Properties

Label 6840.2.a.bg.1.3
Level $6840$
Weight $2$
Character 6840.1
Self dual yes
Analytic conductor $54.618$
Analytic rank $1$
Dimension $3$
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] = [6840,2,Mod(1,6840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6840.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: \(54.6176749826\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 760)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.11491\) of defining polynomial
Character \(\chi\) \(=\) 6840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} +3.58774 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +3.58774 q^{7} +1.35793 q^{13} -5.58774 q^{17} +1.00000 q^{19} +4.87189 q^{23} +1.00000 q^{25} -9.58774 q^{29} -7.17548 q^{31} -3.58774 q^{35} -0.945668 q^{37} -10.4596 q^{41} -2.71585 q^{43} +5.89134 q^{47} +5.87189 q^{49} -9.81756 q^{53} -10.1560 q^{59} +3.28415 q^{61} -1.35793 q^{65} +10.3859 q^{67} -14.3510 q^{71} -4.15604 q^{73} +1.28415 q^{79} +11.1755 q^{83} +5.58774 q^{85} +6.45963 q^{89} +4.87189 q^{91} -1.00000 q^{95} -13.4053 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} - q^{7} + 5 q^{13} - 5 q^{17} + 3 q^{19} + q^{23} + 3 q^{25} - 17 q^{29} + 2 q^{31} + q^{35} + 8 q^{37} - 6 q^{41} - 10 q^{43} - 4 q^{47} + 4 q^{49} - 5 q^{53} - 15 q^{59} + 8 q^{61} - 5 q^{65} + 3 q^{67} + 4 q^{71} + 3 q^{73} + 2 q^{79} + 10 q^{83} + 5 q^{85} - 6 q^{89} + q^{91} - 3 q^{95} - 4 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.58774 1.35604 0.678019 0.735044i \(-0.262839\pi\)
0.678019 + 0.735044i \(0.262839\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 1.35793 0.376621 0.188311 0.982110i \(-0.439699\pi\)
0.188311 + 0.982110i \(0.439699\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.58774 −1.35523 −0.677613 0.735419i \(-0.736986\pi\)
−0.677613 + 0.735419i \(0.736986\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.87189 1.01586 0.507930 0.861399i \(-0.330411\pi\)
0.507930 + 0.861399i \(0.330411\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −9.58774 −1.78040 −0.890199 0.455571i \(-0.849435\pi\)
−0.890199 + 0.455571i \(0.849435\pi\)
\(30\) 0 0
\(31\) −7.17548 −1.28875 −0.644377 0.764708i \(-0.722883\pi\)
−0.644377 + 0.764708i \(0.722883\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.58774 −0.606439
\(36\) 0 0
\(37\) −0.945668 −0.155467 −0.0777334 0.996974i \(-0.524768\pi\)
−0.0777334 + 0.996974i \(0.524768\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.4596 −1.63352 −0.816760 0.576978i \(-0.804232\pi\)
−0.816760 + 0.576978i \(0.804232\pi\)
\(42\) 0 0
\(43\) −2.71585 −0.414164 −0.207082 0.978324i \(-0.566397\pi\)
−0.207082 + 0.978324i \(0.566397\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.89134 0.859340 0.429670 0.902986i \(-0.358630\pi\)
0.429670 + 0.902986i \(0.358630\pi\)
\(48\) 0 0
\(49\) 5.87189 0.838841
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.81756 −1.34855 −0.674273 0.738483i \(-0.735543\pi\)
−0.674273 + 0.738483i \(0.735543\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.1560 −1.32220 −0.661102 0.750296i \(-0.729911\pi\)
−0.661102 + 0.750296i \(0.729911\pi\)
\(60\) 0 0
\(61\) 3.28415 0.420492 0.210246 0.977649i \(-0.432574\pi\)
0.210246 + 0.977649i \(0.432574\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.35793 −0.168430
\(66\) 0 0
\(67\) 10.3859 1.26883 0.634417 0.772991i \(-0.281240\pi\)
0.634417 + 0.772991i \(0.281240\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.3510 −1.70315 −0.851573 0.524236i \(-0.824351\pi\)
−0.851573 + 0.524236i \(0.824351\pi\)
\(72\) 0 0
\(73\) −4.15604 −0.486427 −0.243214 0.969973i \(-0.578202\pi\)
−0.243214 + 0.969973i \(0.578202\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.28415 0.144478 0.0722389 0.997387i \(-0.476986\pi\)
0.0722389 + 0.997387i \(0.476986\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 11.1755 1.22667 0.613334 0.789823i \(-0.289828\pi\)
0.613334 + 0.789823i \(0.289828\pi\)
\(84\) 0 0
\(85\) 5.58774 0.606076
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.45963 0.684719 0.342360 0.939569i \(-0.388774\pi\)
0.342360 + 0.939569i \(0.388774\pi\)
\(90\) 0 0
\(91\) 4.87189 0.510713
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −13.4053 −1.36110 −0.680551 0.732701i \(-0.738259\pi\)
−0.680551 + 0.732701i \(0.738259\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 19.0668 1.89722 0.948610 0.316449i \(-0.102490\pi\)
0.948610 + 0.316449i \(0.102490\pi\)
\(102\) 0 0
\(103\) −9.05433 −0.892150 −0.446075 0.894996i \(-0.647179\pi\)
−0.446075 + 0.894996i \(0.647179\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.24926 0.507465 0.253733 0.967274i \(-0.418342\pi\)
0.253733 + 0.967274i \(0.418342\pi\)
\(108\) 0 0
\(109\) −6.04737 −0.579233 −0.289617 0.957143i \(-0.593528\pi\)
−0.289617 + 0.957143i \(0.593528\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.22982 0.586052 0.293026 0.956105i \(-0.405338\pi\)
0.293026 + 0.956105i \(0.405338\pi\)
\(114\) 0 0
\(115\) −4.87189 −0.454306
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −20.0474 −1.83774
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −14.1212 −1.25305 −0.626525 0.779402i \(-0.715523\pi\)
−0.626525 + 0.779402i \(0.715523\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 18.3510 1.60333 0.801666 0.597773i \(-0.203947\pi\)
0.801666 + 0.597773i \(0.203947\pi\)
\(132\) 0 0
\(133\) 3.58774 0.311097
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −3.01945 −0.257969 −0.128984 0.991647i \(-0.541172\pi\)
−0.128984 + 0.991647i \(0.541172\pi\)
\(138\) 0 0
\(139\) 11.7827 0.999393 0.499697 0.866201i \(-0.333445\pi\)
0.499697 + 0.866201i \(0.333445\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 9.58774 0.796219
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −20.4985 −1.67930 −0.839652 0.543124i \(-0.817241\pi\)
−0.839652 + 0.543124i \(0.817241\pi\)
\(150\) 0 0
\(151\) −3.85244 −0.313507 −0.156754 0.987638i \(-0.550103\pi\)
−0.156754 + 0.987638i \(0.550103\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 7.17548 0.576349
\(156\) 0 0
\(157\) 3.89134 0.310562 0.155281 0.987870i \(-0.450372\pi\)
0.155281 + 0.987870i \(0.450372\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 17.4791 1.37754
\(162\) 0 0
\(163\) −18.7159 −1.46594 −0.732969 0.680262i \(-0.761866\pi\)
−0.732969 + 0.680262i \(0.761866\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.09323 0.239361 0.119681 0.992812i \(-0.461813\pi\)
0.119681 + 0.992812i \(0.461813\pi\)
\(168\) 0 0
\(169\) −11.1560 −0.858157
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −7.51396 −0.571276 −0.285638 0.958338i \(-0.592205\pi\)
−0.285638 + 0.958338i \(0.592205\pi\)
\(174\) 0 0
\(175\) 3.58774 0.271208
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.45963 0.333328 0.166664 0.986014i \(-0.446700\pi\)
0.166664 + 0.986014i \(0.446700\pi\)
\(180\) 0 0
\(181\) 8.56829 0.636876 0.318438 0.947944i \(-0.396842\pi\)
0.318438 + 0.947944i \(0.396842\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.945668 0.0695269
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −7.73530 −0.559707 −0.279853 0.960043i \(-0.590286\pi\)
−0.279853 + 0.960043i \(0.590286\pi\)
\(192\) 0 0
\(193\) 6.08226 0.437810 0.218905 0.975746i \(-0.429751\pi\)
0.218905 + 0.975746i \(0.429751\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) −6.91078 −0.489892 −0.244946 0.969537i \(-0.578770\pi\)
−0.244946 + 0.969537i \(0.578770\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −34.3983 −2.41429
\(204\) 0 0
\(205\) 10.4596 0.730532
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −12.1949 −0.839534 −0.419767 0.907632i \(-0.637888\pi\)
−0.419767 + 0.907632i \(0.637888\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.71585 0.185220
\(216\) 0 0
\(217\) −25.7438 −1.74760
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −7.58774 −0.510407
\(222\) 0 0
\(223\) −1.87885 −0.125817 −0.0629085 0.998019i \(-0.520038\pi\)
−0.0629085 + 0.998019i \(0.520038\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −7.35793 −0.488363 −0.244181 0.969730i \(-0.578519\pi\)
−0.244181 + 0.969730i \(0.578519\pi\)
\(228\) 0 0
\(229\) 13.0279 0.860909 0.430455 0.902612i \(-0.358353\pi\)
0.430455 + 0.902612i \(0.358353\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 26.7019 1.74930 0.874651 0.484753i \(-0.161091\pi\)
0.874651 + 0.484753i \(0.161091\pi\)
\(234\) 0 0
\(235\) −5.89134 −0.384308
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 9.47908 0.613151 0.306575 0.951846i \(-0.400817\pi\)
0.306575 + 0.951846i \(0.400817\pi\)
\(240\) 0 0
\(241\) −24.2034 −1.55908 −0.779539 0.626353i \(-0.784547\pi\)
−0.779539 + 0.626353i \(0.784547\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −5.87189 −0.375141
\(246\) 0 0
\(247\) 1.35793 0.0864028
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −9.43171 −0.595324 −0.297662 0.954671i \(-0.596207\pi\)
−0.297662 + 0.954671i \(0.596207\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −25.1102 −1.56633 −0.783165 0.621814i \(-0.786396\pi\)
−0.783165 + 0.621814i \(0.786396\pi\)
\(258\) 0 0
\(259\) −3.39281 −0.210819
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 28.7019 1.76984 0.884918 0.465746i \(-0.154214\pi\)
0.884918 + 0.465746i \(0.154214\pi\)
\(264\) 0 0
\(265\) 9.81756 0.603088
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.350966 0.0213988 0.0106994 0.999943i \(-0.496594\pi\)
0.0106994 + 0.999943i \(0.496594\pi\)
\(270\) 0 0
\(271\) 16.8719 1.02489 0.512447 0.858719i \(-0.328739\pi\)
0.512447 + 0.858719i \(0.328739\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −6.14756 −0.369371 −0.184685 0.982798i \(-0.559127\pi\)
−0.184685 + 0.982798i \(0.559127\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 17.0279 1.01580 0.507900 0.861416i \(-0.330422\pi\)
0.507900 + 0.861416i \(0.330422\pi\)
\(282\) 0 0
\(283\) 5.74378 0.341432 0.170716 0.985320i \(-0.445392\pi\)
0.170716 + 0.985320i \(0.445392\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −37.5264 −2.21512
\(288\) 0 0
\(289\) 14.2229 0.836639
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 14.2772 0.834082 0.417041 0.908888i \(-0.363067\pi\)
0.417041 + 0.908888i \(0.363067\pi\)
\(294\) 0 0
\(295\) 10.1560 0.591307
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.61567 0.382594
\(300\) 0 0
\(301\) −9.74378 −0.561622
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3.28415 −0.188050
\(306\) 0 0
\(307\) 8.83700 0.504354 0.252177 0.967681i \(-0.418853\pi\)
0.252177 + 0.967681i \(0.418853\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −23.2229 −1.31685 −0.658424 0.752648i \(-0.728776\pi\)
−0.658424 + 0.752648i \(0.728776\pi\)
\(312\) 0 0
\(313\) 30.1421 1.70373 0.851867 0.523759i \(-0.175471\pi\)
0.851867 + 0.523759i \(0.175471\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.96511 −0.335034 −0.167517 0.985869i \(-0.553575\pi\)
−0.167517 + 0.985869i \(0.553575\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.58774 −0.310910
\(324\) 0 0
\(325\) 1.35793 0.0753242
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 21.1366 1.16530
\(330\) 0 0
\(331\) −2.15604 −0.118506 −0.0592532 0.998243i \(-0.518872\pi\)
−0.0592532 + 0.998243i \(0.518872\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −10.3859 −0.567440
\(336\) 0 0
\(337\) 0.945668 0.0515138 0.0257569 0.999668i \(-0.491800\pi\)
0.0257569 + 0.999668i \(0.491800\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −4.04737 −0.218538
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −10.2562 −0.550583 −0.275291 0.961361i \(-0.588774\pi\)
−0.275291 + 0.961361i \(0.588774\pi\)
\(348\) 0 0
\(349\) −24.0558 −1.28768 −0.643840 0.765160i \(-0.722660\pi\)
−0.643840 + 0.765160i \(0.722660\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −15.3315 −0.816014 −0.408007 0.912979i \(-0.633776\pi\)
−0.408007 + 0.912979i \(0.633776\pi\)
\(354\) 0 0
\(355\) 14.3510 0.761670
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −15.1281 −0.798431 −0.399216 0.916857i \(-0.630718\pi\)
−0.399216 + 0.916857i \(0.630718\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.15604 0.217537
\(366\) 0 0
\(367\) −33.6740 −1.75777 −0.878884 0.477035i \(-0.841712\pi\)
−0.878884 + 0.477035i \(0.841712\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −35.2229 −1.82868
\(372\) 0 0
\(373\) 16.0738 0.832269 0.416134 0.909303i \(-0.363385\pi\)
0.416134 + 0.909303i \(0.363385\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −13.0194 −0.670536
\(378\) 0 0
\(379\) −6.00848 −0.308635 −0.154317 0.988021i \(-0.549318\pi\)
−0.154317 + 0.988021i \(0.549318\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 15.8649 0.810660 0.405330 0.914170i \(-0.367157\pi\)
0.405330 + 0.914170i \(0.367157\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −15.7438 −0.798241 −0.399121 0.916898i \(-0.630685\pi\)
−0.399121 + 0.916898i \(0.630685\pi\)
\(390\) 0 0
\(391\) −27.2229 −1.37672
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.28415 −0.0646125
\(396\) 0 0
\(397\) −23.9861 −1.20383 −0.601913 0.798561i \(-0.705595\pi\)
−0.601913 + 0.798561i \(0.705595\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 24.5683 1.22688 0.613441 0.789741i \(-0.289785\pi\)
0.613441 + 0.789741i \(0.289785\pi\)
\(402\) 0 0
\(403\) −9.74378 −0.485372
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −14.9193 −0.737710 −0.368855 0.929487i \(-0.620250\pi\)
−0.368855 + 0.929487i \(0.620250\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −36.4372 −1.79296
\(414\) 0 0
\(415\) −11.1755 −0.548583
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −14.0389 −0.685845 −0.342922 0.939364i \(-0.611417\pi\)
−0.342922 + 0.939364i \(0.611417\pi\)
\(420\) 0 0
\(421\) 39.9387 1.94649 0.973247 0.229762i \(-0.0737949\pi\)
0.973247 + 0.229762i \(0.0737949\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5.58774 −0.271045
\(426\) 0 0
\(427\) 11.7827 0.570203
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −27.0279 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(432\) 0 0
\(433\) −40.0683 −1.92556 −0.962781 0.270284i \(-0.912882\pi\)
−0.962781 + 0.270284i \(0.912882\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.87189 0.233054
\(438\) 0 0
\(439\) 6.42074 0.306445 0.153223 0.988192i \(-0.451035\pi\)
0.153223 + 0.988192i \(0.451035\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 35.2702 1.67574 0.837870 0.545871i \(-0.183801\pi\)
0.837870 + 0.545871i \(0.183801\pi\)
\(444\) 0 0
\(445\) −6.45963 −0.306216
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −5.78267 −0.272901 −0.136451 0.990647i \(-0.543569\pi\)
−0.136451 + 0.990647i \(0.543569\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4.87189 −0.228398
\(456\) 0 0
\(457\) −9.12811 −0.426995 −0.213498 0.976944i \(-0.568486\pi\)
−0.213498 + 0.976944i \(0.568486\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −8.86341 −0.412810 −0.206405 0.978467i \(-0.566176\pi\)
−0.206405 + 0.978467i \(0.566176\pi\)
\(462\) 0 0
\(463\) −21.1366 −0.982301 −0.491150 0.871075i \(-0.663423\pi\)
−0.491150 + 0.871075i \(0.663423\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7.78267 −0.360139 −0.180070 0.983654i \(-0.557632\pi\)
−0.180070 + 0.983654i \(0.557632\pi\)
\(468\) 0 0
\(469\) 37.2617 1.72059
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −40.4068 −1.84623 −0.923117 0.384519i \(-0.874367\pi\)
−0.923117 + 0.384519i \(0.874367\pi\)
\(480\) 0 0
\(481\) −1.28415 −0.0585521
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 13.4053 0.608703
\(486\) 0 0
\(487\) 5.82603 0.264003 0.132001 0.991250i \(-0.457860\pi\)
0.132001 + 0.991250i \(0.457860\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −22.6630 −1.02277 −0.511384 0.859352i \(-0.670867\pi\)
−0.511384 + 0.859352i \(0.670867\pi\)
\(492\) 0 0
\(493\) 53.5738 2.41284
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −51.4876 −2.30953
\(498\) 0 0
\(499\) −25.7438 −1.15245 −0.576225 0.817291i \(-0.695475\pi\)
−0.576225 + 0.817291i \(0.695475\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −25.4791 −1.13606 −0.568028 0.823009i \(-0.692293\pi\)
−0.568028 + 0.823009i \(0.692293\pi\)
\(504\) 0 0
\(505\) −19.0668 −0.848462
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 9.54037 0.422869 0.211435 0.977392i \(-0.432186\pi\)
0.211435 + 0.977392i \(0.432186\pi\)
\(510\) 0 0
\(511\) −14.9108 −0.659614
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 9.05433 0.398982
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.31207 −0.276537 −0.138268 0.990395i \(-0.544154\pi\)
−0.138268 + 0.990395i \(0.544154\pi\)
\(522\) 0 0
\(523\) 8.93719 0.390796 0.195398 0.980724i \(-0.437400\pi\)
0.195398 + 0.980724i \(0.437400\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 40.0947 1.74655
\(528\) 0 0
\(529\) 0.735300 0.0319696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −14.2034 −0.615218
\(534\) 0 0
\(535\) −5.24926 −0.226945
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.71585 0.374724 0.187362 0.982291i \(-0.440006\pi\)
0.187362 + 0.982291i \(0.440006\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.04737 0.259041
\(546\) 0 0
\(547\) −26.9457 −1.15211 −0.576057 0.817410i \(-0.695409\pi\)
−0.576057 + 0.817410i \(0.695409\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −9.58774 −0.408452
\(552\) 0 0
\(553\) 4.60719 0.195918
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 23.3619 0.989877 0.494938 0.868928i \(-0.335191\pi\)
0.494938 + 0.868928i \(0.335191\pi\)
\(558\) 0 0
\(559\) −3.68793 −0.155983
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −31.6476 −1.33379 −0.666894 0.745153i \(-0.732376\pi\)
−0.666894 + 0.745153i \(0.732376\pi\)
\(564\) 0 0
\(565\) −6.22982 −0.262090
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 20.8804 0.875350 0.437675 0.899133i \(-0.355802\pi\)
0.437675 + 0.899133i \(0.355802\pi\)
\(570\) 0 0
\(571\) 42.6630 1.78539 0.892696 0.450659i \(-0.148811\pi\)
0.892696 + 0.450659i \(0.148811\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.87189 0.203172
\(576\) 0 0
\(577\) −8.59871 −0.357969 −0.178985 0.983852i \(-0.557281\pi\)
−0.178985 + 0.983852i \(0.557281\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 40.0947 1.66341
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −29.7438 −1.22766 −0.613829 0.789439i \(-0.710371\pi\)
−0.613829 + 0.789439i \(0.710371\pi\)
\(588\) 0 0
\(589\) −7.17548 −0.295661
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −4.56829 −0.187597 −0.0937987 0.995591i \(-0.529901\pi\)
−0.0937987 + 0.995591i \(0.529901\pi\)
\(594\) 0 0
\(595\) 20.0474 0.821862
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −14.7159 −0.601273 −0.300637 0.953739i \(-0.597199\pi\)
−0.300637 + 0.953739i \(0.597199\pi\)
\(600\) 0 0
\(601\) 37.3230 1.52244 0.761219 0.648495i \(-0.224601\pi\)
0.761219 + 0.648495i \(0.224601\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 11.0000 0.447214
\(606\) 0 0
\(607\) 23.0404 0.935181 0.467591 0.883945i \(-0.345122\pi\)
0.467591 + 0.883945i \(0.345122\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) 32.4985 1.31260 0.656302 0.754499i \(-0.272120\pi\)
0.656302 + 0.754499i \(0.272120\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 32.2982 1.30027 0.650137 0.759817i \(-0.274711\pi\)
0.650137 + 0.759817i \(0.274711\pi\)
\(618\) 0 0
\(619\) −34.5683 −1.38942 −0.694709 0.719291i \(-0.744467\pi\)
−0.694709 + 0.719291i \(0.744467\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 23.1755 0.928506
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.28415 0.210693
\(630\) 0 0
\(631\) −19.2702 −0.767136 −0.383568 0.923513i \(-0.625305\pi\)
−0.383568 + 0.923513i \(0.625305\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 14.1212 0.560381
\(636\) 0 0
\(637\) 7.97359 0.315925
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.06682 0.121132 0.0605660 0.998164i \(-0.480709\pi\)
0.0605660 + 0.998164i \(0.480709\pi\)
\(642\) 0 0
\(643\) −25.5264 −1.00666 −0.503332 0.864093i \(-0.667893\pi\)
−0.503332 + 0.864093i \(0.667893\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −25.4791 −1.00169 −0.500843 0.865538i \(-0.666977\pi\)
−0.500843 + 0.865538i \(0.666977\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 36.2034 1.41675 0.708374 0.705837i \(-0.249429\pi\)
0.708374 + 0.705837i \(0.249429\pi\)
\(654\) 0 0
\(655\) −18.3510 −0.717032
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 42.6406 1.66104 0.830522 0.556986i \(-0.188042\pi\)
0.830522 + 0.556986i \(0.188042\pi\)
\(660\) 0 0
\(661\) 24.4681 0.951699 0.475850 0.879527i \(-0.342141\pi\)
0.475850 + 0.879527i \(0.342141\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.58774 −0.139127
\(666\) 0 0
\(667\) −46.7104 −1.80863
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −26.7672 −1.03180 −0.515901 0.856649i \(-0.672543\pi\)
−0.515901 + 0.856649i \(0.672543\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5.65304 0.217264 0.108632 0.994082i \(-0.465353\pi\)
0.108632 + 0.994082i \(0.465353\pi\)
\(678\) 0 0
\(679\) −48.0947 −1.84571
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 11.1102 0.425119 0.212560 0.977148i \(-0.431820\pi\)
0.212560 + 0.977148i \(0.431820\pi\)
\(684\) 0 0
\(685\) 3.01945 0.115367
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −13.3315 −0.507890
\(690\) 0 0
\(691\) −5.96111 −0.226771 −0.113386 0.993551i \(-0.536170\pi\)
−0.113386 + 0.993551i \(0.536170\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −11.7827 −0.446942
\(696\) 0 0
\(697\) 58.4457 2.21379
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 25.8524 0.976433 0.488217 0.872722i \(-0.337648\pi\)
0.488217 + 0.872722i \(0.337648\pi\)
\(702\) 0 0
\(703\) −0.945668 −0.0356665
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 68.4068 2.57270
\(708\) 0 0
\(709\) −23.4317 −0.879996 −0.439998 0.897999i \(-0.645021\pi\)
−0.439998 + 0.897999i \(0.645021\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −34.9582 −1.30919
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 6.20885 0.231551 0.115776 0.993275i \(-0.463065\pi\)
0.115776 + 0.993275i \(0.463065\pi\)
\(720\) 0 0
\(721\) −32.4846 −1.20979
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −9.58774 −0.356080
\(726\) 0 0
\(727\) 29.7214 1.10230 0.551152 0.834405i \(-0.314188\pi\)
0.551152 + 0.834405i \(0.314188\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 15.1755 0.561286
\(732\) 0 0
\(733\) −7.52645 −0.277996 −0.138998 0.990293i \(-0.544388\pi\)
−0.138998 + 0.990293i \(0.544388\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 22.9582 0.844529 0.422265 0.906473i \(-0.361235\pi\)
0.422265 + 0.906473i \(0.361235\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −0.524931 −0.0192579 −0.00962893 0.999954i \(-0.503065\pi\)
−0.00962893 + 0.999954i \(0.503065\pi\)
\(744\) 0 0
\(745\) 20.4985 0.751008
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 18.8330 0.688143
\(750\) 0 0
\(751\) 21.3619 0.779508 0.389754 0.920919i \(-0.372560\pi\)
0.389754 + 0.920919i \(0.372560\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 3.85244 0.140205
\(756\) 0 0
\(757\) 50.4068 1.83207 0.916033 0.401102i \(-0.131373\pi\)
0.916033 + 0.401102i \(0.131373\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 20.2338 0.733476 0.366738 0.930324i \(-0.380475\pi\)
0.366738 + 0.930324i \(0.380475\pi\)
\(762\) 0 0
\(763\) −21.6964 −0.785463
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −13.7911 −0.497970
\(768\) 0 0
\(769\) −27.1142 −0.977763 −0.488881 0.872350i \(-0.662595\pi\)
−0.488881 + 0.872350i \(0.662595\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 12.1685 0.437671 0.218836 0.975762i \(-0.429774\pi\)
0.218836 + 0.975762i \(0.429774\pi\)
\(774\) 0 0
\(775\) −7.17548 −0.257751
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −10.4596 −0.374755
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.89134 −0.138888
\(786\) 0 0
\(787\) −36.1296 −1.28788 −0.643941 0.765075i \(-0.722702\pi\)
−0.643941 + 0.765075i \(0.722702\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 22.3510 0.794709
\(792\) 0 0
\(793\) 4.45963 0.158366
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −42.4417 −1.50336 −0.751681 0.659527i \(-0.770757\pi\)
−0.751681 + 0.659527i \(0.770757\pi\)
\(798\) 0 0
\(799\) −32.9193 −1.16460
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −17.4791 −0.616057
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 37.3006 1.31142 0.655710 0.755013i \(-0.272369\pi\)
0.655710 + 0.755013i \(0.272369\pi\)
\(810\) 0 0
\(811\) 27.4402 0.963555 0.481778 0.876294i \(-0.339991\pi\)
0.481778 + 0.876294i \(0.339991\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 18.7159 0.655588
\(816\) 0 0
\(817\) −2.71585 −0.0950157
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −47.0140 −1.64080 −0.820400 0.571790i \(-0.806249\pi\)
−0.820400 + 0.571790i \(0.806249\pi\)
\(822\) 0 0
\(823\) 35.9776 1.25410 0.627050 0.778979i \(-0.284262\pi\)
0.627050 + 0.778979i \(0.284262\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −14.6032 −0.507802 −0.253901 0.967230i \(-0.581714\pi\)
−0.253901 + 0.967230i \(0.581714\pi\)
\(828\) 0 0
\(829\) −6.96663 −0.241961 −0.120981 0.992655i \(-0.538604\pi\)
−0.120981 + 0.992655i \(0.538604\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −32.8106 −1.13682
\(834\) 0 0
\(835\) −3.09323 −0.107046
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0.459630 0.0158682 0.00793410 0.999969i \(-0.497474\pi\)
0.00793410 + 0.999969i \(0.497474\pi\)
\(840\) 0 0
\(841\) 62.9248 2.16982
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 11.1560 0.383779
\(846\) 0 0
\(847\) −39.4652 −1.35604
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −4.60719 −0.157932
\(852\) 0 0
\(853\) −5.48755 −0.187890 −0.0939451 0.995577i \(-0.529948\pi\)
−0.0939451 + 0.995577i \(0.529948\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −52.6586 −1.79878 −0.899391 0.437145i \(-0.855990\pi\)
−0.899391 + 0.437145i \(0.855990\pi\)
\(858\) 0 0
\(859\) −4.29512 −0.146547 −0.0732737 0.997312i \(-0.523345\pi\)
−0.0732737 + 0.997312i \(0.523345\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 14.6506 0.498711 0.249355 0.968412i \(-0.419781\pi\)
0.249355 + 0.968412i \(0.419781\pi\)
\(864\) 0 0
\(865\) 7.51396 0.255482
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 14.1032 0.477869
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.58774 −0.121288
\(876\) 0 0
\(877\) −19.7089 −0.665522 −0.332761 0.943011i \(-0.607980\pi\)
−0.332761 + 0.943011i \(0.607980\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.32304 −0.0445744 −0.0222872 0.999752i \(-0.507095\pi\)
−0.0222872 + 0.999752i \(0.507095\pi\)
\(882\) 0 0
\(883\) −25.0668 −0.843566 −0.421783 0.906697i \(-0.638596\pi\)
−0.421783 + 0.906697i \(0.638596\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 36.5669 1.22779 0.613897 0.789386i \(-0.289601\pi\)
0.613897 + 0.789386i \(0.289601\pi\)
\(888\) 0 0
\(889\) −50.6630 −1.69918
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5.89134 0.197146
\(894\) 0 0
\(895\) −4.45963 −0.149069
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 68.7967 2.29450
\(900\) 0 0
\(901\) 54.8580 1.82758
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8.56829 −0.284820
\(906\) 0 0
\(907\) −40.7368 −1.35264 −0.676322 0.736606i \(-0.736427\pi\)
−0.676322 + 0.736606i \(0.736427\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 58.2982 1.93150 0.965752 0.259467i \(-0.0835469\pi\)
0.965752 + 0.259467i \(0.0835469\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 65.8385 2.17418
\(918\) 0 0
\(919\) −14.5209 −0.479001 −0.239501 0.970896i \(-0.576984\pi\)
−0.239501 + 0.970896i \(0.576984\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −19.4876 −0.641441
\(924\) 0 0
\(925\) −0.945668 −0.0310934
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −13.7523 −0.451197 −0.225598 0.974220i \(-0.572434\pi\)
−0.225598 + 0.974220i \(0.572434\pi\)
\(930\) 0 0
\(931\) 5.87189 0.192443
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −12.4512 −0.406761 −0.203381 0.979100i \(-0.565193\pi\)
−0.203381 + 0.979100i \(0.565193\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 34.3594 1.12009 0.560043 0.828464i \(-0.310785\pi\)
0.560043 + 0.828464i \(0.310785\pi\)
\(942\) 0 0
\(943\) −50.9582 −1.65943
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −23.4876 −0.763243 −0.381621 0.924319i \(-0.624634\pi\)
−0.381621 + 0.924319i \(0.624634\pi\)
\(948\) 0 0
\(949\) −5.64359 −0.183199
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −48.5808 −1.57369 −0.786843 0.617153i \(-0.788286\pi\)
−0.786843 + 0.617153i \(0.788286\pi\)
\(954\) 0 0
\(955\) 7.73530 0.250308
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −10.8330 −0.349816
\(960\) 0 0
\(961\) 20.4876 0.660889
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −6.08226 −0.195795
\(966\) 0 0
\(967\) 53.3261 1.71485 0.857425 0.514608i \(-0.172063\pi\)
0.857425 + 0.514608i \(0.172063\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 33.1366 1.06340 0.531702 0.846932i \(-0.321553\pi\)
0.531702 + 0.846932i \(0.321553\pi\)
\(972\) 0 0
\(973\) 42.2732 1.35522
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −5.62263 −0.179884 −0.0899419 0.995947i \(-0.528668\pi\)
−0.0899419 + 0.995947i \(0.528668\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −21.3664 −0.681482 −0.340741 0.940157i \(-0.610678\pi\)
−0.340741 + 0.940157i \(0.610678\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −13.2313 −0.420732
\(990\) 0 0
\(991\) 41.1446 1.30700 0.653501 0.756926i \(-0.273300\pi\)
0.653501 + 0.756926i \(0.273300\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 6.91078 0.219087
\(996\) 0 0
\(997\) −48.5155 −1.53650 −0.768250 0.640150i \(-0.778872\pi\)
−0.768250 + 0.640150i \(0.778872\pi\)
\(998\) 0 0
\(999\) 0 0
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 6840.2.a.bg.1.3 3
3.2 odd 2 760.2.a.j.1.2 3
12.11 even 2 1520.2.a.s.1.2 3
15.2 even 4 3800.2.d.l.3649.4 6
15.8 even 4 3800.2.d.l.3649.3 6
15.14 odd 2 3800.2.a.x.1.2 3
24.5 odd 2 6080.2.a.bv.1.2 3
24.11 even 2 6080.2.a.bq.1.2 3
60.59 even 2 7600.2.a.bq.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.j.1.2 3 3.2 odd 2
1520.2.a.s.1.2 3 12.11 even 2
3800.2.a.x.1.2 3 15.14 odd 2
3800.2.d.l.3649.3 6 15.8 even 4
3800.2.d.l.3649.4 6 15.2 even 4
6080.2.a.bq.1.2 3 24.11 even 2
6080.2.a.bv.1.2 3 24.5 odd 2
6840.2.a.bg.1.3 3 1.1 even 1 trivial
7600.2.a.bq.1.2 3 60.59 even 2