Properties

Label 6840.2.a.bg.1.2
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 1.2
Root \(-1.86081\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{5} -1.39821 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -1.39821 q^{7} +4.32340 q^{13} -0.601793 q^{17} +1.00000 q^{19} -6.04502 q^{23} +1.00000 q^{25} -4.60179 q^{29} +2.79641 q^{31} +1.39821 q^{35} +1.07480 q^{37} +5.44322 q^{41} -8.64681 q^{43} +1.85039 q^{47} -5.04502 q^{49} +3.11982 q^{53} +6.69182 q^{59} -2.64681 q^{61} -4.32340 q^{65} -14.4134 q^{67} +5.59283 q^{71} +12.6918 q^{73} -4.64681 q^{79} +1.20359 q^{83} +0.601793 q^{85} -9.44322 q^{89} -6.04502 q^{91} -1.00000 q^{95} +4.51803 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} - q^{7}+O(q^{10}) \) 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
\(7\) −1.39821 −0.528473 −0.264236 0.964458i \(-0.585120\pi\)
−0.264236 + 0.964458i \(0.585120\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 4.32340 1.19910 0.599548 0.800339i \(-0.295347\pi\)
0.599548 + 0.800339i \(0.295347\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.601793 −0.145956 −0.0729781 0.997334i \(-0.523250\pi\)
−0.0729781 + 0.997334i \(0.523250\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.04502 −1.26047 −0.630236 0.776403i \(-0.717042\pi\)
−0.630236 + 0.776403i \(0.717042\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.60179 −0.854531 −0.427266 0.904126i \(-0.640523\pi\)
−0.427266 + 0.904126i \(0.640523\pi\)
\(30\) 0 0
\(31\) 2.79641 0.502251 0.251125 0.967955i \(-0.419199\pi\)
0.251125 + 0.967955i \(0.419199\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.39821 0.236340
\(36\) 0 0
\(37\) 1.07480 0.176697 0.0883483 0.996090i \(-0.471841\pi\)
0.0883483 + 0.996090i \(0.471841\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.44322 0.850089 0.425044 0.905173i \(-0.360259\pi\)
0.425044 + 0.905173i \(0.360259\pi\)
\(42\) 0 0
\(43\) −8.64681 −1.31863 −0.659313 0.751869i \(-0.729153\pi\)
−0.659313 + 0.751869i \(0.729153\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.85039 0.269908 0.134954 0.990852i \(-0.456911\pi\)
0.134954 + 0.990852i \(0.456911\pi\)
\(48\) 0 0
\(49\) −5.04502 −0.720717
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.11982 0.428540 0.214270 0.976774i \(-0.431263\pi\)
0.214270 + 0.976774i \(0.431263\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.69182 0.871201 0.435601 0.900140i \(-0.356536\pi\)
0.435601 + 0.900140i \(0.356536\pi\)
\(60\) 0 0
\(61\) −2.64681 −0.338889 −0.169445 0.985540i \(-0.554197\pi\)
−0.169445 + 0.985540i \(0.554197\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.32340 −0.536252
\(66\) 0 0
\(67\) −14.4134 −1.76088 −0.880441 0.474156i \(-0.842753\pi\)
−0.880441 + 0.474156i \(0.842753\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.59283 0.663747 0.331873 0.943324i \(-0.392319\pi\)
0.331873 + 0.943324i \(0.392319\pi\)
\(72\) 0 0
\(73\) 12.6918 1.48547 0.742733 0.669588i \(-0.233529\pi\)
0.742733 + 0.669588i \(0.233529\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.64681 −0.522807 −0.261403 0.965230i \(-0.584185\pi\)
−0.261403 + 0.965230i \(0.584185\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.20359 0.132111 0.0660553 0.997816i \(-0.478959\pi\)
0.0660553 + 0.997816i \(0.478959\pi\)
\(84\) 0 0
\(85\) 0.601793 0.0652736
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.44322 −1.00098 −0.500490 0.865742i \(-0.666847\pi\)
−0.500490 + 0.865742i \(0.666847\pi\)
\(90\) 0 0
\(91\) −6.04502 −0.633690
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) 4.51803 0.458736 0.229368 0.973340i \(-0.426334\pi\)
0.229368 + 0.973340i \(0.426334\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.05398 0.502890 0.251445 0.967872i \(-0.419094\pi\)
0.251445 + 0.967872i \(0.419094\pi\)
\(102\) 0 0
\(103\) −11.0748 −1.09123 −0.545616 0.838035i \(-0.683704\pi\)
−0.545616 + 0.838035i \(0.683704\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.17380 0.403496 0.201748 0.979437i \(-0.435338\pi\)
0.201748 + 0.979437i \(0.435338\pi\)
\(108\) 0 0
\(109\) 14.8414 1.42155 0.710776 0.703419i \(-0.248344\pi\)
0.710776 + 0.703419i \(0.248344\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.72161 −0.161956 −0.0809778 0.996716i \(-0.525804\pi\)
−0.0809778 + 0.996716i \(0.525804\pi\)
\(114\) 0 0
\(115\) 6.04502 0.563701
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.841431 0.0771338
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −2.12878 −0.188899 −0.0944494 0.995530i \(-0.530109\pi\)
−0.0944494 + 0.995530i \(0.530109\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.59283 −0.139166 −0.0695831 0.997576i \(-0.522167\pi\)
−0.0695831 + 0.997576i \(0.522167\pi\)
\(132\) 0 0
\(133\) −1.39821 −0.121240
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.89541 −0.845422 −0.422711 0.906265i \(-0.638921\pi\)
−0.422711 + 0.906265i \(0.638921\pi\)
\(138\) 0 0
\(139\) 3.70079 0.313897 0.156948 0.987607i \(-0.449834\pi\)
0.156948 + 0.987607i \(0.449834\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 4.60179 0.382158
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −18.3476 −1.50309 −0.751547 0.659680i \(-0.770692\pi\)
−0.751547 + 0.659680i \(0.770692\pi\)
\(150\) 0 0
\(151\) 13.9404 1.13446 0.567228 0.823561i \(-0.308016\pi\)
0.567228 + 0.823561i \(0.308016\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.79641 −0.224613
\(156\) 0 0
\(157\) −0.149606 −0.0119399 −0.00596994 0.999982i \(-0.501900\pi\)
−0.00596994 + 0.999982i \(0.501900\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.45219 0.666126
\(162\) 0 0
\(163\) −24.6468 −1.93049 −0.965244 0.261352i \(-0.915832\pi\)
−0.965244 + 0.261352i \(0.915832\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 18.8656 1.45987 0.729933 0.683519i \(-0.239551\pi\)
0.729933 + 0.683519i \(0.239551\pi\)
\(168\) 0 0
\(169\) 5.69182 0.437833
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.36842 0.484182 0.242091 0.970254i \(-0.422167\pi\)
0.242091 + 0.970254i \(0.422167\pi\)
\(174\) 0 0
\(175\) −1.39821 −0.105695
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −11.4432 −0.855307 −0.427653 0.903943i \(-0.640660\pi\)
−0.427653 + 0.903943i \(0.640660\pi\)
\(180\) 0 0
\(181\) −3.29362 −0.244813 −0.122406 0.992480i \(-0.539061\pi\)
−0.122406 + 0.992480i \(0.539061\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.07480 −0.0790211
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −20.5422 −1.48638 −0.743191 0.669079i \(-0.766689\pi\)
−0.743191 + 0.669079i \(0.766689\pi\)
\(192\) 0 0
\(193\) −19.6620 −1.41530 −0.707652 0.706561i \(-0.750246\pi\)
−0.707652 + 0.706561i \(0.750246\pi\)
\(194\) 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
\(199\) −9.74580 −0.690862 −0.345431 0.938444i \(-0.612267\pi\)
−0.345431 + 0.938444i \(0.612267\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.43426 0.451597
\(204\) 0 0
\(205\) −5.44322 −0.380171
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −9.09899 −0.626401 −0.313200 0.949687i \(-0.601401\pi\)
−0.313200 + 0.949687i \(0.601401\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 8.64681 0.589707
\(216\) 0 0
\(217\) −3.90997 −0.265426
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.60179 −0.175016
\(222\) 0 0
\(223\) −13.8712 −0.928885 −0.464443 0.885603i \(-0.653745\pi\)
−0.464443 + 0.885603i \(0.653745\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.3234 −0.685188 −0.342594 0.939483i \(-0.611306\pi\)
−0.342594 + 0.939483i \(0.611306\pi\)
\(228\) 0 0
\(229\) −14.7368 −0.973838 −0.486919 0.873447i \(-0.661879\pi\)
−0.486919 + 0.873447i \(0.661879\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −13.1857 −0.863821 −0.431911 0.901916i \(-0.642160\pi\)
−0.431911 + 0.901916i \(0.642160\pi\)
\(234\) 0 0
\(235\) −1.85039 −0.120706
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0.452186 0.0292495 0.0146247 0.999893i \(-0.495345\pi\)
0.0146247 + 0.999893i \(0.495345\pi\)
\(240\) 0 0
\(241\) 13.5333 0.871754 0.435877 0.900006i \(-0.356438\pi\)
0.435877 + 0.900006i \(0.356438\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5.04502 0.322314
\(246\) 0 0
\(247\) 4.32340 0.275092
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −21.2936 −1.34404 −0.672021 0.740532i \(-0.734573\pi\)
−0.672021 + 0.740532i \(0.734573\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 28.3989 1.77147 0.885737 0.464188i \(-0.153654\pi\)
0.885737 + 0.464188i \(0.153654\pi\)
\(258\) 0 0
\(259\) −1.50280 −0.0933793
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −11.1857 −0.689737 −0.344869 0.938651i \(-0.612077\pi\)
−0.344869 + 0.938651i \(0.612077\pi\)
\(264\) 0 0
\(265\) −3.11982 −0.191649
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −19.5928 −1.19460 −0.597298 0.802019i \(-0.703759\pi\)
−0.597298 + 0.802019i \(0.703759\pi\)
\(270\) 0 0
\(271\) 5.95498 0.361740 0.180870 0.983507i \(-0.442109\pi\)
0.180870 + 0.983507i \(0.442109\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −23.9404 −1.43844 −0.719220 0.694782i \(-0.755501\pi\)
−0.719220 + 0.694782i \(0.755501\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −10.7368 −0.640506 −0.320253 0.947332i \(-0.603768\pi\)
−0.320253 + 0.947332i \(0.603768\pi\)
\(282\) 0 0
\(283\) −16.0900 −0.956453 −0.478227 0.878237i \(-0.658720\pi\)
−0.478227 + 0.878237i \(0.658720\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −7.61076 −0.449249
\(288\) 0 0
\(289\) −16.6378 −0.978697
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −14.5630 −0.850782 −0.425391 0.905010i \(-0.639863\pi\)
−0.425391 + 0.905010i \(0.639863\pi\)
\(294\) 0 0
\(295\) −6.69182 −0.389613
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −26.1350 −1.51143
\(300\) 0 0
\(301\) 12.0900 0.696858
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.64681 0.151556
\(306\) 0 0
\(307\) 2.77559 0.158411 0.0792057 0.996858i \(-0.474762\pi\)
0.0792057 + 0.996858i \(0.474762\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7.63785 0.433102 0.216551 0.976271i \(-0.430519\pi\)
0.216551 + 0.976271i \(0.430519\pi\)
\(312\) 0 0
\(313\) −32.5243 −1.83838 −0.919191 0.393812i \(-0.871156\pi\)
−0.919191 + 0.393812i \(0.871156\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10.8206 −0.607746 −0.303873 0.952713i \(-0.598280\pi\)
−0.303873 + 0.952713i \(0.598280\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −0.601793 −0.0334846
\(324\) 0 0
\(325\) 4.32340 0.239819
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2.58723 −0.142639
\(330\) 0 0
\(331\) 14.6918 0.807536 0.403768 0.914861i \(-0.367700\pi\)
0.403768 + 0.914861i \(0.367700\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 14.4134 0.787490
\(336\) 0 0
\(337\) −1.07480 −0.0585483 −0.0292741 0.999571i \(-0.509320\pi\)
−0.0292741 + 0.999571i \(0.509320\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 16.8414 0.909352
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −32.0900 −1.72268 −0.861342 0.508026i \(-0.830375\pi\)
−0.861342 + 0.508026i \(0.830375\pi\)
\(348\) 0 0
\(349\) 31.4737 1.68475 0.842374 0.538894i \(-0.181158\pi\)
0.842374 + 0.538894i \(0.181158\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 11.4882 0.611457 0.305729 0.952119i \(-0.401100\pi\)
0.305729 + 0.952119i \(0.401100\pi\)
\(354\) 0 0
\(355\) −5.59283 −0.296837
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −26.0450 −1.37460 −0.687302 0.726372i \(-0.741205\pi\)
−0.687302 + 0.726372i \(0.741205\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −12.6918 −0.664320
\(366\) 0 0
\(367\) −21.5512 −1.12496 −0.562481 0.826810i \(-0.690153\pi\)
−0.562481 + 0.826810i \(0.690153\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.36215 −0.226472
\(372\) 0 0
\(373\) 24.9702 1.29291 0.646454 0.762953i \(-0.276251\pi\)
0.646454 + 0.762953i \(0.276251\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −19.8954 −1.02467
\(378\) 0 0
\(379\) 28.6323 1.47074 0.735370 0.677666i \(-0.237008\pi\)
0.735370 + 0.677666i \(0.237008\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −17.9612 −0.917777 −0.458889 0.888494i \(-0.651752\pi\)
−0.458889 + 0.888494i \(0.651752\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 6.09003 0.308777 0.154388 0.988010i \(-0.450659\pi\)
0.154388 + 0.988010i \(0.450659\pi\)
\(390\) 0 0
\(391\) 3.63785 0.183974
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.64681 0.233806
\(396\) 0 0
\(397\) 21.8325 1.09574 0.547870 0.836563i \(-0.315439\pi\)
0.547870 + 0.836563i \(0.315439\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.7064 0.634526 0.317263 0.948338i \(-0.397236\pi\)
0.317263 + 0.948338i \(0.397236\pi\)
\(402\) 0 0
\(403\) 12.0900 0.602247
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 16.8864 0.834981 0.417491 0.908681i \(-0.362910\pi\)
0.417491 + 0.908681i \(0.362910\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −9.35656 −0.460406
\(414\) 0 0
\(415\) −1.20359 −0.0590817
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −27.7908 −1.35767 −0.678835 0.734291i \(-0.737515\pi\)
−0.678835 + 0.734291i \(0.737515\pi\)
\(420\) 0 0
\(421\) 15.0090 0.731492 0.365746 0.930715i \(-0.380814\pi\)
0.365746 + 0.930715i \(0.380814\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −0.601793 −0.0291912
\(426\) 0 0
\(427\) 3.70079 0.179094
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0.736841 0.0354924 0.0177462 0.999843i \(-0.494351\pi\)
0.0177462 + 0.999843i \(0.494351\pi\)
\(432\) 0 0
\(433\) 31.4945 1.51353 0.756765 0.653687i \(-0.226779\pi\)
0.756765 + 0.653687i \(0.226779\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.04502 −0.289172
\(438\) 0 0
\(439\) −23.2340 −1.10890 −0.554450 0.832217i \(-0.687071\pi\)
−0.554450 + 0.832217i \(0.687071\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −16.4793 −0.782954 −0.391477 0.920188i \(-0.628036\pi\)
−0.391477 + 0.920188i \(0.628036\pi\)
\(444\) 0 0
\(445\) 9.44322 0.447652
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.29921 0.108507 0.0542533 0.998527i \(-0.482722\pi\)
0.0542533 + 0.998527i \(0.482722\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 6.04502 0.283395
\(456\) 0 0
\(457\) −20.0450 −0.937666 −0.468833 0.883287i \(-0.655325\pi\)
−0.468833 + 0.883287i \(0.655325\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −32.5872 −1.51774 −0.758869 0.651243i \(-0.774248\pi\)
−0.758869 + 0.651243i \(0.774248\pi\)
\(462\) 0 0
\(463\) 2.58723 0.120239 0.0601195 0.998191i \(-0.480852\pi\)
0.0601195 + 0.998191i \(0.480852\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0.299213 0.0138459 0.00692295 0.999976i \(-0.497796\pi\)
0.00692295 + 0.999976i \(0.497796\pi\)
\(468\) 0 0
\(469\) 20.1530 0.930578
\(470\) 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
\(479\) 35.0665 1.60223 0.801115 0.598511i \(-0.204241\pi\)
0.801115 + 0.598511i \(0.204241\pi\)
\(480\) 0 0
\(481\) 4.64681 0.211876
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.51803 −0.205153
\(486\) 0 0
\(487\) −41.7521 −1.89197 −0.945983 0.324215i \(-0.894900\pi\)
−0.945983 + 0.324215i \(0.894900\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 30.9765 1.39795 0.698974 0.715147i \(-0.253640\pi\)
0.698974 + 0.715147i \(0.253640\pi\)
\(492\) 0 0
\(493\) 2.76932 0.124724
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7.81994 −0.350772
\(498\) 0 0
\(499\) −3.90997 −0.175034 −0.0875171 0.996163i \(-0.527893\pi\)
−0.0875171 + 0.996163i \(0.527893\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16.4522 −0.733567 −0.366783 0.930306i \(-0.619541\pi\)
−0.366783 + 0.930306i \(0.619541\pi\)
\(504\) 0 0
\(505\) −5.05398 −0.224899
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 25.4432 1.12775 0.563876 0.825860i \(-0.309310\pi\)
0.563876 + 0.825860i \(0.309310\pi\)
\(510\) 0 0
\(511\) −17.7458 −0.785028
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 11.0748 0.488014
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 27.3836 1.19970 0.599850 0.800113i \(-0.295227\pi\)
0.599850 + 0.800113i \(0.295227\pi\)
\(522\) 0 0
\(523\) 41.5574 1.81718 0.908590 0.417689i \(-0.137160\pi\)
0.908590 + 0.417689i \(0.137160\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.68286 −0.0733066
\(528\) 0 0
\(529\) 13.5422 0.588792
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 23.5333 1.01934
\(534\) 0 0
\(535\) −4.17380 −0.180449
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 14.6468 0.629715 0.314858 0.949139i \(-0.398043\pi\)
0.314858 + 0.949139i \(0.398043\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −14.8414 −0.635737
\(546\) 0 0
\(547\) −24.9252 −1.06572 −0.532862 0.846202i \(-0.678884\pi\)
−0.532862 + 0.846202i \(0.678884\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −4.60179 −0.196043
\(552\) 0 0
\(553\) 6.49720 0.276289
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 44.9348 1.90395 0.951975 0.306176i \(-0.0990496\pi\)
0.951975 + 0.306176i \(0.0990496\pi\)
\(558\) 0 0
\(559\) −37.3836 −1.58116
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10.2605 0.432427 0.216213 0.976346i \(-0.430629\pi\)
0.216213 + 0.976346i \(0.430629\pi\)
\(564\) 0 0
\(565\) 1.72161 0.0724287
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −24.6773 −1.03452 −0.517262 0.855827i \(-0.673049\pi\)
−0.517262 + 0.855827i \(0.673049\pi\)
\(570\) 0 0
\(571\) −10.9765 −0.459351 −0.229676 0.973267i \(-0.573767\pi\)
−0.229676 + 0.973267i \(0.573767\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.04502 −0.252095
\(576\) 0 0
\(577\) −45.1295 −1.87876 −0.939382 0.342873i \(-0.888600\pi\)
−0.939382 + 0.342873i \(0.888600\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1.68286 −0.0698169
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −7.90997 −0.326479 −0.163240 0.986586i \(-0.552194\pi\)
−0.163240 + 0.986586i \(0.552194\pi\)
\(588\) 0 0
\(589\) 2.79641 0.115224
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 7.29362 0.299513 0.149756 0.988723i \(-0.452151\pi\)
0.149756 + 0.988723i \(0.452151\pi\)
\(594\) 0 0
\(595\) −0.841431 −0.0344953
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −20.6468 −0.843606 −0.421803 0.906688i \(-0.638603\pi\)
−0.421803 + 0.906688i \(0.638603\pi\)
\(600\) 0 0
\(601\) 45.1440 1.84146 0.920731 0.390197i \(-0.127593\pi\)
0.920731 + 0.390197i \(0.127593\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 11.0000 0.447214
\(606\) 0 0
\(607\) −20.7577 −0.842528 −0.421264 0.906938i \(-0.638413\pi\)
−0.421264 + 0.906938i \(0.638413\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) 30.3476 1.22573 0.612864 0.790188i \(-0.290017\pi\)
0.612864 + 0.790188i \(0.290017\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −47.2161 −1.90085 −0.950425 0.310955i \(-0.899351\pi\)
−0.950425 + 0.310955i \(0.899351\pi\)
\(618\) 0 0
\(619\) −22.7064 −0.912647 −0.456323 0.889814i \(-0.650834\pi\)
−0.456323 + 0.889814i \(0.650834\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 13.2036 0.528990
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −0.646809 −0.0257899
\(630\) 0 0
\(631\) 32.4793 1.29298 0.646490 0.762923i \(-0.276236\pi\)
0.646490 + 0.762923i \(0.276236\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.12878 0.0844781
\(636\) 0 0
\(637\) −21.8116 −0.864209
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −10.9460 −0.432342 −0.216171 0.976356i \(-0.569357\pi\)
−0.216171 + 0.976356i \(0.569357\pi\)
\(642\) 0 0
\(643\) 4.38924 0.173095 0.0865475 0.996248i \(-0.472417\pi\)
0.0865475 + 0.996248i \(0.472417\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −16.4522 −0.646802 −0.323401 0.946262i \(-0.604826\pi\)
−0.323401 + 0.946262i \(0.604826\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.53326 −0.0600009 −0.0300005 0.999550i \(-0.509551\pi\)
−0.0300005 + 0.999550i \(0.509551\pi\)
\(654\) 0 0
\(655\) 1.59283 0.0622370
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −22.1767 −0.863881 −0.431941 0.901902i \(-0.642171\pi\)
−0.431941 + 0.901902i \(0.642171\pi\)
\(660\) 0 0
\(661\) −26.0755 −1.01422 −0.507109 0.861882i \(-0.669286\pi\)
−0.507109 + 0.861882i \(0.669286\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.39821 0.0542202
\(666\) 0 0
\(667\) 27.8179 1.07711
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −30.4168 −1.17248 −0.586241 0.810137i \(-0.699393\pi\)
−0.586241 + 0.810137i \(0.699393\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 44.2043 1.69891 0.849454 0.527663i \(-0.176932\pi\)
0.849454 + 0.527663i \(0.176932\pi\)
\(678\) 0 0
\(679\) −6.31714 −0.242430
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −42.3989 −1.62235 −0.811174 0.584805i \(-0.801171\pi\)
−0.811174 + 0.584805i \(0.801171\pi\)
\(684\) 0 0
\(685\) 9.89541 0.378084
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 13.4882 0.513861
\(690\) 0 0
\(691\) 7.79082 0.296377 0.148188 0.988959i \(-0.452656\pi\)
0.148188 + 0.988959i \(0.452656\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.70079 −0.140379
\(696\) 0 0
\(697\) −3.27569 −0.124076
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 8.05957 0.304406 0.152203 0.988349i \(-0.451363\pi\)
0.152203 + 0.988349i \(0.451363\pi\)
\(702\) 0 0
\(703\) 1.07480 0.0405370
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −7.06651 −0.265763
\(708\) 0 0
\(709\) −35.2936 −1.32548 −0.662740 0.748850i \(-0.730606\pi\)
−0.662740 + 0.748850i \(0.730606\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −16.9044 −0.633074
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 48.9315 1.82484 0.912418 0.409260i \(-0.134213\pi\)
0.912418 + 0.409260i \(0.134213\pi\)
\(720\) 0 0
\(721\) 15.4849 0.576687
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.60179 −0.170906
\(726\) 0 0
\(727\) −3.29025 −0.122029 −0.0610143 0.998137i \(-0.519434\pi\)
−0.0610143 + 0.998137i \(0.519434\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 5.20359 0.192462
\(732\) 0 0
\(733\) 22.3892 0.826966 0.413483 0.910512i \(-0.364312\pi\)
0.413483 + 0.910512i \(0.364312\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 4.90437 0.180410 0.0902051 0.995923i \(-0.471248\pi\)
0.0902051 + 0.995923i \(0.471248\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −28.1592 −1.03306 −0.516531 0.856268i \(-0.672777\pi\)
−0.516531 + 0.856268i \(0.672777\pi\)
\(744\) 0 0
\(745\) 18.3476 0.672204
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −5.83584 −0.213237
\(750\) 0 0
\(751\) 42.9348 1.56671 0.783357 0.621572i \(-0.213506\pi\)
0.783357 + 0.621572i \(0.213506\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −13.9404 −0.507344
\(756\) 0 0
\(757\) −25.0665 −0.911058 −0.455529 0.890221i \(-0.650550\pi\)
−0.455529 + 0.890221i \(0.650550\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 30.8898 1.11975 0.559877 0.828575i \(-0.310848\pi\)
0.559877 + 0.828575i \(0.310848\pi\)
\(762\) 0 0
\(763\) −20.7514 −0.751251
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 28.9315 1.04465
\(768\) 0 0
\(769\) 7.78745 0.280823 0.140411 0.990093i \(-0.455157\pi\)
0.140411 + 0.990093i \(0.455157\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −20.7126 −0.744982 −0.372491 0.928036i \(-0.621496\pi\)
−0.372491 + 0.928036i \(0.621496\pi\)
\(774\) 0 0
\(775\) 2.79641 0.100450
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.44322 0.195024
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.149606 0.00533968
\(786\) 0 0
\(787\) 10.5035 0.374408 0.187204 0.982321i \(-0.440057\pi\)
0.187204 + 0.982321i \(0.440057\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.40717 0.0855891
\(792\) 0 0
\(793\) −11.4432 −0.406361
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 37.8871 1.34203 0.671015 0.741444i \(-0.265858\pi\)
0.671015 + 0.741444i \(0.265858\pi\)
\(798\) 0 0
\(799\) −1.11355 −0.0393947
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −8.45219 −0.297900
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 33.9438 1.19340 0.596700 0.802464i \(-0.296478\pi\)
0.596700 + 0.802464i \(0.296478\pi\)
\(810\) 0 0
\(811\) 4.66137 0.163683 0.0818414 0.996645i \(-0.473920\pi\)
0.0818414 + 0.996645i \(0.473920\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 24.6468 0.863340
\(816\) 0 0
\(817\) −8.64681 −0.302514
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 26.5693 0.927275 0.463638 0.886025i \(-0.346544\pi\)
0.463638 + 0.886025i \(0.346544\pi\)
\(822\) 0 0
\(823\) 24.7998 0.864466 0.432233 0.901762i \(-0.357726\pi\)
0.432233 + 0.901762i \(0.357726\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.11422 0.0735188 0.0367594 0.999324i \(-0.488296\pi\)
0.0367594 + 0.999324i \(0.488296\pi\)
\(828\) 0 0
\(829\) 45.7279 1.58819 0.794097 0.607790i \(-0.207944\pi\)
0.794097 + 0.607790i \(0.207944\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.03605 0.105193
\(834\) 0 0
\(835\) −18.8656 −0.652872
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −15.4432 −0.533159 −0.266580 0.963813i \(-0.585894\pi\)
−0.266580 + 0.963813i \(0.585894\pi\)
\(840\) 0 0
\(841\) −7.82351 −0.269776
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −5.69182 −0.195805
\(846\) 0 0
\(847\) 15.3803 0.528473
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −6.49720 −0.222721
\(852\) 0 0
\(853\) 38.1801 1.30726 0.653630 0.756814i \(-0.273245\pi\)
0.653630 + 0.756814i \(0.273245\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −52.2672 −1.78541 −0.892707 0.450638i \(-0.851196\pi\)
−0.892707 + 0.450638i \(0.851196\pi\)
\(858\) 0 0
\(859\) −39.8809 −1.36072 −0.680359 0.732879i \(-0.738176\pi\)
−0.680359 + 0.732879i \(0.738176\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −22.9557 −0.781420 −0.390710 0.920514i \(-0.627770\pi\)
−0.390710 + 0.920514i \(0.627770\pi\)
\(864\) 0 0
\(865\) −6.36842 −0.216533
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −62.3151 −2.11147
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.39821 0.0472680
\(876\) 0 0
\(877\) −2.73057 −0.0922050 −0.0461025 0.998937i \(-0.514680\pi\)
−0.0461025 + 0.998937i \(0.514680\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −9.14401 −0.308070 −0.154035 0.988065i \(-0.549227\pi\)
−0.154035 + 0.988065i \(0.549227\pi\)
\(882\) 0 0
\(883\) −11.0540 −0.371996 −0.185998 0.982550i \(-0.559552\pi\)
−0.185998 + 0.982550i \(0.559552\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −37.1469 −1.24727 −0.623636 0.781715i \(-0.714345\pi\)
−0.623636 + 0.781715i \(0.714345\pi\)
\(888\) 0 0
\(889\) 2.97648 0.0998279
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.85039 0.0619211
\(894\) 0 0
\(895\) 11.4432 0.382505
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −12.8685 −0.429189
\(900\) 0 0
\(901\) −1.87748 −0.0625481
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3.29362 0.109484
\(906\) 0 0
\(907\) 4.00627 0.133026 0.0665129 0.997786i \(-0.478813\pi\)
0.0665129 + 0.997786i \(0.478813\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −21.2161 −0.702921 −0.351461 0.936203i \(-0.614315\pi\)
−0.351461 + 0.936203i \(0.614315\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.22711 0.0735455
\(918\) 0 0
\(919\) −23.5478 −0.776771 −0.388385 0.921497i \(-0.626967\pi\)
−0.388385 + 0.921497i \(0.626967\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 24.1801 0.795896
\(924\) 0 0
\(925\) 1.07480 0.0353393
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 42.7223 1.40167 0.700836 0.713322i \(-0.252810\pi\)
0.700836 + 0.713322i \(0.252810\pi\)
\(930\) 0 0
\(931\) −5.04502 −0.165344
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −31.1890 −1.01890 −0.509451 0.860500i \(-0.670151\pi\)
−0.509451 + 0.860500i \(0.670151\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −20.2251 −0.659319 −0.329659 0.944100i \(-0.606934\pi\)
−0.329659 + 0.944100i \(0.606934\pi\)
\(942\) 0 0
\(943\) −32.9044 −1.07151
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 20.1801 0.655764 0.327882 0.944719i \(-0.393665\pi\)
0.327882 + 0.944719i \(0.393665\pi\)
\(948\) 0 0
\(949\) 54.8719 1.78122
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −20.6856 −0.670071 −0.335035 0.942206i \(-0.608748\pi\)
−0.335035 + 0.942206i \(0.608748\pi\)
\(954\) 0 0
\(955\) 20.5422 0.664731
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 13.8358 0.446782
\(960\) 0 0
\(961\) −23.1801 −0.747744
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 19.6620 0.632943
\(966\) 0 0
\(967\) −53.9530 −1.73501 −0.867505 0.497428i \(-0.834278\pi\)
−0.867505 + 0.497428i \(0.834278\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 9.41277 0.302070 0.151035 0.988528i \(-0.451739\pi\)
0.151035 + 0.988528i \(0.451739\pi\)
\(972\) 0 0
\(973\) −5.17447 −0.165886
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4.21881 0.134972 0.0674859 0.997720i \(-0.478502\pi\)
0.0674859 + 0.997720i \(0.478502\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 10.3088 0.328801 0.164401 0.986394i \(-0.447431\pi\)
0.164401 + 0.986394i \(0.447431\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 52.2701 1.66209
\(990\) 0 0
\(991\) 54.6356 1.73556 0.867779 0.496951i \(-0.165547\pi\)
0.867779 + 0.496951i \(0.165547\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 9.74580 0.308963
\(996\) 0 0
\(997\) 22.9169 0.725786 0.362893 0.931831i \(-0.381789\pi\)
0.362893 + 0.931831i \(0.381789\pi\)
\(998\) 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.1.2 3
3.2 odd 2 760.2.a.j.1.3 3
12.11 even 2 1520.2.a.s.1.1 3
15.2 even 4 3800.2.d.l.3649.2 6
15.8 even 4 3800.2.d.l.3649.5 6
15.14 odd 2 3800.2.a.x.1.1 3
24.5 odd 2 6080.2.a.bv.1.1 3
24.11 even 2 6080.2.a.bq.1.3 3
60.59 even 2 7600.2.a.bq.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.j.1.3 3 3.2 odd 2
1520.2.a.s.1.1 3 12.11 even 2
3800.2.a.x.1.1 3 15.14 odd 2
3800.2.d.l.3649.2 6 15.2 even 4
3800.2.d.l.3649.5 6 15.8 even 4
6080.2.a.bq.1.3 3 24.11 even 2
6080.2.a.bv.1.1 3 24.5 odd 2
6840.2.a.bg.1.2 3 1.1 even 1 trivial
7600.2.a.bq.1.3 3 60.59 even 2