Properties

Label 9680.2.a.dc.1.5
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $1$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.25903625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.29082\) 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)
 
\(f(q)\) \(=\) \(q+2.29082 q^{3} -1.00000 q^{5} -4.66366 q^{7} +2.24785 q^{9} +O(q^{10})\) \(q+2.29082 q^{3} -1.00000 q^{5} -4.66366 q^{7} +2.24785 q^{9} +5.94269 q^{13} -2.29082 q^{15} -3.30331 q^{17} +4.19195 q^{19} -10.6836 q^{21} +1.71184 q^{23} +1.00000 q^{25} -1.72304 q^{27} -7.76702 q^{29} +2.97204 q^{31} +4.66366 q^{35} -5.68500 q^{37} +13.6136 q^{39} -6.14758 q^{41} +4.42353 q^{43} -2.24785 q^{45} -4.13243 q^{47} +14.7497 q^{49} -7.56730 q^{51} +8.10104 q^{53} +9.60300 q^{57} +6.21515 q^{59} -1.52758 q^{61} -10.4832 q^{63} -5.94269 q^{65} +7.46447 q^{67} +3.92151 q^{69} +2.23747 q^{71} -7.49400 q^{73} +2.29082 q^{75} -17.5316 q^{79} -10.6907 q^{81} -11.0785 q^{83} +3.30331 q^{85} -17.7928 q^{87} +12.8171 q^{89} -27.7147 q^{91} +6.80840 q^{93} -4.19195 q^{95} -9.53760 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9} - q^{13} - 3 q^{15} - 6 q^{17} - 7 q^{19} - 14 q^{21} + 9 q^{23} + 6 q^{25} + 21 q^{27} - 10 q^{29} - q^{31} + 7 q^{35} + 3 q^{37} - q^{39} + 6 q^{41} + 18 q^{43} - 5 q^{45} + 3 q^{47} + 17 q^{49} + 15 q^{51} - 23 q^{53} - 9 q^{57} + 2 q^{59} - 6 q^{61} - 49 q^{63} + q^{65} + 22 q^{67} + 2 q^{69} + 13 q^{71} - 10 q^{73} + 3 q^{75} - 22 q^{79} + 10 q^{81} + 10 q^{83} + 6 q^{85} - 3 q^{87} - 25 q^{89} - 12 q^{91} + 19 q^{93} + 7 q^{95} - 33 q^{97}+O(q^{100}) \) 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
\(3\) 2.29082 1.32260 0.661302 0.750119i \(-0.270004\pi\)
0.661302 + 0.750119i \(0.270004\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.66366 −1.76270 −0.881348 0.472467i \(-0.843363\pi\)
−0.881348 + 0.472467i \(0.843363\pi\)
\(8\) 0 0
\(9\) 2.24785 0.749284
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 5.94269 1.64821 0.824103 0.566440i \(-0.191680\pi\)
0.824103 + 0.566440i \(0.191680\pi\)
\(14\) 0 0
\(15\) −2.29082 −0.591487
\(16\) 0 0
\(17\) −3.30331 −0.801171 −0.400586 0.916259i \(-0.631193\pi\)
−0.400586 + 0.916259i \(0.631193\pi\)
\(18\) 0 0
\(19\) 4.19195 0.961699 0.480850 0.876803i \(-0.340328\pi\)
0.480850 + 0.876803i \(0.340328\pi\)
\(20\) 0 0
\(21\) −10.6836 −2.33135
\(22\) 0 0
\(23\) 1.71184 0.356942 0.178471 0.983945i \(-0.442885\pi\)
0.178471 + 0.983945i \(0.442885\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.72304 −0.331598
\(28\) 0 0
\(29\) −7.76702 −1.44230 −0.721149 0.692780i \(-0.756386\pi\)
−0.721149 + 0.692780i \(0.756386\pi\)
\(30\) 0 0
\(31\) 2.97204 0.533794 0.266897 0.963725i \(-0.414002\pi\)
0.266897 + 0.963725i \(0.414002\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.66366 0.788302
\(36\) 0 0
\(37\) −5.68500 −0.934608 −0.467304 0.884097i \(-0.654775\pi\)
−0.467304 + 0.884097i \(0.654775\pi\)
\(38\) 0 0
\(39\) 13.6136 2.17993
\(40\) 0 0
\(41\) −6.14758 −0.960090 −0.480045 0.877244i \(-0.659380\pi\)
−0.480045 + 0.877244i \(0.659380\pi\)
\(42\) 0 0
\(43\) 4.42353 0.674582 0.337291 0.941401i \(-0.390489\pi\)
0.337291 + 0.941401i \(0.390489\pi\)
\(44\) 0 0
\(45\) −2.24785 −0.335090
\(46\) 0 0
\(47\) −4.13243 −0.602776 −0.301388 0.953502i \(-0.597450\pi\)
−0.301388 + 0.953502i \(0.597450\pi\)
\(48\) 0 0
\(49\) 14.7497 2.10710
\(50\) 0 0
\(51\) −7.56730 −1.05963
\(52\) 0 0
\(53\) 8.10104 1.11276 0.556381 0.830927i \(-0.312189\pi\)
0.556381 + 0.830927i \(0.312189\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 9.60300 1.27195
\(58\) 0 0
\(59\) 6.21515 0.809144 0.404572 0.914506i \(-0.367420\pi\)
0.404572 + 0.914506i \(0.367420\pi\)
\(60\) 0 0
\(61\) −1.52758 −0.195587 −0.0977934 0.995207i \(-0.531178\pi\)
−0.0977934 + 0.995207i \(0.531178\pi\)
\(62\) 0 0
\(63\) −10.4832 −1.32076
\(64\) 0 0
\(65\) −5.94269 −0.737100
\(66\) 0 0
\(67\) 7.46447 0.911931 0.455965 0.889998i \(-0.349294\pi\)
0.455965 + 0.889998i \(0.349294\pi\)
\(68\) 0 0
\(69\) 3.92151 0.472094
\(70\) 0 0
\(71\) 2.23747 0.265539 0.132770 0.991147i \(-0.457613\pi\)
0.132770 + 0.991147i \(0.457613\pi\)
\(72\) 0 0
\(73\) −7.49400 −0.877106 −0.438553 0.898705i \(-0.644509\pi\)
−0.438553 + 0.898705i \(0.644509\pi\)
\(74\) 0 0
\(75\) 2.29082 0.264521
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −17.5316 −1.97246 −0.986230 0.165378i \(-0.947116\pi\)
−0.986230 + 0.165378i \(0.947116\pi\)
\(80\) 0 0
\(81\) −10.6907 −1.18786
\(82\) 0 0
\(83\) −11.0785 −1.21603 −0.608014 0.793926i \(-0.708034\pi\)
−0.608014 + 0.793926i \(0.708034\pi\)
\(84\) 0 0
\(85\) 3.30331 0.358295
\(86\) 0 0
\(87\) −17.7928 −1.90759
\(88\) 0 0
\(89\) 12.8171 1.35861 0.679305 0.733856i \(-0.262281\pi\)
0.679305 + 0.733856i \(0.262281\pi\)
\(90\) 0 0
\(91\) −27.7147 −2.90529
\(92\) 0 0
\(93\) 6.80840 0.705998
\(94\) 0 0
\(95\) −4.19195 −0.430085
\(96\) 0 0
\(97\) −9.53760 −0.968396 −0.484198 0.874958i \(-0.660889\pi\)
−0.484198 + 0.874958i \(0.660889\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −10.7259 −1.06727 −0.533635 0.845715i \(-0.679174\pi\)
−0.533635 + 0.845715i \(0.679174\pi\)
\(102\) 0 0
\(103\) −2.57865 −0.254082 −0.127041 0.991897i \(-0.540548\pi\)
−0.127041 + 0.991897i \(0.540548\pi\)
\(104\) 0 0
\(105\) 10.6836 1.04261
\(106\) 0 0
\(107\) 17.6297 1.70433 0.852164 0.523275i \(-0.175290\pi\)
0.852164 + 0.523275i \(0.175290\pi\)
\(108\) 0 0
\(109\) −15.3476 −1.47004 −0.735019 0.678047i \(-0.762827\pi\)
−0.735019 + 0.678047i \(0.762827\pi\)
\(110\) 0 0
\(111\) −13.0233 −1.23612
\(112\) 0 0
\(113\) 0.316934 0.0298147 0.0149073 0.999889i \(-0.495255\pi\)
0.0149073 + 0.999889i \(0.495255\pi\)
\(114\) 0 0
\(115\) −1.71184 −0.159630
\(116\) 0 0
\(117\) 13.3583 1.23497
\(118\) 0 0
\(119\) 15.4055 1.41222
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −14.0830 −1.26982
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −18.0671 −1.60320 −0.801599 0.597862i \(-0.796017\pi\)
−0.801599 + 0.597862i \(0.796017\pi\)
\(128\) 0 0
\(129\) 10.1335 0.892205
\(130\) 0 0
\(131\) 7.95176 0.694749 0.347374 0.937727i \(-0.387073\pi\)
0.347374 + 0.937727i \(0.387073\pi\)
\(132\) 0 0
\(133\) −19.5498 −1.69518
\(134\) 0 0
\(135\) 1.72304 0.148295
\(136\) 0 0
\(137\) −8.07748 −0.690106 −0.345053 0.938583i \(-0.612139\pi\)
−0.345053 + 0.938583i \(0.612139\pi\)
\(138\) 0 0
\(139\) −11.3240 −0.960486 −0.480243 0.877135i \(-0.659452\pi\)
−0.480243 + 0.877135i \(0.659452\pi\)
\(140\) 0 0
\(141\) −9.46664 −0.797235
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 7.76702 0.645016
\(146\) 0 0
\(147\) 33.7889 2.78686
\(148\) 0 0
\(149\) 2.35175 0.192663 0.0963314 0.995349i \(-0.469289\pi\)
0.0963314 + 0.995349i \(0.469289\pi\)
\(150\) 0 0
\(151\) −14.7268 −1.19845 −0.599227 0.800579i \(-0.704525\pi\)
−0.599227 + 0.800579i \(0.704525\pi\)
\(152\) 0 0
\(153\) −7.42536 −0.600305
\(154\) 0 0
\(155\) −2.97204 −0.238720
\(156\) 0 0
\(157\) −16.4544 −1.31320 −0.656600 0.754239i \(-0.728006\pi\)
−0.656600 + 0.754239i \(0.728006\pi\)
\(158\) 0 0
\(159\) 18.5580 1.47175
\(160\) 0 0
\(161\) −7.98341 −0.629181
\(162\) 0 0
\(163\) −5.26242 −0.412184 −0.206092 0.978533i \(-0.566075\pi\)
−0.206092 + 0.978533i \(0.566075\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.5495 1.66755 0.833776 0.552102i \(-0.186174\pi\)
0.833776 + 0.552102i \(0.186174\pi\)
\(168\) 0 0
\(169\) 22.3156 1.71658
\(170\) 0 0
\(171\) 9.42288 0.720586
\(172\) 0 0
\(173\) −7.08425 −0.538605 −0.269303 0.963056i \(-0.586793\pi\)
−0.269303 + 0.963056i \(0.586793\pi\)
\(174\) 0 0
\(175\) −4.66366 −0.352539
\(176\) 0 0
\(177\) 14.2378 1.07018
\(178\) 0 0
\(179\) −8.78759 −0.656815 −0.328408 0.944536i \(-0.606512\pi\)
−0.328408 + 0.944536i \(0.606512\pi\)
\(180\) 0 0
\(181\) −12.6451 −0.939904 −0.469952 0.882692i \(-0.655729\pi\)
−0.469952 + 0.882692i \(0.655729\pi\)
\(182\) 0 0
\(183\) −3.49941 −0.258684
\(184\) 0 0
\(185\) 5.68500 0.417969
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 8.03564 0.584507
\(190\) 0 0
\(191\) 17.7224 1.28235 0.641175 0.767394i \(-0.278447\pi\)
0.641175 + 0.767394i \(0.278447\pi\)
\(192\) 0 0
\(193\) −5.03540 −0.362456 −0.181228 0.983441i \(-0.558007\pi\)
−0.181228 + 0.983441i \(0.558007\pi\)
\(194\) 0 0
\(195\) −13.6136 −0.974892
\(196\) 0 0
\(197\) −15.0557 −1.07268 −0.536338 0.844003i \(-0.680193\pi\)
−0.536338 + 0.844003i \(0.680193\pi\)
\(198\) 0 0
\(199\) −2.56534 −0.181852 −0.0909259 0.995858i \(-0.528983\pi\)
−0.0909259 + 0.995858i \(0.528983\pi\)
\(200\) 0 0
\(201\) 17.0998 1.20612
\(202\) 0 0
\(203\) 36.2227 2.54233
\(204\) 0 0
\(205\) 6.14758 0.429365
\(206\) 0 0
\(207\) 3.84795 0.267451
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −9.71897 −0.669082 −0.334541 0.942381i \(-0.608581\pi\)
−0.334541 + 0.942381i \(0.608581\pi\)
\(212\) 0 0
\(213\) 5.12565 0.351204
\(214\) 0 0
\(215\) −4.42353 −0.301682
\(216\) 0 0
\(217\) −13.8606 −0.940916
\(218\) 0 0
\(219\) −17.1674 −1.16006
\(220\) 0 0
\(221\) −19.6306 −1.32050
\(222\) 0 0
\(223\) −0.543504 −0.0363957 −0.0181979 0.999834i \(-0.505793\pi\)
−0.0181979 + 0.999834i \(0.505793\pi\)
\(224\) 0 0
\(225\) 2.24785 0.149857
\(226\) 0 0
\(227\) −5.19727 −0.344955 −0.172478 0.985013i \(-0.555177\pi\)
−0.172478 + 0.985013i \(0.555177\pi\)
\(228\) 0 0
\(229\) −4.09116 −0.270352 −0.135176 0.990822i \(-0.543160\pi\)
−0.135176 + 0.990822i \(0.543160\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.9446 0.913541 0.456771 0.889584i \(-0.349006\pi\)
0.456771 + 0.889584i \(0.349006\pi\)
\(234\) 0 0
\(235\) 4.13243 0.269570
\(236\) 0 0
\(237\) −40.1618 −2.60879
\(238\) 0 0
\(239\) −0.523568 −0.0338668 −0.0169334 0.999857i \(-0.505390\pi\)
−0.0169334 + 0.999857i \(0.505390\pi\)
\(240\) 0 0
\(241\) −6.38011 −0.410979 −0.205489 0.978659i \(-0.565879\pi\)
−0.205489 + 0.978659i \(0.565879\pi\)
\(242\) 0 0
\(243\) −19.3214 −1.23947
\(244\) 0 0
\(245\) −14.7497 −0.942323
\(246\) 0 0
\(247\) 24.9115 1.58508
\(248\) 0 0
\(249\) −25.3789 −1.60832
\(250\) 0 0
\(251\) −13.6413 −0.861030 −0.430515 0.902583i \(-0.641668\pi\)
−0.430515 + 0.902583i \(0.641668\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 7.56730 0.473882
\(256\) 0 0
\(257\) −2.40705 −0.150148 −0.0750738 0.997178i \(-0.523919\pi\)
−0.0750738 + 0.997178i \(0.523919\pi\)
\(258\) 0 0
\(259\) 26.5129 1.64743
\(260\) 0 0
\(261\) −17.4591 −1.08069
\(262\) 0 0
\(263\) 11.7841 0.726641 0.363321 0.931664i \(-0.381643\pi\)
0.363321 + 0.931664i \(0.381643\pi\)
\(264\) 0 0
\(265\) −8.10104 −0.497643
\(266\) 0 0
\(267\) 29.3617 1.79691
\(268\) 0 0
\(269\) −15.1436 −0.923324 −0.461662 0.887056i \(-0.652747\pi\)
−0.461662 + 0.887056i \(0.652747\pi\)
\(270\) 0 0
\(271\) 10.5739 0.642316 0.321158 0.947026i \(-0.395928\pi\)
0.321158 + 0.947026i \(0.395928\pi\)
\(272\) 0 0
\(273\) −63.4893 −3.84255
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −25.1516 −1.51121 −0.755606 0.655026i \(-0.772658\pi\)
−0.755606 + 0.655026i \(0.772658\pi\)
\(278\) 0 0
\(279\) 6.68070 0.399963
\(280\) 0 0
\(281\) −0.322966 −0.0192666 −0.00963328 0.999954i \(-0.503066\pi\)
−0.00963328 + 0.999954i \(0.503066\pi\)
\(282\) 0 0
\(283\) −21.3723 −1.27045 −0.635227 0.772325i \(-0.719094\pi\)
−0.635227 + 0.772325i \(0.719094\pi\)
\(284\) 0 0
\(285\) −9.60300 −0.568832
\(286\) 0 0
\(287\) 28.6702 1.69235
\(288\) 0 0
\(289\) −6.08811 −0.358124
\(290\) 0 0
\(291\) −21.8489 −1.28081
\(292\) 0 0
\(293\) −10.5001 −0.613424 −0.306712 0.951802i \(-0.599229\pi\)
−0.306712 + 0.951802i \(0.599229\pi\)
\(294\) 0 0
\(295\) −6.21515 −0.361860
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10.1729 0.588315
\(300\) 0 0
\(301\) −20.6298 −1.18908
\(302\) 0 0
\(303\) −24.5712 −1.41158
\(304\) 0 0
\(305\) 1.52758 0.0874691
\(306\) 0 0
\(307\) 29.0598 1.65853 0.829266 0.558854i \(-0.188759\pi\)
0.829266 + 0.558854i \(0.188759\pi\)
\(308\) 0 0
\(309\) −5.90722 −0.336050
\(310\) 0 0
\(311\) 19.9483 1.13116 0.565582 0.824692i \(-0.308652\pi\)
0.565582 + 0.824692i \(0.308652\pi\)
\(312\) 0 0
\(313\) 17.4800 0.988030 0.494015 0.869453i \(-0.335529\pi\)
0.494015 + 0.869453i \(0.335529\pi\)
\(314\) 0 0
\(315\) 10.4832 0.590662
\(316\) 0 0
\(317\) −29.8295 −1.67539 −0.837696 0.546136i \(-0.816098\pi\)
−0.837696 + 0.546136i \(0.816098\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 40.3865 2.25415
\(322\) 0 0
\(323\) −13.8473 −0.770486
\(324\) 0 0
\(325\) 5.94269 0.329641
\(326\) 0 0
\(327\) −35.1587 −1.94428
\(328\) 0 0
\(329\) 19.2722 1.06251
\(330\) 0 0
\(331\) −29.4740 −1.62004 −0.810018 0.586405i \(-0.800543\pi\)
−0.810018 + 0.586405i \(0.800543\pi\)
\(332\) 0 0
\(333\) −12.7790 −0.700287
\(334\) 0 0
\(335\) −7.46447 −0.407828
\(336\) 0 0
\(337\) 20.9682 1.14221 0.571106 0.820877i \(-0.306515\pi\)
0.571106 + 0.820877i \(0.306515\pi\)
\(338\) 0 0
\(339\) 0.726039 0.0394330
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −36.1419 −1.95148
\(344\) 0 0
\(345\) −3.92151 −0.211127
\(346\) 0 0
\(347\) 7.27059 0.390306 0.195153 0.980773i \(-0.437480\pi\)
0.195153 + 0.980773i \(0.437480\pi\)
\(348\) 0 0
\(349\) 19.2432 1.03006 0.515032 0.857171i \(-0.327780\pi\)
0.515032 + 0.857171i \(0.327780\pi\)
\(350\) 0 0
\(351\) −10.2395 −0.546542
\(352\) 0 0
\(353\) 14.5882 0.776454 0.388227 0.921564i \(-0.373088\pi\)
0.388227 + 0.921564i \(0.373088\pi\)
\(354\) 0 0
\(355\) −2.23747 −0.118753
\(356\) 0 0
\(357\) 35.2913 1.86781
\(358\) 0 0
\(359\) −2.56617 −0.135437 −0.0677187 0.997704i \(-0.521572\pi\)
−0.0677187 + 0.997704i \(0.521572\pi\)
\(360\) 0 0
\(361\) −1.42756 −0.0751348
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 7.49400 0.392254
\(366\) 0 0
\(367\) 21.7911 1.13748 0.568742 0.822516i \(-0.307430\pi\)
0.568742 + 0.822516i \(0.307430\pi\)
\(368\) 0 0
\(369\) −13.8188 −0.719380
\(370\) 0 0
\(371\) −37.7804 −1.96146
\(372\) 0 0
\(373\) 19.7684 1.02357 0.511783 0.859115i \(-0.328985\pi\)
0.511783 + 0.859115i \(0.328985\pi\)
\(374\) 0 0
\(375\) −2.29082 −0.118297
\(376\) 0 0
\(377\) −46.1570 −2.37721
\(378\) 0 0
\(379\) 13.5393 0.695467 0.347733 0.937594i \(-0.386951\pi\)
0.347733 + 0.937594i \(0.386951\pi\)
\(380\) 0 0
\(381\) −41.3885 −2.12040
\(382\) 0 0
\(383\) −14.3937 −0.735483 −0.367742 0.929928i \(-0.619869\pi\)
−0.367742 + 0.929928i \(0.619869\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 9.94343 0.505453
\(388\) 0 0
\(389\) −15.2204 −0.771705 −0.385853 0.922560i \(-0.626093\pi\)
−0.385853 + 0.922560i \(0.626093\pi\)
\(390\) 0 0
\(391\) −5.65473 −0.285972
\(392\) 0 0
\(393\) 18.2160 0.918878
\(394\) 0 0
\(395\) 17.5316 0.882111
\(396\) 0 0
\(397\) −37.0013 −1.85704 −0.928520 0.371282i \(-0.878918\pi\)
−0.928520 + 0.371282i \(0.878918\pi\)
\(398\) 0 0
\(399\) −44.7851 −2.24206
\(400\) 0 0
\(401\) −24.6104 −1.22898 −0.614492 0.788923i \(-0.710639\pi\)
−0.614492 + 0.788923i \(0.710639\pi\)
\(402\) 0 0
\(403\) 17.6619 0.879802
\(404\) 0 0
\(405\) 10.6907 0.531226
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 18.3346 0.906586 0.453293 0.891362i \(-0.350249\pi\)
0.453293 + 0.891362i \(0.350249\pi\)
\(410\) 0 0
\(411\) −18.5040 −0.912737
\(412\) 0 0
\(413\) −28.9853 −1.42628
\(414\) 0 0
\(415\) 11.0785 0.543824
\(416\) 0 0
\(417\) −25.9411 −1.27034
\(418\) 0 0
\(419\) −23.6524 −1.15550 −0.577748 0.816215i \(-0.696068\pi\)
−0.577748 + 0.816215i \(0.696068\pi\)
\(420\) 0 0
\(421\) 11.6921 0.569840 0.284920 0.958551i \(-0.408033\pi\)
0.284920 + 0.958551i \(0.408033\pi\)
\(422\) 0 0
\(423\) −9.28908 −0.451651
\(424\) 0 0
\(425\) −3.30331 −0.160234
\(426\) 0 0
\(427\) 7.12412 0.344760
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.42118 0.261129 0.130565 0.991440i \(-0.458321\pi\)
0.130565 + 0.991440i \(0.458321\pi\)
\(432\) 0 0
\(433\) −14.0482 −0.675116 −0.337558 0.941305i \(-0.609601\pi\)
−0.337558 + 0.941305i \(0.609601\pi\)
\(434\) 0 0
\(435\) 17.7928 0.853101
\(436\) 0 0
\(437\) 7.17593 0.343271
\(438\) 0 0
\(439\) −13.4133 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(440\) 0 0
\(441\) 33.1551 1.57881
\(442\) 0 0
\(443\) 28.1807 1.33891 0.669453 0.742855i \(-0.266529\pi\)
0.669453 + 0.742855i \(0.266529\pi\)
\(444\) 0 0
\(445\) −12.8171 −0.607589
\(446\) 0 0
\(447\) 5.38743 0.254817
\(448\) 0 0
\(449\) −37.5023 −1.76984 −0.884921 0.465741i \(-0.845788\pi\)
−0.884921 + 0.465741i \(0.845788\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −33.7365 −1.58508
\(454\) 0 0
\(455\) 27.7147 1.29928
\(456\) 0 0
\(457\) 15.5343 0.726666 0.363333 0.931659i \(-0.381639\pi\)
0.363333 + 0.931659i \(0.381639\pi\)
\(458\) 0 0
\(459\) 5.69173 0.265667
\(460\) 0 0
\(461\) −20.4407 −0.952018 −0.476009 0.879440i \(-0.657917\pi\)
−0.476009 + 0.879440i \(0.657917\pi\)
\(462\) 0 0
\(463\) 9.64734 0.448350 0.224175 0.974549i \(-0.428031\pi\)
0.224175 + 0.974549i \(0.428031\pi\)
\(464\) 0 0
\(465\) −6.80840 −0.315732
\(466\) 0 0
\(467\) −13.9773 −0.646790 −0.323395 0.946264i \(-0.604824\pi\)
−0.323395 + 0.946264i \(0.604824\pi\)
\(468\) 0 0
\(469\) −34.8117 −1.60746
\(470\) 0 0
\(471\) −37.6939 −1.73685
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.19195 0.192340
\(476\) 0 0
\(477\) 18.2099 0.833775
\(478\) 0 0
\(479\) −28.7475 −1.31351 −0.656754 0.754105i \(-0.728071\pi\)
−0.656754 + 0.754105i \(0.728071\pi\)
\(480\) 0 0
\(481\) −33.7842 −1.54043
\(482\) 0 0
\(483\) −18.2886 −0.832158
\(484\) 0 0
\(485\) 9.53760 0.433080
\(486\) 0 0
\(487\) −10.2994 −0.466709 −0.233355 0.972392i \(-0.574970\pi\)
−0.233355 + 0.972392i \(0.574970\pi\)
\(488\) 0 0
\(489\) −12.0552 −0.545157
\(490\) 0 0
\(491\) 7.35859 0.332088 0.166044 0.986118i \(-0.446901\pi\)
0.166044 + 0.986118i \(0.446901\pi\)
\(492\) 0 0
\(493\) 25.6569 1.15553
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −10.4348 −0.468065
\(498\) 0 0
\(499\) −2.83645 −0.126977 −0.0634885 0.997983i \(-0.520223\pi\)
−0.0634885 + 0.997983i \(0.520223\pi\)
\(500\) 0 0
\(501\) 49.3661 2.20551
\(502\) 0 0
\(503\) 13.8353 0.616887 0.308444 0.951243i \(-0.400192\pi\)
0.308444 + 0.951243i \(0.400192\pi\)
\(504\) 0 0
\(505\) 10.7259 0.477297
\(506\) 0 0
\(507\) 51.1209 2.27036
\(508\) 0 0
\(509\) 24.9515 1.10596 0.552978 0.833196i \(-0.313491\pi\)
0.552978 + 0.833196i \(0.313491\pi\)
\(510\) 0 0
\(511\) 34.9494 1.54607
\(512\) 0 0
\(513\) −7.22288 −0.318898
\(514\) 0 0
\(515\) 2.57865 0.113629
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −16.2287 −0.712362
\(520\) 0 0
\(521\) 1.72550 0.0755955 0.0377978 0.999285i \(-0.487966\pi\)
0.0377978 + 0.999285i \(0.487966\pi\)
\(522\) 0 0
\(523\) 14.9448 0.653489 0.326745 0.945113i \(-0.394048\pi\)
0.326745 + 0.945113i \(0.394048\pi\)
\(524\) 0 0
\(525\) −10.6836 −0.466270
\(526\) 0 0
\(527\) −9.81758 −0.427660
\(528\) 0 0
\(529\) −20.0696 −0.872592
\(530\) 0 0
\(531\) 13.9707 0.606279
\(532\) 0 0
\(533\) −36.5331 −1.58243
\(534\) 0 0
\(535\) −17.6297 −0.762199
\(536\) 0 0
\(537\) −20.1308 −0.868707
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −40.0200 −1.72059 −0.860297 0.509793i \(-0.829722\pi\)
−0.860297 + 0.509793i \(0.829722\pi\)
\(542\) 0 0
\(543\) −28.9677 −1.24312
\(544\) 0 0
\(545\) 15.3476 0.657421
\(546\) 0 0
\(547\) −22.9990 −0.983368 −0.491684 0.870774i \(-0.663618\pi\)
−0.491684 + 0.870774i \(0.663618\pi\)
\(548\) 0 0
\(549\) −3.43378 −0.146550
\(550\) 0 0
\(551\) −32.5589 −1.38706
\(552\) 0 0
\(553\) 81.7614 3.47685
\(554\) 0 0
\(555\) 13.0233 0.552808
\(556\) 0 0
\(557\) 11.9261 0.505323 0.252662 0.967555i \(-0.418694\pi\)
0.252662 + 0.967555i \(0.418694\pi\)
\(558\) 0 0
\(559\) 26.2877 1.11185
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 39.2951 1.65609 0.828046 0.560660i \(-0.189453\pi\)
0.828046 + 0.560660i \(0.189453\pi\)
\(564\) 0 0
\(565\) −0.316934 −0.0133335
\(566\) 0 0
\(567\) 49.8578 2.09383
\(568\) 0 0
\(569\) −17.1895 −0.720621 −0.360310 0.932832i \(-0.617329\pi\)
−0.360310 + 0.932832i \(0.617329\pi\)
\(570\) 0 0
\(571\) 26.4297 1.10605 0.553025 0.833165i \(-0.313473\pi\)
0.553025 + 0.833165i \(0.313473\pi\)
\(572\) 0 0
\(573\) 40.5989 1.69604
\(574\) 0 0
\(575\) 1.71184 0.0713885
\(576\) 0 0
\(577\) 10.3838 0.432284 0.216142 0.976362i \(-0.430653\pi\)
0.216142 + 0.976362i \(0.430653\pi\)
\(578\) 0 0
\(579\) −11.5352 −0.479386
\(580\) 0 0
\(581\) 51.6665 2.14349
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −13.3583 −0.552297
\(586\) 0 0
\(587\) 21.2848 0.878517 0.439258 0.898361i \(-0.355241\pi\)
0.439258 + 0.898361i \(0.355241\pi\)
\(588\) 0 0
\(589\) 12.4586 0.513349
\(590\) 0 0
\(591\) −34.4900 −1.41873
\(592\) 0 0
\(593\) −6.74652 −0.277046 −0.138523 0.990359i \(-0.544236\pi\)
−0.138523 + 0.990359i \(0.544236\pi\)
\(594\) 0 0
\(595\) −15.4055 −0.631565
\(596\) 0 0
\(597\) −5.87672 −0.240518
\(598\) 0 0
\(599\) 4.09390 0.167272 0.0836361 0.996496i \(-0.473347\pi\)
0.0836361 + 0.996496i \(0.473347\pi\)
\(600\) 0 0
\(601\) 2.46697 0.100630 0.0503149 0.998733i \(-0.483978\pi\)
0.0503149 + 0.998733i \(0.483978\pi\)
\(602\) 0 0
\(603\) 16.7790 0.683295
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −27.9423 −1.13414 −0.567072 0.823668i \(-0.691924\pi\)
−0.567072 + 0.823668i \(0.691924\pi\)
\(608\) 0 0
\(609\) 82.9796 3.36250
\(610\) 0 0
\(611\) −24.5577 −0.993499
\(612\) 0 0
\(613\) −38.9146 −1.57174 −0.785872 0.618389i \(-0.787786\pi\)
−0.785872 + 0.618389i \(0.787786\pi\)
\(614\) 0 0
\(615\) 14.0830 0.567881
\(616\) 0 0
\(617\) −0.639294 −0.0257370 −0.0128685 0.999917i \(-0.504096\pi\)
−0.0128685 + 0.999917i \(0.504096\pi\)
\(618\) 0 0
\(619\) 19.2927 0.775440 0.387720 0.921777i \(-0.373263\pi\)
0.387720 + 0.921777i \(0.373263\pi\)
\(620\) 0 0
\(621\) −2.94955 −0.118361
\(622\) 0 0
\(623\) −59.7746 −2.39482
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 18.7793 0.748781
\(630\) 0 0
\(631\) −16.3865 −0.652336 −0.326168 0.945312i \(-0.605758\pi\)
−0.326168 + 0.945312i \(0.605758\pi\)
\(632\) 0 0
\(633\) −22.2644 −0.884931
\(634\) 0 0
\(635\) 18.0671 0.716972
\(636\) 0 0
\(637\) 87.6528 3.47293
\(638\) 0 0
\(639\) 5.02951 0.198964
\(640\) 0 0
\(641\) −9.13171 −0.360681 −0.180341 0.983604i \(-0.557720\pi\)
−0.180341 + 0.983604i \(0.557720\pi\)
\(642\) 0 0
\(643\) −39.5059 −1.55796 −0.778981 0.627047i \(-0.784263\pi\)
−0.778981 + 0.627047i \(0.784263\pi\)
\(644\) 0 0
\(645\) −10.1335 −0.399006
\(646\) 0 0
\(647\) −4.26785 −0.167786 −0.0838932 0.996475i \(-0.526735\pi\)
−0.0838932 + 0.996475i \(0.526735\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −31.7520 −1.24446
\(652\) 0 0
\(653\) −4.36846 −0.170951 −0.0854755 0.996340i \(-0.527241\pi\)
−0.0854755 + 0.996340i \(0.527241\pi\)
\(654\) 0 0
\(655\) −7.95176 −0.310701
\(656\) 0 0
\(657\) −16.8454 −0.657201
\(658\) 0 0
\(659\) 0.146275 0.00569808 0.00284904 0.999996i \(-0.499093\pi\)
0.00284904 + 0.999996i \(0.499093\pi\)
\(660\) 0 0
\(661\) 41.7505 1.62391 0.811953 0.583723i \(-0.198405\pi\)
0.811953 + 0.583723i \(0.198405\pi\)
\(662\) 0 0
\(663\) −44.9701 −1.74649
\(664\) 0 0
\(665\) 19.5498 0.758109
\(666\) 0 0
\(667\) −13.2959 −0.514818
\(668\) 0 0
\(669\) −1.24507 −0.0481372
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 2.58869 0.0997866 0.0498933 0.998755i \(-0.484112\pi\)
0.0498933 + 0.998755i \(0.484112\pi\)
\(674\) 0 0
\(675\) −1.72304 −0.0663197
\(676\) 0 0
\(677\) 45.0415 1.73108 0.865542 0.500836i \(-0.166974\pi\)
0.865542 + 0.500836i \(0.166974\pi\)
\(678\) 0 0
\(679\) 44.4801 1.70699
\(680\) 0 0
\(681\) −11.9060 −0.456239
\(682\) 0 0
\(683\) 34.4943 1.31989 0.659944 0.751314i \(-0.270580\pi\)
0.659944 + 0.751314i \(0.270580\pi\)
\(684\) 0 0
\(685\) 8.07748 0.308625
\(686\) 0 0
\(687\) −9.37212 −0.357569
\(688\) 0 0
\(689\) 48.1420 1.83406
\(690\) 0 0
\(691\) −22.2062 −0.844764 −0.422382 0.906418i \(-0.638806\pi\)
−0.422382 + 0.906418i \(0.638806\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 11.3240 0.429542
\(696\) 0 0
\(697\) 20.3074 0.769197
\(698\) 0 0
\(699\) 31.9446 1.20825
\(700\) 0 0
\(701\) 8.01026 0.302543 0.151272 0.988492i \(-0.451663\pi\)
0.151272 + 0.988492i \(0.451663\pi\)
\(702\) 0 0
\(703\) −23.8312 −0.898812
\(704\) 0 0
\(705\) 9.46664 0.356534
\(706\) 0 0
\(707\) 50.0220 1.88127
\(708\) 0 0
\(709\) 12.7422 0.478544 0.239272 0.970953i \(-0.423091\pi\)
0.239272 + 0.970953i \(0.423091\pi\)
\(710\) 0 0
\(711\) −39.4085 −1.47793
\(712\) 0 0
\(713\) 5.08764 0.190534
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −1.19940 −0.0447924
\(718\) 0 0
\(719\) 6.15517 0.229549 0.114775 0.993392i \(-0.463385\pi\)
0.114775 + 0.993392i \(0.463385\pi\)
\(720\) 0 0
\(721\) 12.0259 0.447869
\(722\) 0 0
\(723\) −14.6157 −0.543563
\(724\) 0 0
\(725\) −7.76702 −0.288460
\(726\) 0 0
\(727\) 9.77647 0.362589 0.181295 0.983429i \(-0.441971\pi\)
0.181295 + 0.983429i \(0.441971\pi\)
\(728\) 0 0
\(729\) −12.1897 −0.451469
\(730\) 0 0
\(731\) −14.6123 −0.540455
\(732\) 0 0
\(733\) −22.1966 −0.819849 −0.409925 0.912119i \(-0.634445\pi\)
−0.409925 + 0.912119i \(0.634445\pi\)
\(734\) 0 0
\(735\) −33.7889 −1.24632
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −20.7597 −0.763657 −0.381828 0.924233i \(-0.624705\pi\)
−0.381828 + 0.924233i \(0.624705\pi\)
\(740\) 0 0
\(741\) 57.0676 2.09643
\(742\) 0 0
\(743\) 34.6946 1.27282 0.636410 0.771351i \(-0.280419\pi\)
0.636410 + 0.771351i \(0.280419\pi\)
\(744\) 0 0
\(745\) −2.35175 −0.0861614
\(746\) 0 0
\(747\) −24.9029 −0.911150
\(748\) 0 0
\(749\) −82.2189 −3.00421
\(750\) 0 0
\(751\) 20.6367 0.753043 0.376521 0.926408i \(-0.377120\pi\)
0.376521 + 0.926408i \(0.377120\pi\)
\(752\) 0 0
\(753\) −31.2497 −1.13880
\(754\) 0 0
\(755\) 14.7268 0.535965
\(756\) 0 0
\(757\) −23.6794 −0.860642 −0.430321 0.902676i \(-0.641600\pi\)
−0.430321 + 0.902676i \(0.641600\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 46.5039 1.68576 0.842882 0.538098i \(-0.180857\pi\)
0.842882 + 0.538098i \(0.180857\pi\)
\(762\) 0 0
\(763\) 71.5761 2.59123
\(764\) 0 0
\(765\) 7.42536 0.268465
\(766\) 0 0
\(767\) 36.9347 1.33364
\(768\) 0 0
\(769\) 8.48780 0.306078 0.153039 0.988220i \(-0.451094\pi\)
0.153039 + 0.988220i \(0.451094\pi\)
\(770\) 0 0
\(771\) −5.51412 −0.198586
\(772\) 0 0
\(773\) 44.8095 1.61168 0.805842 0.592130i \(-0.201713\pi\)
0.805842 + 0.592130i \(0.201713\pi\)
\(774\) 0 0
\(775\) 2.97204 0.106759
\(776\) 0 0
\(777\) 60.7362 2.17890
\(778\) 0 0
\(779\) −25.7703 −0.923318
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 13.3828 0.478264
\(784\) 0 0
\(785\) 16.4544 0.587281
\(786\) 0 0
\(787\) −4.50207 −0.160481 −0.0802407 0.996776i \(-0.525569\pi\)
−0.0802407 + 0.996776i \(0.525569\pi\)
\(788\) 0 0
\(789\) 26.9953 0.961059
\(790\) 0 0
\(791\) −1.47807 −0.0525542
\(792\) 0 0
\(793\) −9.07795 −0.322367
\(794\) 0 0
\(795\) −18.5580 −0.658185
\(796\) 0 0
\(797\) 34.7713 1.23166 0.615832 0.787878i \(-0.288820\pi\)
0.615832 + 0.787878i \(0.288820\pi\)
\(798\) 0 0
\(799\) 13.6507 0.482927
\(800\) 0 0
\(801\) 28.8110 1.01799
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 7.98341 0.281378
\(806\) 0 0
\(807\) −34.6913 −1.22119
\(808\) 0 0
\(809\) 53.5499 1.88271 0.941357 0.337413i \(-0.109552\pi\)
0.941357 + 0.337413i \(0.109552\pi\)
\(810\) 0 0
\(811\) 17.5535 0.616387 0.308193 0.951324i \(-0.400276\pi\)
0.308193 + 0.951324i \(0.400276\pi\)
\(812\) 0 0
\(813\) 24.2228 0.849530
\(814\) 0 0
\(815\) 5.26242 0.184334
\(816\) 0 0
\(817\) 18.5432 0.648744
\(818\) 0 0
\(819\) −62.2985 −2.17688
\(820\) 0 0
\(821\) 12.3663 0.431586 0.215793 0.976439i \(-0.430766\pi\)
0.215793 + 0.976439i \(0.430766\pi\)
\(822\) 0 0
\(823\) 14.0358 0.489258 0.244629 0.969617i \(-0.421334\pi\)
0.244629 + 0.969617i \(0.421334\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 25.8085 0.897450 0.448725 0.893670i \(-0.351878\pi\)
0.448725 + 0.893670i \(0.351878\pi\)
\(828\) 0 0
\(829\) 14.0081 0.486522 0.243261 0.969961i \(-0.421783\pi\)
0.243261 + 0.969961i \(0.421783\pi\)
\(830\) 0 0
\(831\) −57.6177 −1.99874
\(832\) 0 0
\(833\) −48.7229 −1.68815
\(834\) 0 0
\(835\) −21.5495 −0.745752
\(836\) 0 0
\(837\) −5.12093 −0.177005
\(838\) 0 0
\(839\) −6.80179 −0.234824 −0.117412 0.993083i \(-0.537460\pi\)
−0.117412 + 0.993083i \(0.537460\pi\)
\(840\) 0 0
\(841\) 31.3265 1.08023
\(842\) 0 0
\(843\) −0.739857 −0.0254820
\(844\) 0 0
\(845\) −22.3156 −0.767679
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −48.9602 −1.68031
\(850\) 0 0
\(851\) −9.73178 −0.333601
\(852\) 0 0
\(853\) −46.8963 −1.60570 −0.802849 0.596183i \(-0.796683\pi\)
−0.802849 + 0.596183i \(0.796683\pi\)
\(854\) 0 0
\(855\) −9.42288 −0.322256
\(856\) 0 0
\(857\) −2.79309 −0.0954100 −0.0477050 0.998861i \(-0.515191\pi\)
−0.0477050 + 0.998861i \(0.515191\pi\)
\(858\) 0 0
\(859\) −5.83184 −0.198980 −0.0994898 0.995039i \(-0.531721\pi\)
−0.0994898 + 0.995039i \(0.531721\pi\)
\(860\) 0 0
\(861\) 65.6782 2.23831
\(862\) 0 0
\(863\) −44.8495 −1.52670 −0.763348 0.645987i \(-0.776446\pi\)
−0.763348 + 0.645987i \(0.776446\pi\)
\(864\) 0 0
\(865\) 7.08425 0.240872
\(866\) 0 0
\(867\) −13.9468 −0.473657
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 44.3591 1.50305
\(872\) 0 0
\(873\) −21.4391 −0.725604
\(874\) 0 0
\(875\) 4.66366 0.157660
\(876\) 0 0
\(877\) −15.8686 −0.535846 −0.267923 0.963440i \(-0.586337\pi\)
−0.267923 + 0.963440i \(0.586337\pi\)
\(878\) 0 0
\(879\) −24.0539 −0.811318
\(880\) 0 0
\(881\) −30.1411 −1.01548 −0.507740 0.861510i \(-0.669519\pi\)
−0.507740 + 0.861510i \(0.669519\pi\)
\(882\) 0 0
\(883\) 42.7708 1.43935 0.719676 0.694310i \(-0.244290\pi\)
0.719676 + 0.694310i \(0.244290\pi\)
\(884\) 0 0
\(885\) −14.2378 −0.478598
\(886\) 0 0
\(887\) −6.66076 −0.223647 −0.111823 0.993728i \(-0.535669\pi\)
−0.111823 + 0.993728i \(0.535669\pi\)
\(888\) 0 0
\(889\) 84.2588 2.82595
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −17.3229 −0.579689
\(894\) 0 0
\(895\) 8.78759 0.293737
\(896\) 0 0
\(897\) 23.3043 0.778108
\(898\) 0 0
\(899\) −23.0839 −0.769890
\(900\) 0 0
\(901\) −26.7603 −0.891514
\(902\) 0 0
\(903\) −47.2592 −1.57269
\(904\) 0 0
\(905\) 12.6451 0.420338
\(906\) 0 0
\(907\) 37.3310 1.23955 0.619777 0.784778i \(-0.287223\pi\)
0.619777 + 0.784778i \(0.287223\pi\)
\(908\) 0 0
\(909\) −24.1103 −0.799688
\(910\) 0 0
\(911\) 14.7247 0.487852 0.243926 0.969794i \(-0.421565\pi\)
0.243926 + 0.969794i \(0.421565\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 3.49941 0.115687
\(916\) 0 0
\(917\) −37.0843 −1.22463
\(918\) 0 0
\(919\) 34.0203 1.12222 0.561112 0.827740i \(-0.310374\pi\)
0.561112 + 0.827740i \(0.310374\pi\)
\(920\) 0 0
\(921\) 66.5708 2.19358
\(922\) 0 0
\(923\) 13.2966 0.437664
\(924\) 0 0
\(925\) −5.68500 −0.186922
\(926\) 0 0
\(927\) −5.79643 −0.190380
\(928\) 0 0
\(929\) 18.6992 0.613502 0.306751 0.951790i \(-0.400758\pi\)
0.306751 + 0.951790i \(0.400758\pi\)
\(930\) 0 0
\(931\) 61.8299 2.02639
\(932\) 0 0
\(933\) 45.6979 1.49608
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0.122357 0.00399723 0.00199862 0.999998i \(-0.499364\pi\)
0.00199862 + 0.999998i \(0.499364\pi\)
\(938\) 0 0
\(939\) 40.0436 1.30677
\(940\) 0 0
\(941\) 29.8311 0.972465 0.486233 0.873829i \(-0.338371\pi\)
0.486233 + 0.873829i \(0.338371\pi\)
\(942\) 0 0
\(943\) −10.5236 −0.342697
\(944\) 0 0
\(945\) −8.03564 −0.261399
\(946\) 0 0
\(947\) 49.0368 1.59348 0.796741 0.604320i \(-0.206555\pi\)
0.796741 + 0.604320i \(0.206555\pi\)
\(948\) 0 0
\(949\) −44.5345 −1.44565
\(950\) 0 0
\(951\) −68.3340 −2.21588
\(952\) 0 0
\(953\) 33.5129 1.08559 0.542794 0.839866i \(-0.317366\pi\)
0.542794 + 0.839866i \(0.317366\pi\)
\(954\) 0 0
\(955\) −17.7224 −0.573485
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 37.6706 1.21645
\(960\) 0 0
\(961\) −22.1670 −0.715064
\(962\) 0 0
\(963\) 39.6290 1.27703
\(964\) 0 0
\(965\) 5.03540 0.162095
\(966\) 0 0
\(967\) −52.5565 −1.69010 −0.845052 0.534685i \(-0.820430\pi\)
−0.845052 + 0.534685i \(0.820430\pi\)
\(968\) 0 0
\(969\) −31.7217 −1.01905
\(970\) 0 0
\(971\) −4.28868 −0.137630 −0.0688152 0.997629i \(-0.521922\pi\)
−0.0688152 + 0.997629i \(0.521922\pi\)
\(972\) 0 0
\(973\) 52.8111 1.69305
\(974\) 0 0
\(975\) 13.6136 0.435985
\(976\) 0 0
\(977\) −21.5573 −0.689678 −0.344839 0.938662i \(-0.612066\pi\)
−0.344839 + 0.938662i \(0.612066\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −34.4992 −1.10148
\(982\) 0 0
\(983\) −26.3667 −0.840966 −0.420483 0.907300i \(-0.638139\pi\)
−0.420483 + 0.907300i \(0.638139\pi\)
\(984\) 0 0
\(985\) 15.0557 0.479716
\(986\) 0 0
\(987\) 44.1491 1.40528
\(988\) 0 0
\(989\) 7.57235 0.240787
\(990\) 0 0
\(991\) 49.3520 1.56772 0.783858 0.620940i \(-0.213249\pi\)
0.783858 + 0.620940i \(0.213249\pi\)
\(992\) 0 0
\(993\) −67.5195 −2.14267
\(994\) 0 0
\(995\) 2.56534 0.0813266
\(996\) 0 0
\(997\) −2.83009 −0.0896300 −0.0448150 0.998995i \(-0.514270\pi\)
−0.0448150 + 0.998995i \(0.514270\pi\)
\(998\) 0 0
\(999\) 9.79545 0.309914
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.dc.1.5 6
4.3 odd 2 4840.2.a.bb.1.2 6
11.3 even 5 880.2.bo.i.801.1 12
11.4 even 5 880.2.bo.i.401.1 12
11.10 odd 2 9680.2.a.dd.1.5 6
44.3 odd 10 440.2.y.c.361.3 12
44.15 odd 10 440.2.y.c.401.3 yes 12
44.43 even 2 4840.2.a.ba.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.c.361.3 12 44.3 odd 10
440.2.y.c.401.3 yes 12 44.15 odd 10
880.2.bo.i.401.1 12 11.4 even 5
880.2.bo.i.801.1 12 11.3 even 5
4840.2.a.ba.1.2 6 44.43 even 2
4840.2.a.bb.1.2 6 4.3 odd 2
9680.2.a.dc.1.5 6 1.1 even 1 trivial
9680.2.a.dd.1.5 6 11.10 odd 2