647 + 0.550785i \(0.814328\pi\)
\(740\) 0 0
\(741\) −28254.2 −1.40073
\(742\) 0 0
\(743\) −28870.0 −1.42549 −0.712743 0.701425i \(-0.752547\pi\)
−0.712743 + 0.701425i \(0.752547\pi\)
\(744\) 0 0
\(745\) −8880.74 −0.436732
\(746\) 0 0
\(747\) 2130.18 0.104336
\(748\) 0 0
\(749\) 56484.3 2.75553
\(750\) 0 0
\(751\) 33907.3 1.64753 0.823765 0.566931i \(-0.191869\pi\)
0.823765 + 0.566931i \(0.191869\pi\)
\(752\) 0 0
\(753\) −15731.8 −0.761354
\(754\) 0 0
\(755\) −24928.0 −1.20162
\(756\) 0 0
\(757\) 23317.4 1.11953 0.559765 0.828651i \(-0.310891\pi\)
0.559765 + 0.828651i \(0.310891\pi\)
\(758\) 0 0
\(759\) −830.799 −0.0397314
\(760\) 0 0
\(761\) 10183.9 0.485109 0.242554 0.970138i \(-0.422015\pi\)
0.242554 + 0.970138i \(0.422015\pi\)
\(762\) 0 0
\(763\) 35298.1 1.67481
\(764\) 0 0
\(765\) 12107.6 0.572224
\(766\) 0 0
\(767\) 47507.6 2.23650
\(768\) 0 0
\(769\) 17775.1 0.833533 0.416766 0.909014i \(-0.363163\pi\)
0.416766 + 0.909014i \(0.363163\pi\)
\(770\) 0 0
\(771\) 24354.2 1.13761
\(772\) 0 0
\(773\) −42778.7 −1.99048 −0.995241 0.0974480i \(-0.968932\pi\)
−0.995241 + 0.0974480i \(0.968932\pi\)
\(774\) 0 0
\(775\) 197.294 0.00914452
\(776\) 0 0
\(777\) −41469.0 −1.91466
\(778\) 0 0
\(779\) −33360.0 −1.53433
\(780\) 0 0
\(781\) −1096.13 −0.0502208
\(782\) 0 0
\(783\) 30901.1 1.41036
\(784\) 0 0
\(785\) −9274.77 −0.421695
\(786\) 0 0
\(787\) −2161.84 −0.0979178 −0.0489589 0.998801i \(-0.515590\pi\)
−0.0489589 + 0.998801i \(0.515590\pi\)
\(788\) 0 0
\(789\) −16740.6 −0.755361
\(790\) 0 0
\(791\) −58371.8 −2.62385
\(792\) 0 0
\(793\) −39745.0 −1.77981
\(794\) 0 0
\(795\) −26165.7 −1.16730
\(796\) 0 0
\(797\) −28610.2 −1.27155 −0.635775 0.771874i \(-0.719320\pi\)
−0.635775 + 0.771874i \(0.719320\pi\)
\(798\) 0 0
\(799\) −30887.0 −1.36759
\(800\) 0 0
\(801\) 3384.35 0.149288
\(802\) 0 0
\(803\) 3815.92 0.167697
\(804\) 0 0
\(805\) −13826.4 −0.605363
\(806\) 0 0
\(807\) −12581.4 −0.548805
\(808\) 0 0
\(809\) −10133.1 −0.440371 −0.220185 0.975458i \(-0.570666\pi\)
−0.220185 + 0.975458i \(0.570666\pi\)
\(810\) 0 0
\(811\) −24819.3 −1.07463 −0.537315 0.843382i \(-0.680561\pi\)
−0.537315 + 0.843382i \(0.680561\pi\)
\(812\) 0 0
\(813\) 5172.02 0.223113
\(814\) 0 0
\(815\) 26150.9 1.12396
\(816\) 0 0
\(817\) 7509.82 0.321586
\(818\) 0 0
\(819\) −22493.8 −0.959704
\(820\) 0 0
\(821\) 11489.8 0.488426 0.244213 0.969722i \(-0.421470\pi\)
0.244213 + 0.969722i \(0.421470\pi\)
\(822\) 0 0
\(823\) −13011.0 −0.551077 −0.275539 0.961290i \(-0.588856\pi\)
−0.275539 + 0.961290i \(0.588856\pi\)
\(824\) 0 0
\(825\) −124.921 −0.00527173
\(826\) 0 0
\(827\) −21913.5 −0.921411 −0.460706 0.887553i \(-0.652404\pi\)
−0.460706 + 0.887553i \(0.652404\pi\)
\(828\) 0 0
\(829\) 32815.2 1.37481 0.687407 0.726273i \(-0.258749\pi\)
0.687407 + 0.726273i \(0.258749\pi\)
\(830\) 0 0
\(831\) −3359.15 −0.140226
\(832\) 0 0
\(833\) 62141.6 2.58473
\(834\) 0 0
\(835\) 8940.55 0.370539
\(836\) 0 0
\(837\) 4729.71 0.195320
\(838\) 0 0
\(839\) 8028.93 0.330381 0.165190 0.986262i \(-0.447176\pi\)
0.165190 + 0.986262i \(0.447176\pi\)
\(840\) 0 0
\(841\) 16631.5 0.681927
\(842\) 0 0
\(843\) 21623.0 0.883436
\(844\) 0 0
\(845\) −37910.6 −1.54339
\(846\) 0 0
\(847\) 39239.9 1.59185
\(848\) 0 0
\(849\) 7667.02 0.309931
\(850\) 0 0
\(851\) −14175.4 −0.571008
\(852\) 0 0
\(853\) 23398.3 0.939204 0.469602 0.882878i \(-0.344397\pi\)
0.469602 + 0.882878i \(0.344397\pi\)
\(854\) 0 0
\(855\) 9846.64 0.393857
\(856\) 0 0
\(857\) −30105.8 −1.19999 −0.599996 0.800003i \(-0.704831\pi\)
−0.599996 + 0.800003i \(0.704831\pi\)
\(858\) 0 0
\(859\) −18957.5 −0.752995 −0.376497 0.926418i \(-0.622872\pi\)
−0.376497 + 0.926418i \(0.622872\pi\)
\(860\) 0 0
\(861\) 45481.9 1.80025
\(862\) 0 0
\(863\) −22648.6 −0.893358 −0.446679 0.894694i \(-0.647393\pi\)
−0.446679 + 0.894694i \(0.647393\pi\)
\(864\) 0 0
\(865\) 8112.21 0.318871
\(866\) 0 0
\(867\) 31206.3 1.22240
\(868\) 0 0
\(869\) −2163.13 −0.0844411
\(870\) 0 0
\(871\) 51595.7 2.00718
\(872\) 0 0
\(873\) 9905.25 0.384011
\(874\) 0 0
\(875\) −42911.5 −1.65791
\(876\) 0 0
\(877\) −2662.53 −0.102517 −0.0512585 0.998685i \(-0.516323\pi\)
−0.0512585 + 0.998685i \(0.516323\pi\)
\(878\) 0 0
\(879\) 7758.81 0.297723
\(880\) 0 0
\(881\) 23475.1 0.897726 0.448863 0.893601i \(-0.351829\pi\)
0.448863 + 0.893601i \(0.351829\pi\)
\(882\) 0 0
\(883\) −4188.29 −0.159623 −0.0798115 0.996810i \(-0.525432\pi\)
−0.0798115 + 0.996810i \(0.525432\pi\)
\(884\) 0 0
\(885\) 28353.1 1.07693
\(886\) 0 0
\(887\) 27258.1 1.03183 0.515917 0.856638i \(-0.327451\pi\)
0.515917 + 0.856638i \(0.327451\pi\)
\(888\) 0 0
\(889\) −14610.4 −0.551199
\(890\) 0 0
\(891\) −1717.02 −0.0645593
\(892\) 0 0
\(893\) −25119.2 −0.941301
\(894\) 0 0
\(895\) −135.712 −0.00506857
\(896\) 0 0
\(897\) 13167.7 0.490140
\(898\) 0 0
\(899\) 6278.59 0.232929
\(900\) 0 0
\(901\) 64978.4 2.40260
\(902\) 0 0
\(903\) −10238.6 −0.377321
\(904\) 0 0
\(905\) 8781.91 0.322564
\(906\) 0 0
\(907\) 15800.6 0.578447 0.289223 0.957262i \(-0.406603\pi\)
0.289223 + 0.957262i \(0.406603\pi\)
\(908\) 0 0
\(909\) 5831.24 0.212772
\(910\) 0 0
\(911\) 8688.47 0.315985 0.157992 0.987440i \(-0.449498\pi\)
0.157992 + 0.987440i \(0.449498\pi\)
\(912\) 0 0
\(913\) −1017.40 −0.0368797
\(914\) 0 0
\(915\) −23720.3 −0.857017
\(916\) 0 0
\(917\) 13596.7 0.489643
\(918\) 0 0
\(919\) 4601.95 0.165184 0.0825922 0.996583i \(-0.473680\pi\)
0.0825922 + 0.996583i \(0.473680\pi\)
\(920\) 0 0
\(921\) −42134.6 −1.50747
\(922\) 0 0
\(923\) 17372.9 0.619542
\(924\) 0 0
\(925\) −2131.45 −0.0757639
\(926\) 0 0
\(927\) −4901.69 −0.173671
\(928\) 0 0
\(929\) −27866.0 −0.984126 −0.492063 0.870560i \(-0.663757\pi\)
−0.492063 + 0.870560i \(0.663757\pi\)
\(930\) 0 0
\(931\) 50537.4 1.77905
\(932\) 0 0
\(933\) 2805.79 0.0984537
\(934\) 0 0
\(935\) −5782.75 −0.202263
\(936\) 0 0
\(937\) 38874.6 1.35536 0.677682 0.735355i \(-0.262985\pi\)
0.677682 + 0.735355i \(0.262985\pi\)
\(938\) 0 0
\(939\) −2090.09 −0.0726386
\(940\) 0 0
\(941\) 13340.3 0.462147 0.231074 0.972936i \(-0.425776\pi\)
0.231074 + 0.972936i \(0.425776\pi\)
\(942\) 0 0
\(943\) 15547.2 0.536888
\(944\) 0 0
\(945\) −49838.9 −1.71562
\(946\) 0 0
\(947\) −13866.9 −0.475833 −0.237917 0.971286i \(-0.576464\pi\)
−0.237917 + 0.971286i \(0.576464\pi\)
\(948\) 0 0
\(949\) −60480.0 −2.06877
\(950\) 0 0
\(951\) 431.259 0.0147051
\(952\) 0 0
\(953\) −21776.9 −0.740213 −0.370107 0.928989i \(-0.620679\pi\)
−0.370107 + 0.928989i \(0.620679\pi\)
\(954\) 0 0
\(955\) 22501.7 0.762449
\(956\) 0 0
\(957\) −3975.42 −0.134281
\(958\) 0 0
\(959\) 26069.9 0.877832
\(960\) 0 0
\(961\) 961.000 0.0322581
\(962\) 0 0
\(963\) 18747.0 0.627326
\(964\) 0 0
\(965\) 353.142 0.0117804
\(966\) 0 0
\(967\) −16969.3 −0.564319 −0.282160 0.959368i \(-0.591051\pi\)
−0.282160 + 0.959368i \(0.591051\pi\)
\(968\) 0 0
\(969\) 41875.3 1.38826
\(970\) 0 0
\(971\) −15787.3 −0.521768 −0.260884 0.965370i \(-0.584014\pi\)
−0.260884 + 0.965370i \(0.584014\pi\)
\(972\) 0 0
\(973\) 8727.79 0.287564
\(974\) 0 0
\(975\) 1979.92 0.0650339
\(976\) 0 0
\(977\) −33954.1 −1.11186 −0.555930 0.831229i \(-0.687638\pi\)
−0.555930 + 0.831229i \(0.687638\pi\)
\(978\) 0 0
\(979\) −1616.41 −0.0527688
\(980\) 0 0
\(981\) 11715.4 0.381287
\(982\) 0 0
\(983\) 48246.8 1.56545 0.782723 0.622370i \(-0.213830\pi\)
0.782723 + 0.622370i \(0.213830\pi\)
\(984\) 0 0
\(985\) 8454.13 0.273473
\(986\) 0 0
\(987\) 34246.7 1.10444
\(988\) 0 0
\(989\) −3499.90 −0.112528
\(990\) 0 0
\(991\) −5908.32 −0.189388 −0.0946942 0.995506i \(-0.530187\pi\)
−0.0946942 + 0.995506i \(0.530187\pi\)
\(992\) 0 0
\(993\) 33636.7 1.07495
\(994\) 0 0
\(995\) −18321.6 −0.583754
\(996\) 0 0
\(997\) −20202.1 −0.641732 −0.320866 0.947125i \(-0.603974\pi\)
−0.320866 + 0.947125i \(0.603974\pi\)
\(998\) 0 0
\(999\) −51097.0 −1.61826
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 124.4.a.a.1.4 4
3.2 odd 2 1116.4.a.g.1.3 4
4.3 odd 2 496.4.a.h.1.1 4
8.3 odd 2 1984.4.a.k.1.4 4
8.5 even 2 1984.4.a.p.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
124.4.a.a.1.4 4 1.1 even 1 trivial
496.4.a.h.1.1 4 4.3 odd 2
1116.4.a.g.1.3 4 3.2 odd 2
1984.4.a.k.1.4 4 8.3 odd 2
1984.4.a.p.1.1 4 8.5 even 2