Properties

Label 9680.2.a.cr.1.4
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
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] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9680, 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("9680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.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: \(77.2951891566\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4752.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4840)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.49551\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.82684 q^{3} -1.00000 q^{5} +2.09479 q^{7} +4.99102 q^{9} +O(q^{10})\) \(q+2.82684 q^{3} -1.00000 q^{5} +2.09479 q^{7} +4.99102 q^{9} -4.99102 q^{13} -2.82684 q^{15} +2.80144 q^{19} +5.92163 q^{21} -8.45512 q^{23} +1.00000 q^{25} +5.62828 q^{27} -4.00000 q^{29} -9.18059 q^{31} -2.09479 q^{35} -9.98203 q^{37} -14.1088 q^{39} +2.53349 q^{41} +0.233566 q^{43} -4.99102 q^{45} -9.15519 q^{47} -2.61186 q^{49} +1.33734 q^{53} +7.91922 q^{57} -12.1088 q^{59} +1.12019 q^{61} +10.4551 q^{63} +4.99102 q^{65} +1.50152 q^{67} -23.9012 q^{69} +7.72964 q^{71} -8.00000 q^{73} +2.82684 q^{75} -10.1716 q^{79} +0.937188 q^{81} -13.7625 q^{83} -11.3074 q^{87} +8.60288 q^{89} -10.4551 q^{91} -25.9520 q^{93} -2.80144 q^{95} -3.04181 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 4 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 4 q^{5} + 6 q^{7} + 4 q^{9} - 4 q^{13} - 2 q^{15} + 12 q^{21} - 4 q^{23} + 4 q^{25} + 2 q^{27} - 16 q^{29} - 16 q^{31} - 6 q^{35} - 8 q^{37} - 8 q^{39} - 8 q^{41} - 10 q^{43} - 4 q^{45} - 14 q^{47} - 4 q^{49} + 8 q^{53} - 12 q^{57} + 4 q^{61} + 12 q^{63} + 4 q^{65} + 2 q^{67} - 20 q^{69} - 8 q^{71} - 32 q^{73} + 2 q^{75} - 4 q^{79} - 8 q^{81} + 12 q^{83} - 8 q^{87} + 12 q^{89} - 12 q^{91} - 32 q^{93}+O(q^{100}) \) 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
\(3\) 2.82684 1.63208 0.816038 0.577998i \(-0.196166\pi\)
0.816038 + 0.577998i \(0.196166\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.09479 0.791755 0.395878 0.918303i \(-0.370440\pi\)
0.395878 + 0.918303i \(0.370440\pi\)
\(8\) 0 0
\(9\) 4.99102 1.66367
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −4.99102 −1.38426 −0.692129 0.721774i \(-0.743327\pi\)
−0.692129 + 0.721774i \(0.743327\pi\)
\(14\) 0 0
\(15\) −2.82684 −0.729887
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 2.80144 0.642694 0.321347 0.946961i \(-0.395864\pi\)
0.321347 + 0.946961i \(0.395864\pi\)
\(20\) 0 0
\(21\) 5.92163 1.29220
\(22\) 0 0
\(23\) −8.45512 −1.76301 −0.881507 0.472171i \(-0.843470\pi\)
−0.881507 + 0.472171i \(0.843470\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.62828 1.08316
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −9.18059 −1.64888 −0.824441 0.565947i \(-0.808511\pi\)
−0.824441 + 0.565947i \(0.808511\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.09479 −0.354084
\(36\) 0 0
\(37\) −9.98203 −1.64104 −0.820518 0.571621i \(-0.806315\pi\)
−0.820518 + 0.571621i \(0.806315\pi\)
\(38\) 0 0
\(39\) −14.1088 −2.25922
\(40\) 0 0
\(41\) 2.53349 0.395665 0.197832 0.980236i \(-0.436610\pi\)
0.197832 + 0.980236i \(0.436610\pi\)
\(42\) 0 0
\(43\) 0.233566 0.0356185 0.0178093 0.999841i \(-0.494331\pi\)
0.0178093 + 0.999841i \(0.494331\pi\)
\(44\) 0 0
\(45\) −4.99102 −0.744017
\(46\) 0 0
\(47\) −9.15519 −1.33542 −0.667711 0.744421i \(-0.732726\pi\)
−0.667711 + 0.744421i \(0.732726\pi\)
\(48\) 0 0
\(49\) −2.61186 −0.373124
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.33734 0.183698 0.0918488 0.995773i \(-0.470722\pi\)
0.0918488 + 0.995773i \(0.470722\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 7.91922 1.04893
\(58\) 0 0
\(59\) −12.1088 −1.57643 −0.788215 0.615400i \(-0.788995\pi\)
−0.788215 + 0.615400i \(0.788995\pi\)
\(60\) 0 0
\(61\) 1.12019 0.143425 0.0717126 0.997425i \(-0.477154\pi\)
0.0717126 + 0.997425i \(0.477154\pi\)
\(62\) 0 0
\(63\) 10.4551 1.31722
\(64\) 0 0
\(65\) 4.99102 0.619059
\(66\) 0 0
\(67\) 1.50152 0.183439 0.0917197 0.995785i \(-0.470764\pi\)
0.0917197 + 0.995785i \(0.470764\pi\)
\(68\) 0 0
\(69\) −23.9012 −2.87737
\(70\) 0 0
\(71\) 7.72964 0.917340 0.458670 0.888607i \(-0.348326\pi\)
0.458670 + 0.888607i \(0.348326\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 0 0
\(75\) 2.82684 0.326415
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −10.1716 −1.14440 −0.572198 0.820116i \(-0.693909\pi\)
−0.572198 + 0.820116i \(0.693909\pi\)
\(80\) 0 0
\(81\) 0.937188 0.104132
\(82\) 0 0
\(83\) −13.7625 −1.51063 −0.755314 0.655363i \(-0.772515\pi\)
−0.755314 + 0.655363i \(0.772515\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −11.3074 −1.21228
\(88\) 0 0
\(89\) 8.60288 0.911903 0.455952 0.890004i \(-0.349299\pi\)
0.455952 + 0.890004i \(0.349299\pi\)
\(90\) 0 0
\(91\) −10.4551 −1.09599
\(92\) 0 0
\(93\) −25.9520 −2.69110
\(94\) 0 0
\(95\) −2.80144 −0.287422
\(96\) 0 0
\(97\) −3.04181 −0.308849 −0.154425 0.988005i \(-0.549352\pi\)
−0.154425 + 0.988005i \(0.549352\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.38573 0.933915 0.466957 0.884280i \(-0.345350\pi\)
0.466957 + 0.884280i \(0.345350\pi\)
\(102\) 0 0
\(103\) 7.50894 0.739878 0.369939 0.929056i \(-0.379379\pi\)
0.369939 + 0.929056i \(0.379379\pi\)
\(104\) 0 0
\(105\) −5.92163 −0.577892
\(106\) 0 0
\(107\) 18.2216 1.76154 0.880772 0.473541i \(-0.157024\pi\)
0.880772 + 0.473541i \(0.157024\pi\)
\(108\) 0 0
\(109\) 20.7051 1.98319 0.991594 0.129386i \(-0.0413006\pi\)
0.991594 + 0.129386i \(0.0413006\pi\)
\(110\) 0 0
\(111\) −28.2176 −2.67829
\(112\) 0 0
\(113\) 15.2894 1.43830 0.719152 0.694853i \(-0.244530\pi\)
0.719152 + 0.694853i \(0.244530\pi\)
\(114\) 0 0
\(115\) 8.45512 0.788444
\(116\) 0 0
\(117\) −24.9102 −2.30295
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 7.16177 0.645755
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 19.8573 1.76205 0.881023 0.473073i \(-0.156855\pi\)
0.881023 + 0.473073i \(0.156855\pi\)
\(128\) 0 0
\(129\) 0.660254 0.0581321
\(130\) 0 0
\(131\) 4.25239 0.371533 0.185766 0.982594i \(-0.440523\pi\)
0.185766 + 0.982594i \(0.440523\pi\)
\(132\) 0 0
\(133\) 5.86842 0.508857
\(134\) 0 0
\(135\) −5.62828 −0.484405
\(136\) 0 0
\(137\) −8.26554 −0.706173 −0.353086 0.935591i \(-0.614868\pi\)
−0.353086 + 0.935591i \(0.614868\pi\)
\(138\) 0 0
\(139\) −18.9551 −1.60775 −0.803874 0.594799i \(-0.797232\pi\)
−0.803874 + 0.594799i \(0.797232\pi\)
\(140\) 0 0
\(141\) −25.8802 −2.17951
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 0 0
\(147\) −7.38332 −0.608966
\(148\) 0 0
\(149\) −12.4827 −1.02262 −0.511311 0.859396i \(-0.670840\pi\)
−0.511311 + 0.859396i \(0.670840\pi\)
\(150\) 0 0
\(151\) −3.37017 −0.274260 −0.137130 0.990553i \(-0.543788\pi\)
−0.137130 + 0.990553i \(0.543788\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 9.18059 0.737403
\(156\) 0 0
\(157\) 8.65671 0.690880 0.345440 0.938441i \(-0.387730\pi\)
0.345440 + 0.938441i \(0.387730\pi\)
\(158\) 0 0
\(159\) 3.78044 0.299808
\(160\) 0 0
\(161\) −17.7117 −1.39588
\(162\) 0 0
\(163\) 6.70008 0.524790 0.262395 0.964960i \(-0.415488\pi\)
0.262395 + 0.964960i \(0.415488\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.83342 0.683550 0.341775 0.939782i \(-0.388972\pi\)
0.341775 + 0.939782i \(0.388972\pi\)
\(168\) 0 0
\(169\) 11.9102 0.916172
\(170\) 0 0
\(171\) 13.9820 1.06923
\(172\) 0 0
\(173\) 21.2086 1.61246 0.806230 0.591602i \(-0.201504\pi\)
0.806230 + 0.591602i \(0.201504\pi\)
\(174\) 0 0
\(175\) 2.09479 0.158351
\(176\) 0 0
\(177\) −34.2296 −2.57285
\(178\) 0 0
\(179\) 3.25656 0.243407 0.121703 0.992567i \(-0.461164\pi\)
0.121703 + 0.992567i \(0.461164\pi\)
\(180\) 0 0
\(181\) −18.1309 −1.34766 −0.673831 0.738886i \(-0.735352\pi\)
−0.673831 + 0.738886i \(0.735352\pi\)
\(182\) 0 0
\(183\) 3.16658 0.234081
\(184\) 0 0
\(185\) 9.98203 0.733893
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 11.7900 0.857600
\(190\) 0 0
\(191\) −4.47309 −0.323661 −0.161831 0.986819i \(-0.551740\pi\)
−0.161831 + 0.986819i \(0.551740\pi\)
\(192\) 0 0
\(193\) 15.0490 1.08325 0.541626 0.840620i \(-0.317809\pi\)
0.541626 + 0.840620i \(0.317809\pi\)
\(194\) 0 0
\(195\) 14.1088 1.01035
\(196\) 0 0
\(197\) −16.0968 −1.14685 −0.573424 0.819259i \(-0.694385\pi\)
−0.573424 + 0.819259i \(0.694385\pi\)
\(198\) 0 0
\(199\) −11.9492 −0.847057 −0.423528 0.905883i \(-0.639209\pi\)
−0.423528 + 0.905883i \(0.639209\pi\)
\(200\) 0 0
\(201\) 4.24454 0.299387
\(202\) 0 0
\(203\) −8.37915 −0.588101
\(204\) 0 0
\(205\) −2.53349 −0.176947
\(206\) 0 0
\(207\) −42.1996 −2.93308
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 9.07464 0.624724 0.312362 0.949963i \(-0.398880\pi\)
0.312362 + 0.949963i \(0.398880\pi\)
\(212\) 0 0
\(213\) 21.8505 1.49717
\(214\) 0 0
\(215\) −0.233566 −0.0159291
\(216\) 0 0
\(217\) −19.2314 −1.30551
\(218\) 0 0
\(219\) −22.6147 −1.52816
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 16.8669 1.12949 0.564744 0.825266i \(-0.308975\pi\)
0.564744 + 0.825266i \(0.308975\pi\)
\(224\) 0 0
\(225\) 4.99102 0.332734
\(226\) 0 0
\(227\) 10.4411 0.693001 0.346500 0.938050i \(-0.387370\pi\)
0.346500 + 0.938050i \(0.387370\pi\)
\(228\) 0 0
\(229\) −12.9312 −0.854520 −0.427260 0.904129i \(-0.640521\pi\)
−0.427260 + 0.904129i \(0.640521\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −27.8833 −1.82669 −0.913347 0.407182i \(-0.866512\pi\)
−0.913347 + 0.407182i \(0.866512\pi\)
\(234\) 0 0
\(235\) 9.15519 0.597219
\(236\) 0 0
\(237\) −28.7535 −1.86774
\(238\) 0 0
\(239\) 1.51206 0.0978068 0.0489034 0.998804i \(-0.484427\pi\)
0.0489034 + 0.998804i \(0.484427\pi\)
\(240\) 0 0
\(241\) −16.7690 −1.08019 −0.540095 0.841604i \(-0.681612\pi\)
−0.540095 + 0.841604i \(0.681612\pi\)
\(242\) 0 0
\(243\) −14.2356 −0.913211
\(244\) 0 0
\(245\) 2.61186 0.166866
\(246\) 0 0
\(247\) −13.9820 −0.889655
\(248\) 0 0
\(249\) −38.9043 −2.46546
\(250\) 0 0
\(251\) 15.3505 0.968915 0.484457 0.874815i \(-0.339017\pi\)
0.484457 + 0.874815i \(0.339017\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.05978 0.440377 0.220189 0.975457i \(-0.429333\pi\)
0.220189 + 0.975457i \(0.429333\pi\)
\(258\) 0 0
\(259\) −20.9102 −1.29930
\(260\) 0 0
\(261\) −19.9641 −1.23574
\(262\) 0 0
\(263\) 14.8343 0.914720 0.457360 0.889282i \(-0.348795\pi\)
0.457360 + 0.889282i \(0.348795\pi\)
\(264\) 0 0
\(265\) −1.33734 −0.0821521
\(266\) 0 0
\(267\) 24.3190 1.48830
\(268\) 0 0
\(269\) 15.2296 0.928565 0.464283 0.885687i \(-0.346312\pi\)
0.464283 + 0.885687i \(0.346312\pi\)
\(270\) 0 0
\(271\) −0.946173 −0.0574759 −0.0287379 0.999587i \(-0.509149\pi\)
−0.0287379 + 0.999587i \(0.509149\pi\)
\(272\) 0 0
\(273\) −29.5549 −1.78875
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −24.1208 −1.44928 −0.724639 0.689128i \(-0.757994\pi\)
−0.724639 + 0.689128i \(0.757994\pi\)
\(278\) 0 0
\(279\) −45.8205 −2.74320
\(280\) 0 0
\(281\) −13.3253 −0.794922 −0.397461 0.917619i \(-0.630109\pi\)
−0.397461 + 0.917619i \(0.630109\pi\)
\(282\) 0 0
\(283\) 21.8065 1.29626 0.648130 0.761530i \(-0.275551\pi\)
0.648130 + 0.761530i \(0.275551\pi\)
\(284\) 0 0
\(285\) −7.91922 −0.469094
\(286\) 0 0
\(287\) 5.30713 0.313270
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −8.59871 −0.504065
\(292\) 0 0
\(293\) 20.0400 1.17075 0.585375 0.810762i \(-0.300947\pi\)
0.585375 + 0.810762i \(0.300947\pi\)
\(294\) 0 0
\(295\) 12.1088 0.705001
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 42.1996 2.44047
\(300\) 0 0
\(301\) 0.489272 0.0282012
\(302\) 0 0
\(303\) 26.5319 1.52422
\(304\) 0 0
\(305\) −1.12019 −0.0641417
\(306\) 0 0
\(307\) 15.5926 0.889915 0.444957 0.895552i \(-0.353219\pi\)
0.444957 + 0.895552i \(0.353219\pi\)
\(308\) 0 0
\(309\) 21.2266 1.20754
\(310\) 0 0
\(311\) −7.02214 −0.398189 −0.199094 0.979980i \(-0.563800\pi\)
−0.199094 + 0.979980i \(0.563800\pi\)
\(312\) 0 0
\(313\) −17.5909 −0.994294 −0.497147 0.867666i \(-0.665619\pi\)
−0.497147 + 0.867666i \(0.665619\pi\)
\(314\) 0 0
\(315\) −10.4551 −0.589079
\(316\) 0 0
\(317\) 32.6147 1.83182 0.915912 0.401379i \(-0.131469\pi\)
0.915912 + 0.401379i \(0.131469\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 51.5094 2.87497
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −4.99102 −0.276852
\(326\) 0 0
\(327\) 58.5300 3.23671
\(328\) 0 0
\(329\) −19.1782 −1.05733
\(330\) 0 0
\(331\) −11.4013 −0.626672 −0.313336 0.949642i \(-0.601447\pi\)
−0.313336 + 0.949642i \(0.601447\pi\)
\(332\) 0 0
\(333\) −49.8205 −2.73014
\(334\) 0 0
\(335\) −1.50152 −0.0820366
\(336\) 0 0
\(337\) −21.9372 −1.19499 −0.597497 0.801871i \(-0.703838\pi\)
−0.597497 + 0.801871i \(0.703838\pi\)
\(338\) 0 0
\(339\) 43.2206 2.34742
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −20.1348 −1.08718
\(344\) 0 0
\(345\) 23.9012 1.28680
\(346\) 0 0
\(347\) 29.1198 1.56323 0.781616 0.623760i \(-0.214396\pi\)
0.781616 + 0.623760i \(0.214396\pi\)
\(348\) 0 0
\(349\) 36.5967 1.95898 0.979489 0.201497i \(-0.0645805\pi\)
0.979489 + 0.201497i \(0.0645805\pi\)
\(350\) 0 0
\(351\) −28.0908 −1.49938
\(352\) 0 0
\(353\) −24.8802 −1.32424 −0.662121 0.749397i \(-0.730344\pi\)
−0.662121 + 0.749397i \(0.730344\pi\)
\(354\) 0 0
\(355\) −7.72964 −0.410247
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 9.57771 0.505492 0.252746 0.967533i \(-0.418666\pi\)
0.252746 + 0.967533i \(0.418666\pi\)
\(360\) 0 0
\(361\) −11.1519 −0.586944
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 8.00000 0.418739
\(366\) 0 0
\(367\) 8.82684 0.460757 0.230379 0.973101i \(-0.426004\pi\)
0.230379 + 0.973101i \(0.426004\pi\)
\(368\) 0 0
\(369\) 12.6447 0.658256
\(370\) 0 0
\(371\) 2.80144 0.145444
\(372\) 0 0
\(373\) 2.59871 0.134556 0.0672781 0.997734i \(-0.478569\pi\)
0.0672781 + 0.997734i \(0.478569\pi\)
\(374\) 0 0
\(375\) −2.82684 −0.145977
\(376\) 0 0
\(377\) 19.9641 1.02820
\(378\) 0 0
\(379\) −4.64185 −0.238436 −0.119218 0.992868i \(-0.538039\pi\)
−0.119218 + 0.992868i \(0.538039\pi\)
\(380\) 0 0
\(381\) 56.1333 2.87579
\(382\) 0 0
\(383\) 4.73267 0.241828 0.120914 0.992663i \(-0.461417\pi\)
0.120914 + 0.992663i \(0.461417\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.16573 0.0592575
\(388\) 0 0
\(389\) −28.9341 −1.46702 −0.733508 0.679681i \(-0.762118\pi\)
−0.733508 + 0.679681i \(0.762118\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 12.0208 0.606370
\(394\) 0 0
\(395\) 10.1716 0.511789
\(396\) 0 0
\(397\) 13.8205 0.693629 0.346815 0.937934i \(-0.387263\pi\)
0.346815 + 0.937934i \(0.387263\pi\)
\(398\) 0 0
\(399\) 16.5891 0.830493
\(400\) 0 0
\(401\) −15.6506 −0.781556 −0.390778 0.920485i \(-0.627794\pi\)
−0.390778 + 0.920485i \(0.627794\pi\)
\(402\) 0 0
\(403\) 45.8205 2.28248
\(404\) 0 0
\(405\) −0.937188 −0.0465692
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −10.0992 −0.499373 −0.249686 0.968327i \(-0.580327\pi\)
−0.249686 + 0.968327i \(0.580327\pi\)
\(410\) 0 0
\(411\) −23.3654 −1.15253
\(412\) 0 0
\(413\) −25.3654 −1.24815
\(414\) 0 0
\(415\) 13.7625 0.675573
\(416\) 0 0
\(417\) −53.5829 −2.62397
\(418\) 0 0
\(419\) 0.195529 0.00955222 0.00477611 0.999989i \(-0.498480\pi\)
0.00477611 + 0.999989i \(0.498480\pi\)
\(420\) 0 0
\(421\) −31.6088 −1.54052 −0.770258 0.637732i \(-0.779873\pi\)
−0.770258 + 0.637732i \(0.779873\pi\)
\(422\) 0 0
\(423\) −45.6937 −2.22170
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 2.34655 0.113558
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −5.95402 −0.286795 −0.143397 0.989665i \(-0.545803\pi\)
−0.143397 + 0.989665i \(0.545803\pi\)
\(432\) 0 0
\(433\) 17.6745 0.849382 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(434\) 0 0
\(435\) 11.3074 0.542146
\(436\) 0 0
\(437\) −23.6865 −1.13308
\(438\) 0 0
\(439\) 18.8923 0.901679 0.450839 0.892605i \(-0.351125\pi\)
0.450839 + 0.892605i \(0.351125\pi\)
\(440\) 0 0
\(441\) −13.0359 −0.620755
\(442\) 0 0
\(443\) −23.2820 −1.10616 −0.553080 0.833128i \(-0.686547\pi\)
−0.553080 + 0.833128i \(0.686547\pi\)
\(444\) 0 0
\(445\) −8.60288 −0.407816
\(446\) 0 0
\(447\) −35.2866 −1.66900
\(448\) 0 0
\(449\) −9.32240 −0.439951 −0.219976 0.975505i \(-0.570598\pi\)
−0.219976 + 0.975505i \(0.570598\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −9.52691 −0.447613
\(454\) 0 0
\(455\) 10.4551 0.490143
\(456\) 0 0
\(457\) 15.9744 0.747250 0.373625 0.927580i \(-0.378115\pi\)
0.373625 + 0.927580i \(0.378115\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6.33190 −0.294906 −0.147453 0.989069i \(-0.547108\pi\)
−0.147453 + 0.989069i \(0.547108\pi\)
\(462\) 0 0
\(463\) −36.9536 −1.71738 −0.858690 0.512496i \(-0.828721\pi\)
−0.858690 + 0.512496i \(0.828721\pi\)
\(464\) 0 0
\(465\) 25.9520 1.20350
\(466\) 0 0
\(467\) −5.04754 −0.233572 −0.116786 0.993157i \(-0.537259\pi\)
−0.116786 + 0.993157i \(0.537259\pi\)
\(468\) 0 0
\(469\) 3.14536 0.145239
\(470\) 0 0
\(471\) 24.4711 1.12757
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 2.80144 0.128539
\(476\) 0 0
\(477\) 6.67468 0.305612
\(478\) 0 0
\(479\) −9.66683 −0.441689 −0.220844 0.975309i \(-0.570881\pi\)
−0.220844 + 0.975309i \(0.570881\pi\)
\(480\) 0 0
\(481\) 49.8205 2.27162
\(482\) 0 0
\(483\) −50.0680 −2.27817
\(484\) 0 0
\(485\) 3.04181 0.138122
\(486\) 0 0
\(487\) −30.5268 −1.38330 −0.691651 0.722231i \(-0.743117\pi\)
−0.691651 + 0.722231i \(0.743117\pi\)
\(488\) 0 0
\(489\) 18.9400 0.856498
\(490\) 0 0
\(491\) 2.04598 0.0923339 0.0461669 0.998934i \(-0.485299\pi\)
0.0461669 + 0.998934i \(0.485299\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 16.1920 0.726309
\(498\) 0 0
\(499\) 29.8713 1.33722 0.668611 0.743613i \(-0.266889\pi\)
0.668611 + 0.743613i \(0.266889\pi\)
\(500\) 0 0
\(501\) 24.9706 1.11561
\(502\) 0 0
\(503\) −20.3244 −0.906220 −0.453110 0.891455i \(-0.649685\pi\)
−0.453110 + 0.891455i \(0.649685\pi\)
\(504\) 0 0
\(505\) −9.38573 −0.417659
\(506\) 0 0
\(507\) 33.6683 1.49526
\(508\) 0 0
\(509\) −16.9672 −0.752057 −0.376028 0.926608i \(-0.622710\pi\)
−0.376028 + 0.926608i \(0.622710\pi\)
\(510\) 0 0
\(511\) −16.7583 −0.741344
\(512\) 0 0
\(513\) 15.7673 0.696143
\(514\) 0 0
\(515\) −7.50894 −0.330884
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 59.9533 2.63166
\(520\) 0 0
\(521\) 37.3012 1.63420 0.817098 0.576499i \(-0.195582\pi\)
0.817098 + 0.576499i \(0.195582\pi\)
\(522\) 0 0
\(523\) −35.4669 −1.55086 −0.775431 0.631433i \(-0.782467\pi\)
−0.775431 + 0.631433i \(0.782467\pi\)
\(524\) 0 0
\(525\) 5.92163 0.258441
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 48.4890 2.10822
\(530\) 0 0
\(531\) −60.4352 −2.62266
\(532\) 0 0
\(533\) −12.6447 −0.547702
\(534\) 0 0
\(535\) −18.2216 −0.787786
\(536\) 0 0
\(537\) 9.20576 0.397258
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −23.5853 −1.01401 −0.507007 0.861942i \(-0.669248\pi\)
−0.507007 + 0.861942i \(0.669248\pi\)
\(542\) 0 0
\(543\) −51.2532 −2.19949
\(544\) 0 0
\(545\) −20.7051 −0.886909
\(546\) 0 0
\(547\) −40.7804 −1.74364 −0.871821 0.489824i \(-0.837061\pi\)
−0.871821 + 0.489824i \(0.837061\pi\)
\(548\) 0 0
\(549\) 5.59086 0.238612
\(550\) 0 0
\(551\) −11.2058 −0.477381
\(552\) 0 0
\(553\) −21.3074 −0.906081
\(554\) 0 0
\(555\) 28.2176 1.19777
\(556\) 0 0
\(557\) 16.6208 0.704245 0.352122 0.935954i \(-0.385460\pi\)
0.352122 + 0.935954i \(0.385460\pi\)
\(558\) 0 0
\(559\) −1.16573 −0.0493052
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −38.4572 −1.62078 −0.810389 0.585892i \(-0.800744\pi\)
−0.810389 + 0.585892i \(0.800744\pi\)
\(564\) 0 0
\(565\) −15.2894 −0.643229
\(566\) 0 0
\(567\) 1.96321 0.0824471
\(568\) 0 0
\(569\) 5.84084 0.244861 0.122430 0.992477i \(-0.460931\pi\)
0.122430 + 0.992477i \(0.460931\pi\)
\(570\) 0 0
\(571\) −2.35948 −0.0987411 −0.0493705 0.998781i \(-0.515722\pi\)
−0.0493705 + 0.998781i \(0.515722\pi\)
\(572\) 0 0
\(573\) −12.6447 −0.528240
\(574\) 0 0
\(575\) −8.45512 −0.352603
\(576\) 0 0
\(577\) −9.16744 −0.381645 −0.190823 0.981625i \(-0.561116\pi\)
−0.190823 + 0.981625i \(0.561116\pi\)
\(578\) 0 0
\(579\) 42.5411 1.76795
\(580\) 0 0
\(581\) −28.8295 −1.19605
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 24.9102 1.02991
\(586\) 0 0
\(587\) 26.5745 1.09684 0.548422 0.836201i \(-0.315229\pi\)
0.548422 + 0.836201i \(0.315229\pi\)
\(588\) 0 0
\(589\) −25.7189 −1.05973
\(590\) 0 0
\(591\) −45.5030 −1.87174
\(592\) 0 0
\(593\) −0.925361 −0.0380000 −0.0190000 0.999819i \(-0.506048\pi\)
−0.0190000 + 0.999819i \(0.506048\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −33.7785 −1.38246
\(598\) 0 0
\(599\) 39.7086 1.62245 0.811224 0.584736i \(-0.198802\pi\)
0.811224 + 0.584736i \(0.198802\pi\)
\(600\) 0 0
\(601\) 26.9282 1.09842 0.549212 0.835683i \(-0.314928\pi\)
0.549212 + 0.835683i \(0.314928\pi\)
\(602\) 0 0
\(603\) 7.49409 0.305183
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7.40129 0.300409 0.150205 0.988655i \(-0.452007\pi\)
0.150205 + 0.988655i \(0.452007\pi\)
\(608\) 0 0
\(609\) −23.6865 −0.959826
\(610\) 0 0
\(611\) 45.6937 1.84857
\(612\) 0 0
\(613\) −42.1685 −1.70317 −0.851585 0.524217i \(-0.824358\pi\)
−0.851585 + 0.524217i \(0.824358\pi\)
\(614\) 0 0
\(615\) −7.16177 −0.288790
\(616\) 0 0
\(617\) −20.1219 −0.810079 −0.405040 0.914299i \(-0.632742\pi\)
−0.405040 + 0.914299i \(0.632742\pi\)
\(618\) 0 0
\(619\) 1.85641 0.0746153 0.0373076 0.999304i \(-0.488122\pi\)
0.0373076 + 0.999304i \(0.488122\pi\)
\(620\) 0 0
\(621\) −47.5878 −1.90963
\(622\) 0 0
\(623\) 18.0212 0.722004
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −20.7583 −0.826375 −0.413187 0.910646i \(-0.635585\pi\)
−0.413187 + 0.910646i \(0.635585\pi\)
\(632\) 0 0
\(633\) 25.6525 1.01960
\(634\) 0 0
\(635\) −19.8573 −0.788011
\(636\) 0 0
\(637\) 13.0359 0.516500
\(638\) 0 0
\(639\) 38.5788 1.52615
\(640\) 0 0
\(641\) 27.0381 1.06794 0.533971 0.845503i \(-0.320699\pi\)
0.533971 + 0.845503i \(0.320699\pi\)
\(642\) 0 0
\(643\) −20.9864 −0.827624 −0.413812 0.910362i \(-0.635803\pi\)
−0.413812 + 0.910362i \(0.635803\pi\)
\(644\) 0 0
\(645\) −0.660254 −0.0259975
\(646\) 0 0
\(647\) 4.56300 0.179390 0.0896950 0.995969i \(-0.471411\pi\)
0.0896950 + 0.995969i \(0.471411\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −54.3640 −2.13069
\(652\) 0 0
\(653\) −30.5311 −1.19477 −0.597387 0.801953i \(-0.703794\pi\)
−0.597387 + 0.801953i \(0.703794\pi\)
\(654\) 0 0
\(655\) −4.25239 −0.166155
\(656\) 0 0
\(657\) −39.9281 −1.55774
\(658\) 0 0
\(659\) −34.5411 −1.34553 −0.672766 0.739856i \(-0.734894\pi\)
−0.672766 + 0.739856i \(0.734894\pi\)
\(660\) 0 0
\(661\) −15.0956 −0.587152 −0.293576 0.955936i \(-0.594845\pi\)
−0.293576 + 0.955936i \(0.594845\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −5.86842 −0.227568
\(666\) 0 0
\(667\) 33.8205 1.30953
\(668\) 0 0
\(669\) 47.6799 1.84341
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 13.7445 0.529812 0.264906 0.964274i \(-0.414659\pi\)
0.264906 + 0.964274i \(0.414659\pi\)
\(674\) 0 0
\(675\) 5.62828 0.216633
\(676\) 0 0
\(677\) 2.80258 0.107712 0.0538559 0.998549i \(-0.482849\pi\)
0.0538559 + 0.998549i \(0.482849\pi\)
\(678\) 0 0
\(679\) −6.37195 −0.244533
\(680\) 0 0
\(681\) 29.5153 1.13103
\(682\) 0 0
\(683\) 19.5714 0.748880 0.374440 0.927251i \(-0.377835\pi\)
0.374440 + 0.927251i \(0.377835\pi\)
\(684\) 0 0
\(685\) 8.26554 0.315810
\(686\) 0 0
\(687\) −36.5545 −1.39464
\(688\) 0 0
\(689\) −6.67468 −0.254285
\(690\) 0 0
\(691\) 12.3475 0.469720 0.234860 0.972029i \(-0.424537\pi\)
0.234860 + 0.972029i \(0.424537\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 18.9551 0.719007
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −78.8215 −2.98130
\(700\) 0 0
\(701\) −28.6208 −1.08099 −0.540496 0.841347i \(-0.681763\pi\)
−0.540496 + 0.841347i \(0.681763\pi\)
\(702\) 0 0
\(703\) −27.9641 −1.05468
\(704\) 0 0
\(705\) 25.8802 0.974707
\(706\) 0 0
\(707\) 19.6611 0.739432
\(708\) 0 0
\(709\) 5.05914 0.190000 0.0949999 0.995477i \(-0.469715\pi\)
0.0949999 + 0.995477i \(0.469715\pi\)
\(710\) 0 0
\(711\) −50.7666 −1.90390
\(712\) 0 0
\(713\) 77.6230 2.90700
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 4.27434 0.159628
\(718\) 0 0
\(719\) 32.6547 1.21782 0.608908 0.793241i \(-0.291608\pi\)
0.608908 + 0.793241i \(0.291608\pi\)
\(720\) 0 0
\(721\) 15.7296 0.585803
\(722\) 0 0
\(723\) −47.4034 −1.76295
\(724\) 0 0
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) 39.9623 1.48212 0.741060 0.671439i \(-0.234323\pi\)
0.741060 + 0.671439i \(0.234323\pi\)
\(728\) 0 0
\(729\) −43.0532 −1.59456
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 10.4475 0.385886 0.192943 0.981210i \(-0.438197\pi\)
0.192943 + 0.981210i \(0.438197\pi\)
\(734\) 0 0
\(735\) 7.38332 0.272338
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 21.1238 0.777053 0.388527 0.921438i \(-0.372984\pi\)
0.388527 + 0.921438i \(0.372984\pi\)
\(740\) 0 0
\(741\) −39.5249 −1.45198
\(742\) 0 0
\(743\) −2.82140 −0.103507 −0.0517536 0.998660i \(-0.516481\pi\)
−0.0517536 + 0.998660i \(0.516481\pi\)
\(744\) 0 0
\(745\) 12.4827 0.457331
\(746\) 0 0
\(747\) −68.6887 −2.51319
\(748\) 0 0
\(749\) 38.1703 1.39471
\(750\) 0 0
\(751\) −24.3547 −0.888714 −0.444357 0.895850i \(-0.646568\pi\)
−0.444357 + 0.895850i \(0.646568\pi\)
\(752\) 0 0
\(753\) 43.3934 1.58134
\(754\) 0 0
\(755\) 3.37017 0.122653
\(756\) 0 0
\(757\) 46.2139 1.67967 0.839837 0.542839i \(-0.182651\pi\)
0.839837 + 0.542839i \(0.182651\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −42.8204 −1.55224 −0.776119 0.630587i \(-0.782814\pi\)
−0.776119 + 0.630587i \(0.782814\pi\)
\(762\) 0 0
\(763\) 43.3728 1.57020
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 60.4352 2.18219
\(768\) 0 0
\(769\) −25.5669 −0.921967 −0.460984 0.887409i \(-0.652503\pi\)
−0.460984 + 0.887409i \(0.652503\pi\)
\(770\) 0 0
\(771\) 19.9569 0.718729
\(772\) 0 0
\(773\) −7.40299 −0.266267 −0.133134 0.991098i \(-0.542504\pi\)
−0.133134 + 0.991098i \(0.542504\pi\)
\(774\) 0 0
\(775\) −9.18059 −0.329777
\(776\) 0 0
\(777\) −59.1099 −2.12055
\(778\) 0 0
\(779\) 7.09742 0.254292
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −22.5131 −0.804553
\(784\) 0 0
\(785\) −8.65671 −0.308971
\(786\) 0 0
\(787\) 20.6090 0.734633 0.367316 0.930096i \(-0.380277\pi\)
0.367316 + 0.930096i \(0.380277\pi\)
\(788\) 0 0
\(789\) 41.9341 1.49289
\(790\) 0 0
\(791\) 32.0280 1.13879
\(792\) 0 0
\(793\) −5.59086 −0.198537
\(794\) 0 0
\(795\) −3.78044 −0.134078
\(796\) 0 0
\(797\) 4.53704 0.160710 0.0803551 0.996766i \(-0.474395\pi\)
0.0803551 + 0.996766i \(0.474395\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 42.9371 1.51711
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 17.7117 0.624254
\(806\) 0 0
\(807\) 43.0516 1.51549
\(808\) 0 0
\(809\) −17.4773 −0.614468 −0.307234 0.951634i \(-0.599403\pi\)
−0.307234 + 0.951634i \(0.599403\pi\)
\(810\) 0 0
\(811\) 17.3163 0.608056 0.304028 0.952663i \(-0.401668\pi\)
0.304028 + 0.952663i \(0.401668\pi\)
\(812\) 0 0
\(813\) −2.67468 −0.0938050
\(814\) 0 0
\(815\) −6.70008 −0.234693
\(816\) 0 0
\(817\) 0.654322 0.0228918
\(818\) 0 0
\(819\) −52.1816 −1.82337
\(820\) 0 0
\(821\) −4.23485 −0.147797 −0.0738987 0.997266i \(-0.523544\pi\)
−0.0738987 + 0.997266i \(0.523544\pi\)
\(822\) 0 0
\(823\) −25.8478 −0.900999 −0.450500 0.892777i \(-0.648754\pi\)
−0.450500 + 0.892777i \(0.648754\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 25.9943 0.903912 0.451956 0.892040i \(-0.350726\pi\)
0.451956 + 0.892040i \(0.350726\pi\)
\(828\) 0 0
\(829\) −17.9845 −0.624627 −0.312314 0.949979i \(-0.601104\pi\)
−0.312314 + 0.949979i \(0.601104\pi\)
\(830\) 0 0
\(831\) −68.1856 −2.36533
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −8.83342 −0.305693
\(836\) 0 0
\(837\) −51.6709 −1.78601
\(838\) 0 0
\(839\) −9.84155 −0.339768 −0.169884 0.985464i \(-0.554339\pi\)
−0.169884 + 0.985464i \(0.554339\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) −37.6685 −1.29737
\(844\) 0 0
\(845\) −11.9102 −0.409725
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 61.6433 2.11559
\(850\) 0 0
\(851\) 84.3992 2.89317
\(852\) 0 0
\(853\) 39.9041 1.36629 0.683145 0.730283i \(-0.260612\pi\)
0.683145 + 0.730283i \(0.260612\pi\)
\(854\) 0 0
\(855\) −13.9820 −0.478175
\(856\) 0 0
\(857\) −37.5981 −1.28433 −0.642163 0.766568i \(-0.721963\pi\)
−0.642163 + 0.766568i \(0.721963\pi\)
\(858\) 0 0
\(859\) 6.99811 0.238772 0.119386 0.992848i \(-0.461907\pi\)
0.119386 + 0.992848i \(0.461907\pi\)
\(860\) 0 0
\(861\) 15.0024 0.511280
\(862\) 0 0
\(863\) −6.87157 −0.233911 −0.116956 0.993137i \(-0.537314\pi\)
−0.116956 + 0.993137i \(0.537314\pi\)
\(864\) 0 0
\(865\) −21.2086 −0.721114
\(866\) 0 0
\(867\) −48.0563 −1.63208
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −7.49409 −0.253927
\(872\) 0 0
\(873\) −15.1817 −0.513824
\(874\) 0 0
\(875\) −2.09479 −0.0708167
\(876\) 0 0
\(877\) 40.6727 1.37342 0.686710 0.726932i \(-0.259054\pi\)
0.686710 + 0.726932i \(0.259054\pi\)
\(878\) 0 0
\(879\) 56.6499 1.91075
\(880\) 0 0
\(881\) 13.3224 0.448843 0.224422 0.974492i \(-0.427951\pi\)
0.224422 + 0.974492i \(0.427951\pi\)
\(882\) 0 0
\(883\) −16.5807 −0.557986 −0.278993 0.960293i \(-0.590001\pi\)
−0.278993 + 0.960293i \(0.590001\pi\)
\(884\) 0 0
\(885\) 34.2296 1.15062
\(886\) 0 0
\(887\) 6.53561 0.219444 0.109722 0.993962i \(-0.465004\pi\)
0.109722 + 0.993962i \(0.465004\pi\)
\(888\) 0 0
\(889\) 41.5967 1.39511
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −25.6477 −0.858268
\(894\) 0 0
\(895\) −3.25656 −0.108855
\(896\) 0 0
\(897\) 119.292 3.98303
\(898\) 0 0
\(899\) 36.7224 1.22476
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 1.38309 0.0460264
\(904\) 0 0
\(905\) 18.1309 0.602693
\(906\) 0 0
\(907\) 16.1151 0.535093 0.267546 0.963545i \(-0.413787\pi\)
0.267546 + 0.963545i \(0.413787\pi\)
\(908\) 0 0
\(909\) 46.8443 1.55373
\(910\) 0 0
\(911\) 24.0149 0.795648 0.397824 0.917462i \(-0.369766\pi\)
0.397824 + 0.917462i \(0.369766\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −3.16658 −0.104684
\(916\) 0 0
\(917\) 8.90785 0.294163
\(918\) 0 0
\(919\) −56.2428 −1.85528 −0.927639 0.373479i \(-0.878165\pi\)
−0.927639 + 0.373479i \(0.878165\pi\)
\(920\) 0 0
\(921\) 44.0777 1.45241
\(922\) 0 0
\(923\) −38.5788 −1.26984
\(924\) 0 0
\(925\) −9.98203 −0.328207
\(926\) 0 0
\(927\) 37.4773 1.23091
\(928\) 0 0
\(929\) −19.8445 −0.651077 −0.325538 0.945529i \(-0.605546\pi\)
−0.325538 + 0.945529i \(0.605546\pi\)
\(930\) 0 0
\(931\) −7.31698 −0.239804
\(932\) 0 0
\(933\) −19.8505 −0.649875
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −25.6997 −0.839571 −0.419786 0.907623i \(-0.637895\pi\)
−0.419786 + 0.907623i \(0.637895\pi\)
\(938\) 0 0
\(939\) −49.7265 −1.62276
\(940\) 0 0
\(941\) −3.28289 −0.107019 −0.0535095 0.998567i \(-0.517041\pi\)
−0.0535095 + 0.998567i \(0.517041\pi\)
\(942\) 0 0
\(943\) −21.4210 −0.697563
\(944\) 0 0
\(945\) −11.7900 −0.383530
\(946\) 0 0
\(947\) −7.32107 −0.237903 −0.118951 0.992900i \(-0.537953\pi\)
−0.118951 + 0.992900i \(0.537953\pi\)
\(948\) 0 0
\(949\) 39.9281 1.29612
\(950\) 0 0
\(951\) 92.1965 2.98968
\(952\) 0 0
\(953\) 14.9282 0.483572 0.241786 0.970330i \(-0.422267\pi\)
0.241786 + 0.970330i \(0.422267\pi\)
\(954\) 0 0
\(955\) 4.47309 0.144746
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −17.3146 −0.559116
\(960\) 0 0
\(961\) 53.2832 1.71881
\(962\) 0 0
\(963\) 90.9440 2.93063
\(964\) 0 0
\(965\) −15.0490 −0.484445
\(966\) 0 0
\(967\) −40.2396 −1.29402 −0.647010 0.762482i \(-0.723981\pi\)
−0.647010 + 0.762482i \(0.723981\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −8.69492 −0.279033 −0.139517 0.990220i \(-0.544555\pi\)
−0.139517 + 0.990220i \(0.544555\pi\)
\(972\) 0 0
\(973\) −39.7069 −1.27294
\(974\) 0 0
\(975\) −14.1088 −0.451843
\(976\) 0 0
\(977\) 25.7308 0.823201 0.411600 0.911364i \(-0.364970\pi\)
0.411600 + 0.911364i \(0.364970\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 103.339 3.29937
\(982\) 0 0
\(983\) 35.7454 1.14010 0.570051 0.821609i \(-0.306924\pi\)
0.570051 + 0.821609i \(0.306924\pi\)
\(984\) 0 0
\(985\) 16.0968 0.512886
\(986\) 0 0
\(987\) −54.2136 −1.72564
\(988\) 0 0
\(989\) −1.97483 −0.0627959
\(990\) 0 0
\(991\) −31.1894 −0.990764 −0.495382 0.868675i \(-0.664972\pi\)
−0.495382 + 0.868675i \(0.664972\pi\)
\(992\) 0 0
\(993\) −32.2296 −1.02278
\(994\) 0 0
\(995\) 11.9492 0.378815
\(996\) 0 0
\(997\) −23.1009 −0.731614 −0.365807 0.930691i \(-0.619207\pi\)
−0.365807 + 0.930691i \(0.619207\pi\)
\(998\) 0 0
\(999\) −56.1816 −1.77751
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 9680.2.a.cr.1.4 4
4.3 odd 2 4840.2.a.w.1.1 4
11.10 odd 2 9680.2.a.cq.1.4 4
44.43 even 2 4840.2.a.x.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.w.1.1 4 4.3 odd 2
4840.2.a.x.1.1 yes 4 44.43 even 2
9680.2.a.cq.1.4 4 11.10 odd 2
9680.2.a.cr.1.4 4 1.1 even 1 trivial