Properties

Label 9680.2.a.cq.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 $1$

Related objects

Downloads

Learn more

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 \(-0.629560\pi\)
−0.395878 + 0.918303i \(0.629560\pi\)
\(8\) 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 \(-0.256673\pi\)
0.692129 + 0.721774i \(0.256673\pi\)
\(14\) 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 \(-0.604136\pi\)
−0.321347 + 0.946961i \(0.604136\pi\)
\(20\) 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 \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 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 \(-0.563390\pi\)
−0.197832 + 0.980236i \(0.563390\pi\)
\(42\) 0 0
\(43\) −0.233566 −0.0356185 −0.0178093 0.999841i \(-0.505669\pi\)
−0.0178093 + 0.999841i \(0.505669\pi\)
\(44\) 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 \(-0.522846\pi\)
−0.0717126 + 0.997425i \(0.522846\pi\)
\(62\) 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 \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 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 \(-0.306091\pi\)
0.572198 + 0.820116i \(0.306091\pi\)
\(80\) 0 0
\(81\) 0.937188 0.104132
\(82\) 0 0
\(83\) 13.7625 1.51063 0.755314 0.655363i \(-0.227485\pi\)
0.755314 + 0.655363i \(0.227485\pi\)
\(84\) 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 \(-0.654650\pi\)
−0.466957 + 0.884280i \(0.654650\pi\)
\(102\) 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 \(-0.842976\pi\)
−0.880772 + 0.473541i \(0.842976\pi\)
\(108\) 0 0
\(109\) −20.7051 −1.98319 −0.991594 0.129386i \(-0.958699\pi\)
−0.991594 + 0.129386i \(0.958699\pi\)
\(110\) 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 \(-0.843145\pi\)
−0.881023 + 0.473073i \(0.843145\pi\)
\(128\) 0 0
\(129\) −0.660254 −0.0581321
\(130\) 0 0
\(131\) −4.25239 −0.371533 −0.185766 0.982594i \(-0.559477\pi\)
−0.185766 + 0.982594i \(0.559477\pi\)
\(132\) 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 \(-0.202768\pi\)
0.803874 + 0.594799i \(0.202768\pi\)
\(140\) 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 \(-0.329160\pi\)
0.511311 + 0.859396i \(0.329160\pi\)
\(150\) 0 0
\(151\) 3.37017 0.274260 0.137130 0.990553i \(-0.456212\pi\)
0.137130 + 0.990553i \(0.456212\pi\)
\(152\) 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 \(-0.611028\pi\)
−0.341775 + 0.939782i \(0.611028\pi\)
\(168\) 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 \(-0.798496\pi\)
−0.806230 + 0.591602i \(0.798496\pi\)
\(174\) 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 \(-0.682191\pi\)
−0.541626 + 0.840620i \(0.682191\pi\)
\(194\) 0 0
\(195\) −14.1088 −1.01035
\(196\) 0 0
\(197\) 16.0968 1.14685 0.573424 0.819259i \(-0.305615\pi\)
0.573424 + 0.819259i \(0.305615\pi\)
\(198\) 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 \(-0.601120\pi\)
−0.312362 + 0.949963i \(0.601120\pi\)
\(212\) 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 \(-0.612630\pi\)
−0.346500 + 0.938050i \(0.612630\pi\)
\(228\) 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 \(-0.133488\pi\)
0.913347 + 0.407182i \(0.133488\pi\)
\(234\) 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 \(-0.515573\pi\)
−0.0489034 + 0.998804i \(0.515573\pi\)
\(240\) 0 0
\(241\) 16.7690 1.08019 0.540095 0.841604i \(-0.318388\pi\)
0.540095 + 0.841604i \(0.318388\pi\)
\(242\) 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 \(-0.651205\pi\)
−0.457360 + 0.889282i \(0.651205\pi\)
\(264\) 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 \(-0.490851\pi\)
0.0287379 + 0.999587i \(0.490851\pi\)
\(272\) 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 \(-0.242006\pi\)
0.724639 + 0.689128i \(0.242006\pi\)
\(278\) 0 0
\(279\) −45.8205 −2.74320
\(280\) 0 0
\(281\) 13.3253 0.794922 0.397461 0.917619i \(-0.369891\pi\)
0.397461 + 0.917619i \(0.369891\pi\)
\(282\) 0 0
\(283\) −21.8065 −1.29626 −0.648130 0.761530i \(-0.724449\pi\)
−0.648130 + 0.761530i \(0.724449\pi\)
\(284\) 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 \(-0.699053\pi\)
−0.585375 + 0.810762i \(0.699053\pi\)
\(294\) 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 \(-0.646781\pi\)
−0.444957 + 0.895552i \(0.646781\pi\)
\(308\) 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 \(-0.296162\pi\)
0.597497 + 0.801871i \(0.296162\pi\)
\(338\) 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 \(-0.785604\pi\)
−0.781616 + 0.623760i \(0.785604\pi\)
\(348\) 0 0
\(349\) −36.5967 −1.95898 −0.979489 0.201497i \(-0.935420\pi\)
−0.979489 + 0.201497i \(0.935420\pi\)
\(350\) 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 \(-0.581334\pi\)
−0.252746 + 0.967533i \(0.581334\pi\)
\(360\) 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 \(-0.521431\pi\)
−0.0672781 + 0.997734i \(0.521431\pi\)
\(374\) 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 \(-0.419673\pi\)
0.249686 + 0.968327i \(0.419673\pi\)
\(410\) 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 \(-0.454197\pi\)
0.143397 + 0.989665i \(0.454197\pi\)
\(432\) 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 \(-0.648875\pi\)
−0.450839 + 0.892605i \(0.648875\pi\)
\(440\) 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 \(-0.621885\pi\)
−0.373625 + 0.927580i \(0.621885\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.33190 0.294906 0.147453 0.989069i \(-0.452892\pi\)
0.147453 + 0.989069i \(0.452892\pi\)
\(462\) 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 \(-0.429119\pi\)
0.220844 + 0.975309i \(0.429119\pi\)
\(480\) 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 \(-0.514701\pi\)
−0.0461669 + 0.998934i \(0.514701\pi\)
\(492\) 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 \(-0.350315\pi\)
0.453110 + 0.891455i \(0.350315\pi\)
\(504\) 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 \(-0.217533\pi\)
0.775431 + 0.631433i \(0.217533\pi\)
\(524\) 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 \(-0.330752\pi\)
0.507007 + 0.861942i \(0.330752\pi\)
\(542\) 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 \(-0.162939\pi\)
0.871821 + 0.489824i \(0.162939\pi\)
\(548\) 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 \(-0.614540\pi\)
−0.352122 + 0.935954i \(0.614540\pi\)
\(558\) 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 \(-0.199256\pi\)
0.810389 + 0.585892i \(0.199256\pi\)
\(564\) 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 \(-0.539069\pi\)
−0.122430 + 0.992477i \(0.539069\pi\)
\(570\) 0 0
\(571\) 2.35948 0.0987411 0.0493705 0.998781i \(-0.484278\pi\)
0.0493705 + 0.998781i \(0.484278\pi\)
\(572\) 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 \(-0.493952\pi\)
0.0190000 + 0.999819i \(0.493952\pi\)
\(594\) 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 \(-0.685072\pi\)
−0.549212 + 0.835683i \(0.685072\pi\)
\(602\) 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 \(-0.547993\pi\)
−0.150205 + 0.988655i \(0.547993\pi\)
\(608\) 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 \(-0.175642\pi\)
0.851585 + 0.524217i \(0.175642\pi\)
\(614\) 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 \(-0.265106\pi\)
0.672766 + 0.739856i \(0.265106\pi\)
\(660\) 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 \(-0.585341\pi\)
−0.264906 + 0.964274i \(0.585341\pi\)
\(674\) 0 0
\(675\) 5.62828 0.216633
\(676\) 0 0
\(677\) −2.80258 −0.107712 −0.0538559 0.998549i \(-0.517151\pi\)
−0.0538559 + 0.998549i \(0.517151\pi\)
\(678\) 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 \(-0.318237\pi\)
0.540496 + 0.841347i \(0.318237\pi\)
\(702\) 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 \(-0.561803\pi\)
−0.192943 + 0.981210i \(0.561803\pi\)
\(734\) 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 \(-0.627016\pi\)
−0.388527 + 0.921438i \(0.627016\pi\)
\(740\) 0 0
\(741\) −39.5249 −1.45198
\(742\) 0 0
\(743\) 2.82140 0.103507 0.0517536 0.998660i \(-0.483519\pi\)
0.0517536 + 0.998660i \(0.483519\pi\)
\(744\) 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 \(-0.217186\pi\)
0.776119 + 0.630587i \(0.217186\pi\)
\(762\) 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 \(-0.347497\pi\)
0.460984 + 0.887409i \(0.347497\pi\)
\(770\) 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 \(-0.619723\pi\)
−0.367316 + 0.930096i \(0.619723\pi\)
\(788\) 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 \(-0.400597\pi\)
0.307234 + 0.951634i \(0.400597\pi\)
\(810\) 0 0
\(811\) −17.3163 −0.608056 −0.304028 0.952663i \(-0.598332\pi\)
−0.304028 + 0.952663i \(0.598332\pi\)
\(812\) 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 \(-0.476456\pi\)
0.0738987 + 0.997266i \(0.476456\pi\)
\(822\) 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 \(-0.649274\pi\)
−0.451956 + 0.892040i \(0.649274\pi\)
\(828\) 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 \(-0.739388\pi\)
−0.683145 + 0.730283i \(0.739388\pi\)
\(854\) 0 0
\(855\) 13.9820 0.478175
\(856\) 0 0
\(857\) 37.5981 1.28433 0.642163 0.766568i \(-0.278037\pi\)
0.642163 + 0.766568i \(0.278037\pi\)
\(858\) 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 \(-0.740946\pi\)
−0.686710 + 0.726932i \(0.740946\pi\)
\(878\) 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 \(-0.534996\pi\)
−0.109722 + 0.993962i \(0.534996\pi\)
\(888\) 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 \(-0.121835\pi\)
0.927639 + 0.373479i \(0.121835\pi\)
\(920\) 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 \(-0.362105\pi\)
0.419786 + 0.907623i \(0.362105\pi\)
\(938\) 0 0
\(939\) −49.7265 −1.62276
\(940\) 0 0
\(941\) 3.28289 0.107019 0.0535095 0.998567i \(-0.482959\pi\)
0.0535095 + 0.998567i \(0.482959\pi\)
\(942\) 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 \(-0.577733\pi\)
−0.241786 + 0.970330i \(0.577733\pi\)
\(954\) 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 \(-0.276019\pi\)
0.647010 + 0.762482i \(0.276019\pi\)
\(968\) 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 \(-0.380793\pi\)
0.365807 + 0.930691i \(0.380793\pi\)
\(998\) 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.cq.1.4 4
4.3 odd 2 4840.2.a.x.1.1 yes 4
11.10 odd 2 9680.2.a.cr.1.4 4
44.43 even 2 4840.2.a.w.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.w.1.1 4 44.43 even 2
4840.2.a.x.1.1 yes 4 4.3 odd 2
9680.2.a.cq.1.4 4 1.1 even 1 trivial
9680.2.a.cr.1.4 4 11.10 odd 2