Properties

Label 1904.2.a.m.1.2
Level $1904$
Weight $2$
Character 1904.1
Self dual yes
Analytic conductor $15.204$
Analytic rank $1$
Dimension $3$
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] = [1904,2,Mod(1,1904)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1904, 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("1904.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1904 = 2^{4} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1904.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: \(15.2035165449\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 952)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.86081\) of defining polynomial
Character \(\chi\) \(=\) 1904.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.462598 q^{3} +2.86081 q^{5} -1.00000 q^{7} -2.78600 q^{9} +O(q^{10})\) \(q+0.462598 q^{3} +2.86081 q^{5} -1.00000 q^{7} -2.78600 q^{9} -5.72161 q^{11} +1.32340 q^{15} -1.00000 q^{17} -2.92520 q^{19} -0.462598 q^{21} +0.796415 q^{23} +3.18421 q^{25} -2.67660 q^{27} -1.07480 q^{29} -2.86081 q^{31} -2.64681 q^{33} -2.86081 q^{35} -2.92520 q^{37} -10.5526 q^{41} -11.6274 q^{43} -7.97021 q^{45} -5.85039 q^{47} +1.00000 q^{49} -0.462598 q^{51} +9.78600 q^{53} -16.3684 q^{55} -1.35319 q^{57} +8.09003 q^{59} +11.7562 q^{61} +2.78600 q^{63} +13.8760 q^{67} +0.368420 q^{69} +9.16484 q^{71} -6.18421 q^{73} +1.47301 q^{75} +5.72161 q^{77} -8.00000 q^{79} +7.11982 q^{81} -17.8116 q^{83} -2.86081 q^{85} -0.497202 q^{87} +1.20359 q^{89} -1.32340 q^{93} -8.36842 q^{95} +4.06439 q^{97} +15.9404 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9} - 6 q^{11} - 4 q^{15} - 3 q^{17} - 4 q^{19} + q^{21} - 4 q^{23} - 4 q^{25} - 16 q^{27} - 8 q^{29} - 3 q^{31} + 8 q^{33} - 3 q^{35} - 4 q^{37} + 9 q^{41} + q^{43} - 8 q^{47} + 3 q^{49} + q^{51} + 19 q^{53} - 22 q^{55} - 20 q^{57} - 14 q^{59} + q^{61} - 2 q^{63} - 7 q^{67} - 26 q^{69} - 6 q^{71} - 5 q^{73} + 6 q^{75} + 6 q^{77} - 24 q^{79} + 7 q^{81} - 4 q^{83} - 3 q^{85} + 24 q^{87} + 10 q^{89} + 4 q^{93} + 2 q^{95} + 13 q^{97}+O(q^{100}) \) 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
\(3\) 0.462598 0.267081 0.133541 0.991043i \(-0.457365\pi\)
0.133541 + 0.991043i \(0.457365\pi\)
\(4\) 0 0
\(5\) 2.86081 1.27939 0.639696 0.768628i \(-0.279060\pi\)
0.639696 + 0.768628i \(0.279060\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −2.78600 −0.928668
\(10\) 0 0
\(11\) −5.72161 −1.72513 −0.862565 0.505946i \(-0.831144\pi\)
−0.862565 + 0.505946i \(0.831144\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 1.32340 0.341702
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) −2.92520 −0.671086 −0.335543 0.942025i \(-0.608920\pi\)
−0.335543 + 0.942025i \(0.608920\pi\)
\(20\) 0 0
\(21\) −0.462598 −0.100947
\(22\) 0 0
\(23\) 0.796415 0.166064 0.0830320 0.996547i \(-0.473540\pi\)
0.0830320 + 0.996547i \(0.473540\pi\)
\(24\) 0 0
\(25\) 3.18421 0.636842
\(26\) 0 0
\(27\) −2.67660 −0.515111
\(28\) 0 0
\(29\) −1.07480 −0.199586 −0.0997930 0.995008i \(-0.531818\pi\)
−0.0997930 + 0.995008i \(0.531818\pi\)
\(30\) 0 0
\(31\) −2.86081 −0.513816 −0.256908 0.966436i \(-0.582704\pi\)
−0.256908 + 0.966436i \(0.582704\pi\)
\(32\) 0 0
\(33\) −2.64681 −0.460750
\(34\) 0 0
\(35\) −2.86081 −0.483564
\(36\) 0 0
\(37\) −2.92520 −0.480899 −0.240450 0.970662i \(-0.577295\pi\)
−0.240450 + 0.970662i \(0.577295\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.5526 −1.64804 −0.824022 0.566558i \(-0.808275\pi\)
−0.824022 + 0.566558i \(0.808275\pi\)
\(42\) 0 0
\(43\) −11.6274 −1.77317 −0.886583 0.462569i \(-0.846928\pi\)
−0.886583 + 0.462569i \(0.846928\pi\)
\(44\) 0 0
\(45\) −7.97021 −1.18813
\(46\) 0 0
\(47\) −5.85039 −0.853368 −0.426684 0.904401i \(-0.640318\pi\)
−0.426684 + 0.904401i \(0.640318\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −0.462598 −0.0647767
\(52\) 0 0
\(53\) 9.78600 1.34421 0.672105 0.740455i \(-0.265390\pi\)
0.672105 + 0.740455i \(0.265390\pi\)
\(54\) 0 0
\(55\) −16.3684 −2.20712
\(56\) 0 0
\(57\) −1.35319 −0.179235
\(58\) 0 0
\(59\) 8.09003 1.05323 0.526616 0.850103i \(-0.323460\pi\)
0.526616 + 0.850103i \(0.323460\pi\)
\(60\) 0 0
\(61\) 11.7562 1.50523 0.752615 0.658461i \(-0.228792\pi\)
0.752615 + 0.658461i \(0.228792\pi\)
\(62\) 0 0
\(63\) 2.78600 0.351003
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 13.8760 1.69523 0.847614 0.530614i \(-0.178039\pi\)
0.847614 + 0.530614i \(0.178039\pi\)
\(68\) 0 0
\(69\) 0.368420 0.0443526
\(70\) 0 0
\(71\) 9.16484 1.08767 0.543833 0.839194i \(-0.316973\pi\)
0.543833 + 0.839194i \(0.316973\pi\)
\(72\) 0 0
\(73\) −6.18421 −0.723807 −0.361904 0.932216i \(-0.617873\pi\)
−0.361904 + 0.932216i \(0.617873\pi\)
\(74\) 0 0
\(75\) 1.47301 0.170089
\(76\) 0 0
\(77\) 5.72161 0.652038
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 7.11982 0.791091
\(82\) 0 0
\(83\) −17.8116 −1.95508 −0.977541 0.210746i \(-0.932411\pi\)
−0.977541 + 0.210746i \(0.932411\pi\)
\(84\) 0 0
\(85\) −2.86081 −0.310298
\(86\) 0 0
\(87\) −0.497202 −0.0533057
\(88\) 0 0
\(89\) 1.20359 0.127580 0.0637899 0.997963i \(-0.479681\pi\)
0.0637899 + 0.997963i \(0.479681\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.32340 −0.137231
\(94\) 0 0
\(95\) −8.36842 −0.858582
\(96\) 0 0
\(97\) 4.06439 0.412676 0.206338 0.978481i \(-0.433845\pi\)
0.206338 + 0.978481i \(0.433845\pi\)
\(98\) 0 0
\(99\) 15.9404 1.60207
\(100\) 0 0
\(101\) 5.16484 0.513920 0.256960 0.966422i \(-0.417279\pi\)
0.256960 + 0.966422i \(0.417279\pi\)
\(102\) 0 0
\(103\) −0.128782 −0.0126893 −0.00634463 0.999980i \(-0.502020\pi\)
−0.00634463 + 0.999980i \(0.502020\pi\)
\(104\) 0 0
\(105\) −1.32340 −0.129151
\(106\) 0 0
\(107\) −2.14961 −0.207810 −0.103905 0.994587i \(-0.533134\pi\)
−0.103905 + 0.994587i \(0.533134\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) −1.35319 −0.128439
\(112\) 0 0
\(113\) 13.9404 1.31140 0.655702 0.755019i \(-0.272373\pi\)
0.655702 + 0.755019i \(0.272373\pi\)
\(114\) 0 0
\(115\) 2.27839 0.212461
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) 21.7368 1.97608
\(122\) 0 0
\(123\) −4.88163 −0.440162
\(124\) 0 0
\(125\) −5.19462 −0.464621
\(126\) 0 0
\(127\) 5.26798 0.467457 0.233729 0.972302i \(-0.424907\pi\)
0.233729 + 0.972302i \(0.424907\pi\)
\(128\) 0 0
\(129\) −5.37883 −0.473580
\(130\) 0 0
\(131\) 6.49720 0.567663 0.283832 0.958874i \(-0.408394\pi\)
0.283832 + 0.958874i \(0.408394\pi\)
\(132\) 0 0
\(133\) 2.92520 0.253647
\(134\) 0 0
\(135\) −7.65722 −0.659029
\(136\) 0 0
\(137\) −16.4030 −1.40140 −0.700702 0.713454i \(-0.747130\pi\)
−0.700702 + 0.713454i \(0.747130\pi\)
\(138\) 0 0
\(139\) −0.0643910 −0.00546157 −0.00273079 0.999996i \(-0.500869\pi\)
−0.00273079 + 0.999996i \(0.500869\pi\)
\(140\) 0 0
\(141\) −2.70638 −0.227919
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −3.07480 −0.255349
\(146\) 0 0
\(147\) 0.462598 0.0381545
\(148\) 0 0
\(149\) −15.1094 −1.23781 −0.618905 0.785466i \(-0.712424\pi\)
−0.618905 + 0.785466i \(0.712424\pi\)
\(150\) 0 0
\(151\) 13.1094 1.06683 0.533414 0.845854i \(-0.320909\pi\)
0.533414 + 0.845854i \(0.320909\pi\)
\(152\) 0 0
\(153\) 2.78600 0.225235
\(154\) 0 0
\(155\) −8.18421 −0.657372
\(156\) 0 0
\(157\) −20.0692 −1.60170 −0.800849 0.598867i \(-0.795618\pi\)
−0.800849 + 0.598867i \(0.795618\pi\)
\(158\) 0 0
\(159\) 4.52699 0.359014
\(160\) 0 0
\(161\) −0.796415 −0.0627663
\(162\) 0 0
\(163\) 6.09003 0.477008 0.238504 0.971142i \(-0.423343\pi\)
0.238504 + 0.971142i \(0.423343\pi\)
\(164\) 0 0
\(165\) −7.57201 −0.589480
\(166\) 0 0
\(167\) −10.3338 −0.799655 −0.399827 0.916590i \(-0.630930\pi\)
−0.399827 + 0.916590i \(0.630930\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 8.14961 0.623216
\(172\) 0 0
\(173\) −0.980625 −0.0745555 −0.0372778 0.999305i \(-0.511869\pi\)
−0.0372778 + 0.999305i \(0.511869\pi\)
\(174\) 0 0
\(175\) −3.18421 −0.240704
\(176\) 0 0
\(177\) 3.74244 0.281299
\(178\) 0 0
\(179\) 15.8850 1.18730 0.593650 0.804723i \(-0.297686\pi\)
0.593650 + 0.804723i \(0.297686\pi\)
\(180\) 0 0
\(181\) −20.2396 −1.50440 −0.752200 0.658935i \(-0.771007\pi\)
−0.752200 + 0.658935i \(0.771007\pi\)
\(182\) 0 0
\(183\) 5.43841 0.402019
\(184\) 0 0
\(185\) −8.36842 −0.615258
\(186\) 0 0
\(187\) 5.72161 0.418406
\(188\) 0 0
\(189\) 2.67660 0.194694
\(190\) 0 0
\(191\) −6.86081 −0.496430 −0.248215 0.968705i \(-0.579844\pi\)
−0.248215 + 0.968705i \(0.579844\pi\)
\(192\) 0 0
\(193\) −12.3892 −0.891797 −0.445899 0.895083i \(-0.647116\pi\)
−0.445899 + 0.895083i \(0.647116\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −16.8864 −1.20311 −0.601555 0.798832i \(-0.705452\pi\)
−0.601555 + 0.798832i \(0.705452\pi\)
\(198\) 0 0
\(199\) 5.66618 0.401665 0.200833 0.979626i \(-0.435635\pi\)
0.200833 + 0.979626i \(0.435635\pi\)
\(200\) 0 0
\(201\) 6.41903 0.452764
\(202\) 0 0
\(203\) 1.07480 0.0754364
\(204\) 0 0
\(205\) −30.1890 −2.10849
\(206\) 0 0
\(207\) −2.21881 −0.154218
\(208\) 0 0
\(209\) 16.7368 1.15771
\(210\) 0 0
\(211\) 10.5180 0.724091 0.362045 0.932160i \(-0.382079\pi\)
0.362045 + 0.932160i \(0.382079\pi\)
\(212\) 0 0
\(213\) 4.23964 0.290495
\(214\) 0 0
\(215\) −33.2638 −2.26857
\(216\) 0 0
\(217\) 2.86081 0.194204
\(218\) 0 0
\(219\) −2.86081 −0.193315
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 12.6081 0.844298 0.422149 0.906526i \(-0.361276\pi\)
0.422149 + 0.906526i \(0.361276\pi\)
\(224\) 0 0
\(225\) −8.87122 −0.591415
\(226\) 0 0
\(227\) −1.35801 −0.0901342 −0.0450671 0.998984i \(-0.514350\pi\)
−0.0450671 + 0.998984i \(0.514350\pi\)
\(228\) 0 0
\(229\) 2.27839 0.150560 0.0752801 0.997162i \(-0.476015\pi\)
0.0752801 + 0.997162i \(0.476015\pi\)
\(230\) 0 0
\(231\) 2.64681 0.174147
\(232\) 0 0
\(233\) 11.4224 0.748306 0.374153 0.927367i \(-0.377933\pi\)
0.374153 + 0.927367i \(0.377933\pi\)
\(234\) 0 0
\(235\) −16.7368 −1.09179
\(236\) 0 0
\(237\) −3.70079 −0.240392
\(238\) 0 0
\(239\) 23.8158 1.54052 0.770258 0.637733i \(-0.220128\pi\)
0.770258 + 0.637733i \(0.220128\pi\)
\(240\) 0 0
\(241\) 20.8012 1.33993 0.669963 0.742395i \(-0.266310\pi\)
0.669963 + 0.742395i \(0.266310\pi\)
\(242\) 0 0
\(243\) 11.3234 0.726397
\(244\) 0 0
\(245\) 2.86081 0.182770
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −8.23964 −0.522166
\(250\) 0 0
\(251\) −29.8116 −1.88169 −0.940847 0.338831i \(-0.889968\pi\)
−0.940847 + 0.338831i \(0.889968\pi\)
\(252\) 0 0
\(253\) −4.55678 −0.286482
\(254\) 0 0
\(255\) −1.32340 −0.0828748
\(256\) 0 0
\(257\) 20.6468 1.28791 0.643956 0.765062i \(-0.277292\pi\)
0.643956 + 0.765062i \(0.277292\pi\)
\(258\) 0 0
\(259\) 2.92520 0.181763
\(260\) 0 0
\(261\) 2.99440 0.185349
\(262\) 0 0
\(263\) 6.88645 0.424637 0.212318 0.977201i \(-0.431899\pi\)
0.212318 + 0.977201i \(0.431899\pi\)
\(264\) 0 0
\(265\) 27.9959 1.71977
\(266\) 0 0
\(267\) 0.556777 0.0340742
\(268\) 0 0
\(269\) 21.8325 1.33115 0.665575 0.746331i \(-0.268187\pi\)
0.665575 + 0.746331i \(0.268187\pi\)
\(270\) 0 0
\(271\) 1.05398 0.0640247 0.0320123 0.999487i \(-0.489808\pi\)
0.0320123 + 0.999487i \(0.489808\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −18.2188 −1.09864
\(276\) 0 0
\(277\) −19.5928 −1.17722 −0.588610 0.808417i \(-0.700324\pi\)
−0.588610 + 0.808417i \(0.700324\pi\)
\(278\) 0 0
\(279\) 7.97021 0.477164
\(280\) 0 0
\(281\) −20.3732 −1.21537 −0.607683 0.794180i \(-0.707901\pi\)
−0.607683 + 0.794180i \(0.707901\pi\)
\(282\) 0 0
\(283\) −5.48679 −0.326156 −0.163078 0.986613i \(-0.552142\pi\)
−0.163078 + 0.986613i \(0.552142\pi\)
\(284\) 0 0
\(285\) −3.87122 −0.229311
\(286\) 0 0
\(287\) 10.5526 0.622902
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 1.88018 0.110218
\(292\) 0 0
\(293\) −3.74244 −0.218635 −0.109318 0.994007i \(-0.534867\pi\)
−0.109318 + 0.994007i \(0.534867\pi\)
\(294\) 0 0
\(295\) 23.1440 1.34750
\(296\) 0 0
\(297\) 15.3144 0.888634
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 11.6274 0.670194
\(302\) 0 0
\(303\) 2.38924 0.137259
\(304\) 0 0
\(305\) 33.6323 1.92578
\(306\) 0 0
\(307\) 21.0152 1.19940 0.599701 0.800224i \(-0.295286\pi\)
0.599701 + 0.800224i \(0.295286\pi\)
\(308\) 0 0
\(309\) −0.0595743 −0.00338907
\(310\) 0 0
\(311\) 22.0048 1.24778 0.623889 0.781513i \(-0.285552\pi\)
0.623889 + 0.781513i \(0.285552\pi\)
\(312\) 0 0
\(313\) 4.92038 0.278116 0.139058 0.990284i \(-0.455592\pi\)
0.139058 + 0.990284i \(0.455592\pi\)
\(314\) 0 0
\(315\) 7.97021 0.449071
\(316\) 0 0
\(317\) −8.34760 −0.468848 −0.234424 0.972134i \(-0.575320\pi\)
−0.234424 + 0.972134i \(0.575320\pi\)
\(318\) 0 0
\(319\) 6.14961 0.344312
\(320\) 0 0
\(321\) −0.994404 −0.0555022
\(322\) 0 0
\(323\) 2.92520 0.162762
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −6.47638 −0.358145
\(328\) 0 0
\(329\) 5.85039 0.322543
\(330\) 0 0
\(331\) −18.9896 −1.04376 −0.521881 0.853018i \(-0.674770\pi\)
−0.521881 + 0.853018i \(0.674770\pi\)
\(332\) 0 0
\(333\) 8.14961 0.446596
\(334\) 0 0
\(335\) 39.6966 2.16886
\(336\) 0 0
\(337\) −8.34760 −0.454723 −0.227361 0.973810i \(-0.573010\pi\)
−0.227361 + 0.973810i \(0.573010\pi\)
\(338\) 0 0
\(339\) 6.44882 0.350252
\(340\) 0 0
\(341\) 16.3684 0.886400
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 1.05398 0.0567443
\(346\) 0 0
\(347\) −1.05398 −0.0565805 −0.0282903 0.999600i \(-0.509006\pi\)
−0.0282903 + 0.999600i \(0.509006\pi\)
\(348\) 0 0
\(349\) −25.9612 −1.38967 −0.694837 0.719167i \(-0.744523\pi\)
−0.694837 + 0.719167i \(0.744523\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 11.4820 0.611124 0.305562 0.952172i \(-0.401156\pi\)
0.305562 + 0.952172i \(0.401156\pi\)
\(354\) 0 0
\(355\) 26.2188 1.39155
\(356\) 0 0
\(357\) 0.462598 0.0244833
\(358\) 0 0
\(359\) −17.8760 −0.943461 −0.471731 0.881743i \(-0.656370\pi\)
−0.471731 + 0.881743i \(0.656370\pi\)
\(360\) 0 0
\(361\) −10.4432 −0.549643
\(362\) 0 0
\(363\) 10.0554 0.527773
\(364\) 0 0
\(365\) −17.6918 −0.926032
\(366\) 0 0
\(367\) −23.0707 −1.20428 −0.602139 0.798391i \(-0.705685\pi\)
−0.602139 + 0.798391i \(0.705685\pi\)
\(368\) 0 0
\(369\) 29.3997 1.53048
\(370\) 0 0
\(371\) −9.78600 −0.508064
\(372\) 0 0
\(373\) −4.89059 −0.253225 −0.126613 0.991952i \(-0.540411\pi\)
−0.126613 + 0.991952i \(0.540411\pi\)
\(374\) 0 0
\(375\) −2.40302 −0.124092
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −15.5720 −0.799880 −0.399940 0.916541i \(-0.630969\pi\)
−0.399940 + 0.916541i \(0.630969\pi\)
\(380\) 0 0
\(381\) 2.43696 0.124849
\(382\) 0 0
\(383\) −28.0513 −1.43335 −0.716677 0.697405i \(-0.754338\pi\)
−0.716677 + 0.697405i \(0.754338\pi\)
\(384\) 0 0
\(385\) 16.3684 0.834212
\(386\) 0 0
\(387\) 32.3941 1.64668
\(388\) 0 0
\(389\) 13.7860 0.698978 0.349489 0.936940i \(-0.386355\pi\)
0.349489 + 0.936940i \(0.386355\pi\)
\(390\) 0 0
\(391\) −0.796415 −0.0402764
\(392\) 0 0
\(393\) 3.00560 0.151612
\(394\) 0 0
\(395\) −22.8864 −1.15154
\(396\) 0 0
\(397\) 20.8913 1.04850 0.524251 0.851564i \(-0.324345\pi\)
0.524251 + 0.851564i \(0.324345\pi\)
\(398\) 0 0
\(399\) 1.35319 0.0677443
\(400\) 0 0
\(401\) −18.0513 −0.901438 −0.450719 0.892666i \(-0.648832\pi\)
−0.450719 + 0.892666i \(0.648832\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 20.3684 1.01211
\(406\) 0 0
\(407\) 16.7368 0.829614
\(408\) 0 0
\(409\) 5.31444 0.262782 0.131391 0.991331i \(-0.458056\pi\)
0.131391 + 0.991331i \(0.458056\pi\)
\(410\) 0 0
\(411\) −7.58801 −0.374289
\(412\) 0 0
\(413\) −8.09003 −0.398084
\(414\) 0 0
\(415\) −50.9557 −2.50131
\(416\) 0 0
\(417\) −0.0297872 −0.00145868
\(418\) 0 0
\(419\) −10.3428 −0.505278 −0.252639 0.967561i \(-0.581298\pi\)
−0.252639 + 0.967561i \(0.581298\pi\)
\(420\) 0 0
\(421\) −31.2203 −1.52158 −0.760791 0.648997i \(-0.775189\pi\)
−0.760791 + 0.648997i \(0.775189\pi\)
\(422\) 0 0
\(423\) 16.2992 0.792495
\(424\) 0 0
\(425\) −3.18421 −0.154457
\(426\) 0 0
\(427\) −11.7562 −0.568923
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 20.9252 1.00793 0.503966 0.863724i \(-0.331874\pi\)
0.503966 + 0.863724i \(0.331874\pi\)
\(432\) 0 0
\(433\) 15.8504 0.761721 0.380861 0.924632i \(-0.375628\pi\)
0.380861 + 0.924632i \(0.375628\pi\)
\(434\) 0 0
\(435\) −1.42240 −0.0681988
\(436\) 0 0
\(437\) −2.32967 −0.111443
\(438\) 0 0
\(439\) 10.7714 0.514093 0.257046 0.966399i \(-0.417251\pi\)
0.257046 + 0.966399i \(0.417251\pi\)
\(440\) 0 0
\(441\) −2.78600 −0.132667
\(442\) 0 0
\(443\) 7.70079 0.365875 0.182938 0.983125i \(-0.441439\pi\)
0.182938 + 0.983125i \(0.441439\pi\)
\(444\) 0 0
\(445\) 3.44322 0.163224
\(446\) 0 0
\(447\) −6.98959 −0.330596
\(448\) 0 0
\(449\) 10.4793 0.494548 0.247274 0.968946i \(-0.420465\pi\)
0.247274 + 0.968946i \(0.420465\pi\)
\(450\) 0 0
\(451\) 60.3781 2.84309
\(452\) 0 0
\(453\) 6.06439 0.284930
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 16.8429 0.787877 0.393938 0.919137i \(-0.371112\pi\)
0.393938 + 0.919137i \(0.371112\pi\)
\(458\) 0 0
\(459\) 2.67660 0.124933
\(460\) 0 0
\(461\) 36.2396 1.68785 0.843924 0.536463i \(-0.180240\pi\)
0.843924 + 0.536463i \(0.180240\pi\)
\(462\) 0 0
\(463\) 8.55263 0.397474 0.198737 0.980053i \(-0.436316\pi\)
0.198737 + 0.980053i \(0.436316\pi\)
\(464\) 0 0
\(465\) −3.78600 −0.175572
\(466\) 0 0
\(467\) −32.5872 −1.50796 −0.753979 0.656899i \(-0.771868\pi\)
−0.753979 + 0.656899i \(0.771868\pi\)
\(468\) 0 0
\(469\) −13.8760 −0.640736
\(470\) 0 0
\(471\) −9.28398 −0.427783
\(472\) 0 0
\(473\) 66.5277 3.05895
\(474\) 0 0
\(475\) −9.31444 −0.427376
\(476\) 0 0
\(477\) −27.2638 −1.24833
\(478\) 0 0
\(479\) 19.3282 0.883129 0.441564 0.897230i \(-0.354424\pi\)
0.441564 + 0.897230i \(0.354424\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −0.368420 −0.0167637
\(484\) 0 0
\(485\) 11.6274 0.527975
\(486\) 0 0
\(487\) 9.96125 0.451387 0.225694 0.974198i \(-0.427535\pi\)
0.225694 + 0.974198i \(0.427535\pi\)
\(488\) 0 0
\(489\) 2.81724 0.127400
\(490\) 0 0
\(491\) −1.01041 −0.0455993 −0.0227996 0.999740i \(-0.507258\pi\)
−0.0227996 + 0.999740i \(0.507258\pi\)
\(492\) 0 0
\(493\) 1.07480 0.0484067
\(494\) 0 0
\(495\) 45.6025 2.04968
\(496\) 0 0
\(497\) −9.16484 −0.411099
\(498\) 0 0
\(499\) −16.0692 −0.719357 −0.359678 0.933076i \(-0.617114\pi\)
−0.359678 + 0.933076i \(0.617114\pi\)
\(500\) 0 0
\(501\) −4.78041 −0.213573
\(502\) 0 0
\(503\) −32.4536 −1.44704 −0.723518 0.690305i \(-0.757476\pi\)
−0.723518 + 0.690305i \(0.757476\pi\)
\(504\) 0 0
\(505\) 14.7756 0.657505
\(506\) 0 0
\(507\) −6.01378 −0.267081
\(508\) 0 0
\(509\) −29.4916 −1.30719 −0.653596 0.756843i \(-0.726741\pi\)
−0.653596 + 0.756843i \(0.726741\pi\)
\(510\) 0 0
\(511\) 6.18421 0.273573
\(512\) 0 0
\(513\) 7.82957 0.345684
\(514\) 0 0
\(515\) −0.368420 −0.0162345
\(516\) 0 0
\(517\) 33.4737 1.47217
\(518\) 0 0
\(519\) −0.453636 −0.0199124
\(520\) 0 0
\(521\) 13.5582 0.593997 0.296998 0.954878i \(-0.404014\pi\)
0.296998 + 0.954878i \(0.404014\pi\)
\(522\) 0 0
\(523\) 26.9944 1.18038 0.590191 0.807263i \(-0.299052\pi\)
0.590191 + 0.807263i \(0.299052\pi\)
\(524\) 0 0
\(525\) −1.47301 −0.0642875
\(526\) 0 0
\(527\) 2.86081 0.124619
\(528\) 0 0
\(529\) −22.3657 −0.972423
\(530\) 0 0
\(531\) −22.5389 −0.978103
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −6.14961 −0.265871
\(536\) 0 0
\(537\) 7.34837 0.317106
\(538\) 0 0
\(539\) −5.72161 −0.246447
\(540\) 0 0
\(541\) −40.8864 −1.75785 −0.878923 0.476964i \(-0.841737\pi\)
−0.878923 + 0.476964i \(0.841737\pi\)
\(542\) 0 0
\(543\) −9.36282 −0.401797
\(544\) 0 0
\(545\) −40.0513 −1.71561
\(546\) 0 0
\(547\) −42.9557 −1.83665 −0.918326 0.395826i \(-0.870458\pi\)
−0.918326 + 0.395826i \(0.870458\pi\)
\(548\) 0 0
\(549\) −32.7528 −1.39786
\(550\) 0 0
\(551\) 3.14401 0.133939
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) −3.87122 −0.164324
\(556\) 0 0
\(557\) 45.6233 1.93312 0.966560 0.256439i \(-0.0825493\pi\)
0.966560 + 0.256439i \(0.0825493\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 2.64681 0.111748
\(562\) 0 0
\(563\) 37.2549 1.57011 0.785053 0.619429i \(-0.212636\pi\)
0.785053 + 0.619429i \(0.212636\pi\)
\(564\) 0 0
\(565\) 39.8809 1.67780
\(566\) 0 0
\(567\) −7.11982 −0.299004
\(568\) 0 0
\(569\) 24.7714 1.03847 0.519236 0.854631i \(-0.326216\pi\)
0.519236 + 0.854631i \(0.326216\pi\)
\(570\) 0 0
\(571\) 23.1953 0.970693 0.485346 0.874322i \(-0.338694\pi\)
0.485346 + 0.874322i \(0.338694\pi\)
\(572\) 0 0
\(573\) −3.17380 −0.132587
\(574\) 0 0
\(575\) 2.53595 0.105757
\(576\) 0 0
\(577\) −28.6289 −1.19184 −0.595918 0.803045i \(-0.703212\pi\)
−0.595918 + 0.803045i \(0.703212\pi\)
\(578\) 0 0
\(579\) −5.73125 −0.238182
\(580\) 0 0
\(581\) 17.8116 0.738951
\(582\) 0 0
\(583\) −55.9917 −2.31894
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.7424 0.567211 0.283606 0.958941i \(-0.408469\pi\)
0.283606 + 0.958941i \(0.408469\pi\)
\(588\) 0 0
\(589\) 8.36842 0.344815
\(590\) 0 0
\(591\) −7.81164 −0.321328
\(592\) 0 0
\(593\) −1.81164 −0.0743953 −0.0371976 0.999308i \(-0.511843\pi\)
−0.0371976 + 0.999308i \(0.511843\pi\)
\(594\) 0 0
\(595\) 2.86081 0.117282
\(596\) 0 0
\(597\) 2.62117 0.107277
\(598\) 0 0
\(599\) 20.6218 0.842585 0.421293 0.906925i \(-0.361577\pi\)
0.421293 + 0.906925i \(0.361577\pi\)
\(600\) 0 0
\(601\) −15.8504 −0.646551 −0.323276 0.946305i \(-0.604784\pi\)
−0.323276 + 0.946305i \(0.604784\pi\)
\(602\) 0 0
\(603\) −38.6587 −1.57430
\(604\) 0 0
\(605\) 62.1849 2.52817
\(606\) 0 0
\(607\) −10.4745 −0.425145 −0.212573 0.977145i \(-0.568184\pi\)
−0.212573 + 0.977145i \(0.568184\pi\)
\(608\) 0 0
\(609\) 0.497202 0.0201477
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −10.4716 −0.422942 −0.211471 0.977384i \(-0.567825\pi\)
−0.211471 + 0.977384i \(0.567825\pi\)
\(614\) 0 0
\(615\) −13.9654 −0.563139
\(616\) 0 0
\(617\) 6.36842 0.256383 0.128191 0.991749i \(-0.459083\pi\)
0.128191 + 0.991749i \(0.459083\pi\)
\(618\) 0 0
\(619\) 0.0900320 0.00361869 0.00180935 0.999998i \(-0.499424\pi\)
0.00180935 + 0.999998i \(0.499424\pi\)
\(620\) 0 0
\(621\) −2.13168 −0.0855414
\(622\) 0 0
\(623\) −1.20359 −0.0482206
\(624\) 0 0
\(625\) −30.7819 −1.23127
\(626\) 0 0
\(627\) 7.74244 0.309203
\(628\) 0 0
\(629\) 2.92520 0.116635
\(630\) 0 0
\(631\) −10.3040 −0.410197 −0.205098 0.978741i \(-0.565751\pi\)
−0.205098 + 0.978741i \(0.565751\pi\)
\(632\) 0 0
\(633\) 4.86562 0.193391
\(634\) 0 0
\(635\) 15.0707 0.598061
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −25.5333 −1.01008
\(640\) 0 0
\(641\) 30.5664 1.20730 0.603650 0.797249i \(-0.293712\pi\)
0.603650 + 0.797249i \(0.293712\pi\)
\(642\) 0 0
\(643\) 8.57345 0.338104 0.169052 0.985607i \(-0.445929\pi\)
0.169052 + 0.985607i \(0.445929\pi\)
\(644\) 0 0
\(645\) −15.3878 −0.605894
\(646\) 0 0
\(647\) −6.02082 −0.236703 −0.118352 0.992972i \(-0.537761\pi\)
−0.118352 + 0.992972i \(0.537761\pi\)
\(648\) 0 0
\(649\) −46.2880 −1.81696
\(650\) 0 0
\(651\) 1.32340 0.0518683
\(652\) 0 0
\(653\) 13.4432 0.526074 0.263037 0.964786i \(-0.415276\pi\)
0.263037 + 0.964786i \(0.415276\pi\)
\(654\) 0 0
\(655\) 18.5872 0.726263
\(656\) 0 0
\(657\) 17.2292 0.672176
\(658\) 0 0
\(659\) −8.96876 −0.349373 −0.174687 0.984624i \(-0.555891\pi\)
−0.174687 + 0.984624i \(0.555891\pi\)
\(660\) 0 0
\(661\) −23.8116 −0.926166 −0.463083 0.886315i \(-0.653257\pi\)
−0.463083 + 0.886315i \(0.653257\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.36842 0.324513
\(666\) 0 0
\(667\) −0.855989 −0.0331440
\(668\) 0 0
\(669\) 5.83247 0.225496
\(670\) 0 0
\(671\) −67.2645 −2.59672
\(672\) 0 0
\(673\) 41.8809 1.61439 0.807194 0.590286i \(-0.200985\pi\)
0.807194 + 0.590286i \(0.200985\pi\)
\(674\) 0 0
\(675\) −8.52284 −0.328044
\(676\) 0 0
\(677\) −11.1261 −0.427610 −0.213805 0.976876i \(-0.568586\pi\)
−0.213805 + 0.976876i \(0.568586\pi\)
\(678\) 0 0
\(679\) −4.06439 −0.155977
\(680\) 0 0
\(681\) −0.628212 −0.0240732
\(682\) 0 0
\(683\) 1.83247 0.0701174 0.0350587 0.999385i \(-0.488838\pi\)
0.0350587 + 0.999385i \(0.488838\pi\)
\(684\) 0 0
\(685\) −46.9259 −1.79295
\(686\) 0 0
\(687\) 1.05398 0.0402118
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −27.4183 −1.04304 −0.521520 0.853239i \(-0.674635\pi\)
−0.521520 + 0.853239i \(0.674635\pi\)
\(692\) 0 0
\(693\) −15.9404 −0.605527
\(694\) 0 0
\(695\) −0.184210 −0.00698749
\(696\) 0 0
\(697\) 10.5526 0.399709
\(698\) 0 0
\(699\) 5.28398 0.199859
\(700\) 0 0
\(701\) 32.0305 1.20977 0.604887 0.796311i \(-0.293218\pi\)
0.604887 + 0.796311i \(0.293218\pi\)
\(702\) 0 0
\(703\) 8.55678 0.322725
\(704\) 0 0
\(705\) −7.74244 −0.291597
\(706\) 0 0
\(707\) −5.16484 −0.194244
\(708\) 0 0
\(709\) −18.2396 −0.685004 −0.342502 0.939517i \(-0.611274\pi\)
−0.342502 + 0.939517i \(0.611274\pi\)
\(710\) 0 0
\(711\) 22.2880 0.835866
\(712\) 0 0
\(713\) −2.27839 −0.0853263
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 11.0171 0.411443
\(718\) 0 0
\(719\) 30.4030 1.13384 0.566921 0.823772i \(-0.308135\pi\)
0.566921 + 0.823772i \(0.308135\pi\)
\(720\) 0 0
\(721\) 0.128782 0.00479609
\(722\) 0 0
\(723\) 9.62262 0.357869
\(724\) 0 0
\(725\) −3.42240 −0.127105
\(726\) 0 0
\(727\) −8.92520 −0.331017 −0.165509 0.986208i \(-0.552927\pi\)
−0.165509 + 0.986208i \(0.552927\pi\)
\(728\) 0 0
\(729\) −16.1213 −0.597084
\(730\) 0 0
\(731\) 11.6274 0.430056
\(732\) 0 0
\(733\) 1.35319 0.0499813 0.0249906 0.999688i \(-0.492044\pi\)
0.0249906 + 0.999688i \(0.492044\pi\)
\(734\) 0 0
\(735\) 1.32340 0.0488145
\(736\) 0 0
\(737\) −79.3933 −2.92449
\(738\) 0 0
\(739\) −26.0263 −0.957393 −0.478697 0.877980i \(-0.658891\pi\)
−0.478697 + 0.877980i \(0.658891\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 32.9073 1.20725 0.603625 0.797268i \(-0.293722\pi\)
0.603625 + 0.797268i \(0.293722\pi\)
\(744\) 0 0
\(745\) −43.2251 −1.58364
\(746\) 0 0
\(747\) 49.6233 1.81562
\(748\) 0 0
\(749\) 2.14961 0.0785449
\(750\) 0 0
\(751\) 13.6025 0.496361 0.248180 0.968714i \(-0.420167\pi\)
0.248180 + 0.968714i \(0.420167\pi\)
\(752\) 0 0
\(753\) −13.7908 −0.502565
\(754\) 0 0
\(755\) 37.5035 1.36489
\(756\) 0 0
\(757\) 6.25342 0.227284 0.113642 0.993522i \(-0.463748\pi\)
0.113642 + 0.993522i \(0.463748\pi\)
\(758\) 0 0
\(759\) −2.10796 −0.0765140
\(760\) 0 0
\(761\) 25.5541 0.926335 0.463167 0.886271i \(-0.346713\pi\)
0.463167 + 0.886271i \(0.346713\pi\)
\(762\) 0 0
\(763\) 14.0000 0.506834
\(764\) 0 0
\(765\) 7.97021 0.288164
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 25.7521 0.928643 0.464322 0.885667i \(-0.346298\pi\)
0.464322 + 0.885667i \(0.346298\pi\)
\(770\) 0 0
\(771\) 9.55118 0.343977
\(772\) 0 0
\(773\) 14.7064 0.528952 0.264476 0.964392i \(-0.414801\pi\)
0.264476 + 0.964392i \(0.414801\pi\)
\(774\) 0 0
\(775\) −9.10941 −0.327220
\(776\) 0 0
\(777\) 1.35319 0.0485455
\(778\) 0 0
\(779\) 30.8685 1.10598
\(780\) 0 0
\(781\) −52.4376 −1.87637
\(782\) 0 0
\(783\) 2.87681 0.102809
\(784\) 0 0
\(785\) −57.4141 −2.04920
\(786\) 0 0
\(787\) 36.5277 1.30207 0.651035 0.759048i \(-0.274335\pi\)
0.651035 + 0.759048i \(0.274335\pi\)
\(788\) 0 0
\(789\) 3.18566 0.113412
\(790\) 0 0
\(791\) −13.9404 −0.495664
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 12.9508 0.459319
\(796\) 0 0
\(797\) 52.2313 1.85013 0.925065 0.379810i \(-0.124011\pi\)
0.925065 + 0.379810i \(0.124011\pi\)
\(798\) 0 0
\(799\) 5.85039 0.206972
\(800\) 0 0
\(801\) −3.35319 −0.118479
\(802\) 0 0
\(803\) 35.3836 1.24866
\(804\) 0 0
\(805\) −2.27839 −0.0803026
\(806\) 0 0
\(807\) 10.0997 0.355525
\(808\) 0 0
\(809\) 13.7424 0.483158 0.241579 0.970381i \(-0.422335\pi\)
0.241579 + 0.970381i \(0.422335\pi\)
\(810\) 0 0
\(811\) −11.9058 −0.418070 −0.209035 0.977908i \(-0.567032\pi\)
−0.209035 + 0.977908i \(0.567032\pi\)
\(812\) 0 0
\(813\) 0.487569 0.0170998
\(814\) 0 0
\(815\) 17.4224 0.610280
\(816\) 0 0
\(817\) 34.0125 1.18995
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −42.6260 −1.48766 −0.743829 0.668370i \(-0.766992\pi\)
−0.743829 + 0.668370i \(0.766992\pi\)
\(822\) 0 0
\(823\) −17.1648 −0.598329 −0.299164 0.954202i \(-0.596708\pi\)
−0.299164 + 0.954202i \(0.596708\pi\)
\(824\) 0 0
\(825\) −8.42799 −0.293425
\(826\) 0 0
\(827\) −27.6316 −0.960844 −0.480422 0.877037i \(-0.659517\pi\)
−0.480422 + 0.877037i \(0.659517\pi\)
\(828\) 0 0
\(829\) −32.5568 −1.13074 −0.565372 0.824836i \(-0.691267\pi\)
−0.565372 + 0.824836i \(0.691267\pi\)
\(830\) 0 0
\(831\) −9.06361 −0.314413
\(832\) 0 0
\(833\) −1.00000 −0.0346479
\(834\) 0 0
\(835\) −29.5630 −1.02307
\(836\) 0 0
\(837\) 7.65722 0.264672
\(838\) 0 0
\(839\) 29.5928 1.02166 0.510829 0.859682i \(-0.329339\pi\)
0.510829 + 0.859682i \(0.329339\pi\)
\(840\) 0 0
\(841\) −27.8448 −0.960165
\(842\) 0 0
\(843\) −9.42463 −0.324601
\(844\) 0 0
\(845\) −37.1905 −1.27939
\(846\) 0 0
\(847\) −21.7368 −0.746887
\(848\) 0 0
\(849\) −2.53818 −0.0871101
\(850\) 0 0
\(851\) −2.32967 −0.0798601
\(852\) 0 0
\(853\) −56.9348 −1.94941 −0.974706 0.223492i \(-0.928254\pi\)
−0.974706 + 0.223492i \(0.928254\pi\)
\(854\) 0 0
\(855\) 23.3144 0.797337
\(856\) 0 0
\(857\) 34.7804 1.18808 0.594038 0.804437i \(-0.297533\pi\)
0.594038 + 0.804437i \(0.297533\pi\)
\(858\) 0 0
\(859\) −33.5637 −1.14518 −0.572590 0.819842i \(-0.694061\pi\)
−0.572590 + 0.819842i \(0.694061\pi\)
\(860\) 0 0
\(861\) 4.88163 0.166366
\(862\) 0 0
\(863\) 51.0409 1.73745 0.868726 0.495293i \(-0.164939\pi\)
0.868726 + 0.495293i \(0.164939\pi\)
\(864\) 0 0
\(865\) −2.80538 −0.0953857
\(866\) 0 0
\(867\) 0.462598 0.0157107
\(868\) 0 0
\(869\) 45.7729 1.55274
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −11.3234 −0.383239
\(874\) 0 0
\(875\) 5.19462 0.175610
\(876\) 0 0
\(877\) 30.7785 1.03932 0.519658 0.854374i \(-0.326059\pi\)
0.519658 + 0.854374i \(0.326059\pi\)
\(878\) 0 0
\(879\) −1.73125 −0.0583934
\(880\) 0 0
\(881\) −17.3580 −0.584806 −0.292403 0.956295i \(-0.594455\pi\)
−0.292403 + 0.956295i \(0.594455\pi\)
\(882\) 0 0
\(883\) −48.7208 −1.63959 −0.819793 0.572660i \(-0.805912\pi\)
−0.819793 + 0.572660i \(0.805912\pi\)
\(884\) 0 0
\(885\) 10.7064 0.359891
\(886\) 0 0
\(887\) −56.3761 −1.89293 −0.946463 0.322813i \(-0.895371\pi\)
−0.946463 + 0.322813i \(0.895371\pi\)
\(888\) 0 0
\(889\) −5.26798 −0.176682
\(890\) 0 0
\(891\) −40.7368 −1.36474
\(892\) 0 0
\(893\) 17.1136 0.572683
\(894\) 0 0
\(895\) 45.4439 1.51902
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.07480 0.102550
\(900\) 0 0
\(901\) −9.78600 −0.326019
\(902\) 0 0
\(903\) 5.37883 0.178996
\(904\) 0 0
\(905\) −57.9017 −1.92472
\(906\) 0 0
\(907\) −44.4197 −1.47493 −0.737466 0.675384i \(-0.763978\pi\)
−0.737466 + 0.675384i \(0.763978\pi\)
\(908\) 0 0
\(909\) −14.3892 −0.477261
\(910\) 0 0
\(911\) −4.32677 −0.143352 −0.0716762 0.997428i \(-0.522835\pi\)
−0.0716762 + 0.997428i \(0.522835\pi\)
\(912\) 0 0
\(913\) 101.911 3.37277
\(914\) 0 0
\(915\) 15.5582 0.514339
\(916\) 0 0
\(917\) −6.49720 −0.214557
\(918\) 0 0
\(919\) 29.2203 0.963888 0.481944 0.876202i \(-0.339931\pi\)
0.481944 + 0.876202i \(0.339931\pi\)
\(920\) 0 0
\(921\) 9.72161 0.320338
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −9.31444 −0.306257
\(926\) 0 0
\(927\) 0.358787 0.0117841
\(928\) 0 0
\(929\) −1.91478 −0.0628220 −0.0314110 0.999507i \(-0.510000\pi\)
−0.0314110 + 0.999507i \(0.510000\pi\)
\(930\) 0 0
\(931\) −2.92520 −0.0958695
\(932\) 0 0
\(933\) 10.1794 0.333258
\(934\) 0 0
\(935\) 16.3684 0.535305
\(936\) 0 0
\(937\) −20.5693 −0.671970 −0.335985 0.941867i \(-0.609069\pi\)
−0.335985 + 0.941867i \(0.609069\pi\)
\(938\) 0 0
\(939\) 2.27616 0.0742797
\(940\) 0 0
\(941\) 15.7979 0.514996 0.257498 0.966279i \(-0.417102\pi\)
0.257498 + 0.966279i \(0.417102\pi\)
\(942\) 0 0
\(943\) −8.40427 −0.273681
\(944\) 0 0
\(945\) 7.65722 0.249089
\(946\) 0 0
\(947\) 16.6856 0.542208 0.271104 0.962550i \(-0.412611\pi\)
0.271104 + 0.962550i \(0.412611\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −3.86158 −0.125220
\(952\) 0 0
\(953\) 17.7085 0.573635 0.286817 0.957985i \(-0.407403\pi\)
0.286817 + 0.957985i \(0.407403\pi\)
\(954\) 0 0
\(955\) −19.6274 −0.635129
\(956\) 0 0
\(957\) 2.84480 0.0919593
\(958\) 0 0
\(959\) 16.4030 0.529681
\(960\) 0 0
\(961\) −22.8158 −0.735993
\(962\) 0 0
\(963\) 5.98881 0.192987
\(964\) 0 0
\(965\) −35.4432 −1.14096
\(966\) 0 0
\(967\) −12.1426 −0.390478 −0.195239 0.980756i \(-0.562548\pi\)
−0.195239 + 0.980756i \(0.562548\pi\)
\(968\) 0 0
\(969\) 1.35319 0.0434708
\(970\) 0 0
\(971\) 5.07480 0.162858 0.0814291 0.996679i \(-0.474052\pi\)
0.0814291 + 0.996679i \(0.474052\pi\)
\(972\) 0 0
\(973\) 0.0643910 0.00206428
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −33.1876 −1.06176 −0.530882 0.847446i \(-0.678139\pi\)
−0.530882 + 0.847446i \(0.678139\pi\)
\(978\) 0 0
\(979\) −6.88645 −0.220092
\(980\) 0 0
\(981\) 39.0040 1.24530
\(982\) 0 0
\(983\) −40.3761 −1.28780 −0.643899 0.765110i \(-0.722684\pi\)
−0.643899 + 0.765110i \(0.722684\pi\)
\(984\) 0 0
\(985\) −48.3088 −1.53925
\(986\) 0 0
\(987\) 2.70638 0.0861451
\(988\) 0 0
\(989\) −9.26026 −0.294459
\(990\) 0 0
\(991\) −30.9460 −0.983033 −0.491516 0.870868i \(-0.663557\pi\)
−0.491516 + 0.870868i \(0.663557\pi\)
\(992\) 0 0
\(993\) −8.78455 −0.278769
\(994\) 0 0
\(995\) 16.2099 0.513887
\(996\) 0 0
\(997\) −41.9334 −1.32804 −0.664022 0.747713i \(-0.731152\pi\)
−0.664022 + 0.747713i \(0.731152\pi\)
\(998\) 0 0
\(999\) 7.82957 0.247717
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 1904.2.a.m.1.2 3
4.3 odd 2 952.2.a.f.1.2 3
8.3 odd 2 7616.2.a.bc.1.2 3
8.5 even 2 7616.2.a.be.1.2 3
12.11 even 2 8568.2.a.x.1.1 3
28.27 even 2 6664.2.a.j.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.f.1.2 3 4.3 odd 2
1904.2.a.m.1.2 3 1.1 even 1 trivial
6664.2.a.j.1.2 3 28.27 even 2
7616.2.a.bc.1.2 3 8.3 odd 2
7616.2.a.be.1.2 3 8.5 even 2
8568.2.a.x.1.1 3 12.11 even 2