Properties

Label 3800.2.a.z.1.6
Level $3800$
Weight $2$
Character 3800.1
Self dual yes
Analytic conductor $30.343$
Analytic rank $1$
Dimension $6$
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] = [3800,2,Mod(1,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, 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("3800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.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: \(30.3431527681\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.253565184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 11x^{4} + 20x^{3} + 22x^{2} - 32x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-2.84742\) of defining polynomial
Character \(\chi\) \(=\) 3800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.84742 q^{3} +0.145034 q^{7} +5.10782 q^{9} -5.71774 q^{11} -5.24345 q^{13} -7.15378 q^{17} +1.00000 q^{19} +0.412974 q^{21} +0.622762 q^{23} +6.00186 q^{27} -5.46811 q^{29} -3.77513 q^{31} -16.2808 q^{33} +5.03553 q^{37} -14.9303 q^{39} +5.77513 q^{41} +3.32308 q^{43} -5.85683 q^{47} -6.97897 q^{49} -20.3699 q^{51} +6.97219 q^{53} +2.84742 q^{57} +9.09089 q^{59} -8.16707 q^{61} +0.740810 q^{63} -13.6583 q^{67} +1.77327 q^{69} +2.41484 q^{71} -7.44385 q^{73} -0.829270 q^{77} -9.69485 q^{79} +1.76638 q^{81} +2.17116 q^{83} -15.5700 q^{87} +3.90782 q^{89} -0.760481 q^{91} -10.7494 q^{93} -4.98328 q^{97} -29.2052 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 6 q^{7} + 8 q^{9} - 2 q^{11} - 14 q^{13} - 10 q^{17} + 6 q^{19} + 18 q^{21} - 2 q^{23} - 2 q^{27} - 2 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{37} - 18 q^{39} + 4 q^{41} - 4 q^{43} - 4 q^{47}+ \cdots - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.84742 1.64396 0.821980 0.569516i \(-0.192869\pi\)
0.821980 + 0.569516i \(0.192869\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.145034 0.0548179 0.0274089 0.999624i \(-0.491274\pi\)
0.0274089 + 0.999624i \(0.491274\pi\)
\(8\) 0 0
\(9\) 5.10782 1.70261
\(10\) 0 0
\(11\) −5.71774 −1.72396 −0.861982 0.506938i \(-0.830777\pi\)
−0.861982 + 0.506938i \(0.830777\pi\)
\(12\) 0 0
\(13\) −5.24345 −1.45427 −0.727136 0.686493i \(-0.759149\pi\)
−0.727136 + 0.686493i \(0.759149\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.15378 −1.73505 −0.867524 0.497396i \(-0.834290\pi\)
−0.867524 + 0.497396i \(0.834290\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.412974 0.0901184
\(22\) 0 0
\(23\) 0.622762 0.129855 0.0649274 0.997890i \(-0.479318\pi\)
0.0649274 + 0.997890i \(0.479318\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 6.00186 1.15506
\(28\) 0 0
\(29\) −5.46811 −1.01540 −0.507702 0.861533i \(-0.669505\pi\)
−0.507702 + 0.861533i \(0.669505\pi\)
\(30\) 0 0
\(31\) −3.77513 −0.678033 −0.339017 0.940780i \(-0.610094\pi\)
−0.339017 + 0.940780i \(0.610094\pi\)
\(32\) 0 0
\(33\) −16.2808 −2.83413
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.03553 0.827836 0.413918 0.910314i \(-0.364160\pi\)
0.413918 + 0.910314i \(0.364160\pi\)
\(38\) 0 0
\(39\) −14.9303 −2.39077
\(40\) 0 0
\(41\) 5.77513 0.901924 0.450962 0.892543i \(-0.351081\pi\)
0.450962 + 0.892543i \(0.351081\pi\)
\(42\) 0 0
\(43\) 3.32308 0.506765 0.253382 0.967366i \(-0.418457\pi\)
0.253382 + 0.967366i \(0.418457\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.85683 −0.854306 −0.427153 0.904179i \(-0.640483\pi\)
−0.427153 + 0.904179i \(0.640483\pi\)
\(48\) 0 0
\(49\) −6.97897 −0.996995
\(50\) 0 0
\(51\) −20.3699 −2.85235
\(52\) 0 0
\(53\) 6.97219 0.957704 0.478852 0.877896i \(-0.341053\pi\)
0.478852 + 0.877896i \(0.341053\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.84742 0.377150
\(58\) 0 0
\(59\) 9.09089 1.18353 0.591767 0.806109i \(-0.298431\pi\)
0.591767 + 0.806109i \(0.298431\pi\)
\(60\) 0 0
\(61\) −8.16707 −1.04569 −0.522843 0.852429i \(-0.675129\pi\)
−0.522843 + 0.852429i \(0.675129\pi\)
\(62\) 0 0
\(63\) 0.740810 0.0933333
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −13.6583 −1.66863 −0.834313 0.551291i \(-0.814135\pi\)
−0.834313 + 0.551291i \(0.814135\pi\)
\(68\) 0 0
\(69\) 1.77327 0.213476
\(70\) 0 0
\(71\) 2.41484 0.286588 0.143294 0.989680i \(-0.454230\pi\)
0.143294 + 0.989680i \(0.454230\pi\)
\(72\) 0 0
\(73\) −7.44385 −0.871237 −0.435619 0.900131i \(-0.643470\pi\)
−0.435619 + 0.900131i \(0.643470\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.829270 −0.0945041
\(78\) 0 0
\(79\) −9.69485 −1.09076 −0.545378 0.838190i \(-0.683614\pi\)
−0.545378 + 0.838190i \(0.683614\pi\)
\(80\) 0 0
\(81\) 1.76638 0.196264
\(82\) 0 0
\(83\) 2.17116 0.238316 0.119158 0.992875i \(-0.461981\pi\)
0.119158 + 0.992875i \(0.461981\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −15.5700 −1.66928
\(88\) 0 0
\(89\) 3.90782 0.414228 0.207114 0.978317i \(-0.433593\pi\)
0.207114 + 0.978317i \(0.433593\pi\)
\(90\) 0 0
\(91\) −0.760481 −0.0797201
\(92\) 0 0
\(93\) −10.7494 −1.11466
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.98328 −0.505976 −0.252988 0.967469i \(-0.581413\pi\)
−0.252988 + 0.967469i \(0.581413\pi\)
\(98\) 0 0
\(99\) −29.2052 −2.93524
\(100\) 0 0
\(101\) −5.35148 −0.532493 −0.266246 0.963905i \(-0.585783\pi\)
−0.266246 + 0.963905i \(0.585783\pi\)
\(102\) 0 0
\(103\) 3.59495 0.354221 0.177111 0.984191i \(-0.443325\pi\)
0.177111 + 0.984191i \(0.443325\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.3518 1.48412 0.742059 0.670335i \(-0.233850\pi\)
0.742059 + 0.670335i \(0.233850\pi\)
\(108\) 0 0
\(109\) 2.20105 0.210823 0.105411 0.994429i \(-0.466384\pi\)
0.105411 + 0.994429i \(0.466384\pi\)
\(110\) 0 0
\(111\) 14.3383 1.36093
\(112\) 0 0
\(113\) 12.3613 1.16286 0.581428 0.813598i \(-0.302494\pi\)
0.581428 + 0.813598i \(0.302494\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −26.7826 −2.47605
\(118\) 0 0
\(119\) −1.03754 −0.0951116
\(120\) 0 0
\(121\) 21.6926 1.97205
\(122\) 0 0
\(123\) 16.4442 1.48273
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −18.0907 −1.60529 −0.802644 0.596459i \(-0.796574\pi\)
−0.802644 + 0.596459i \(0.796574\pi\)
\(128\) 0 0
\(129\) 9.46222 0.833102
\(130\) 0 0
\(131\) 12.1244 1.05931 0.529655 0.848213i \(-0.322321\pi\)
0.529655 + 0.848213i \(0.322321\pi\)
\(132\) 0 0
\(133\) 0.145034 0.0125761
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 20.0385 1.71201 0.856004 0.516969i \(-0.172940\pi\)
0.856004 + 0.516969i \(0.172940\pi\)
\(138\) 0 0
\(139\) 11.3996 0.966905 0.483452 0.875371i \(-0.339383\pi\)
0.483452 + 0.875371i \(0.339383\pi\)
\(140\) 0 0
\(141\) −16.6769 −1.40445
\(142\) 0 0
\(143\) 29.9807 2.50711
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −19.8721 −1.63902
\(148\) 0 0
\(149\) −15.5818 −1.27651 −0.638257 0.769823i \(-0.720344\pi\)
−0.638257 + 0.769823i \(0.720344\pi\)
\(150\) 0 0
\(151\) 22.3717 1.82058 0.910292 0.413966i \(-0.135857\pi\)
0.910292 + 0.413966i \(0.135857\pi\)
\(152\) 0 0
\(153\) −36.5403 −2.95410
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −23.1047 −1.84395 −0.921976 0.387246i \(-0.873426\pi\)
−0.921976 + 0.387246i \(0.873426\pi\)
\(158\) 0 0
\(159\) 19.8528 1.57443
\(160\) 0 0
\(161\) 0.0903219 0.00711836
\(162\) 0 0
\(163\) −11.3524 −0.889186 −0.444593 0.895733i \(-0.646652\pi\)
−0.444593 + 0.895733i \(0.646652\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.09989 −0.472024 −0.236012 0.971750i \(-0.575840\pi\)
−0.236012 + 0.971750i \(0.575840\pi\)
\(168\) 0 0
\(169\) 14.4938 1.11491
\(170\) 0 0
\(171\) 5.10782 0.390605
\(172\) 0 0
\(173\) 10.8951 0.828338 0.414169 0.910200i \(-0.364072\pi\)
0.414169 + 0.910200i \(0.364072\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 25.8856 1.94568
\(178\) 0 0
\(179\) 4.46123 0.333448 0.166724 0.986004i \(-0.446681\pi\)
0.166724 + 0.986004i \(0.446681\pi\)
\(180\) 0 0
\(181\) 3.58784 0.266682 0.133341 0.991070i \(-0.457429\pi\)
0.133341 + 0.991070i \(0.457429\pi\)
\(182\) 0 0
\(183\) −23.2551 −1.71907
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 40.9035 2.99116
\(188\) 0 0
\(189\) 0.870477 0.0633179
\(190\) 0 0
\(191\) 8.00777 0.579422 0.289711 0.957114i \(-0.406441\pi\)
0.289711 + 0.957114i \(0.406441\pi\)
\(192\) 0 0
\(193\) 1.45641 0.104835 0.0524173 0.998625i \(-0.483307\pi\)
0.0524173 + 0.998625i \(0.483307\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −26.3510 −1.87743 −0.938716 0.344692i \(-0.887983\pi\)
−0.938716 + 0.344692i \(0.887983\pi\)
\(198\) 0 0
\(199\) 12.1739 0.862985 0.431492 0.902117i \(-0.357987\pi\)
0.431492 + 0.902117i \(0.357987\pi\)
\(200\) 0 0
\(201\) −38.8909 −2.74316
\(202\) 0 0
\(203\) −0.793065 −0.0556622
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 3.18096 0.221092
\(208\) 0 0
\(209\) −5.71774 −0.395505
\(210\) 0 0
\(211\) −19.5088 −1.34304 −0.671521 0.740985i \(-0.734359\pi\)
−0.671521 + 0.740985i \(0.734359\pi\)
\(212\) 0 0
\(213\) 6.87606 0.471140
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −0.547524 −0.0371683
\(218\) 0 0
\(219\) −21.1958 −1.43228
\(220\) 0 0
\(221\) 37.5105 2.52323
\(222\) 0 0
\(223\) −14.9208 −0.999168 −0.499584 0.866265i \(-0.666514\pi\)
−0.499584 + 0.866265i \(0.666514\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −13.4422 −0.892192 −0.446096 0.894985i \(-0.647186\pi\)
−0.446096 + 0.894985i \(0.647186\pi\)
\(228\) 0 0
\(229\) −7.36620 −0.486772 −0.243386 0.969929i \(-0.578258\pi\)
−0.243386 + 0.969929i \(0.578258\pi\)
\(230\) 0 0
\(231\) −2.36128 −0.155361
\(232\) 0 0
\(233\) −4.24405 −0.278037 −0.139018 0.990290i \(-0.544395\pi\)
−0.139018 + 0.990290i \(0.544395\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −27.6053 −1.79316
\(238\) 0 0
\(239\) −15.3387 −0.992176 −0.496088 0.868272i \(-0.665231\pi\)
−0.496088 + 0.868272i \(0.665231\pi\)
\(240\) 0 0
\(241\) 26.7026 1.72006 0.860032 0.510240i \(-0.170443\pi\)
0.860032 + 0.510240i \(0.170443\pi\)
\(242\) 0 0
\(243\) −12.9760 −0.832408
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −5.24345 −0.333633
\(248\) 0 0
\(249\) 6.18221 0.391781
\(250\) 0 0
\(251\) 3.64463 0.230047 0.115023 0.993363i \(-0.463306\pi\)
0.115023 + 0.993363i \(0.463306\pi\)
\(252\) 0 0
\(253\) −3.56079 −0.223865
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −9.71103 −0.605757 −0.302879 0.953029i \(-0.597948\pi\)
−0.302879 + 0.953029i \(0.597948\pi\)
\(258\) 0 0
\(259\) 0.730325 0.0453802
\(260\) 0 0
\(261\) −27.9302 −1.72883
\(262\) 0 0
\(263\) 1.82052 0.112258 0.0561290 0.998424i \(-0.482124\pi\)
0.0561290 + 0.998424i \(0.482124\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 11.1272 0.680974
\(268\) 0 0
\(269\) 21.6353 1.31913 0.659563 0.751649i \(-0.270741\pi\)
0.659563 + 0.751649i \(0.270741\pi\)
\(270\) 0 0
\(271\) −7.54436 −0.458287 −0.229144 0.973393i \(-0.573593\pi\)
−0.229144 + 0.973393i \(0.573593\pi\)
\(272\) 0 0
\(273\) −2.16541 −0.131057
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 12.5559 0.754410 0.377205 0.926130i \(-0.376885\pi\)
0.377205 + 0.926130i \(0.376885\pi\)
\(278\) 0 0
\(279\) −19.2827 −1.15442
\(280\) 0 0
\(281\) 22.1947 1.32403 0.662013 0.749493i \(-0.269703\pi\)
0.662013 + 0.749493i \(0.269703\pi\)
\(282\) 0 0
\(283\) −20.6171 −1.22556 −0.612779 0.790254i \(-0.709948\pi\)
−0.612779 + 0.790254i \(0.709948\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.837592 0.0494415
\(288\) 0 0
\(289\) 34.1766 2.01039
\(290\) 0 0
\(291\) −14.1895 −0.831804
\(292\) 0 0
\(293\) −22.1421 −1.29356 −0.646778 0.762679i \(-0.723884\pi\)
−0.646778 + 0.762679i \(0.723884\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −34.3171 −1.99128
\(298\) 0 0
\(299\) −3.26542 −0.188844
\(300\) 0 0
\(301\) 0.481961 0.0277798
\(302\) 0 0
\(303\) −15.2379 −0.875397
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.80060 0.102766 0.0513828 0.998679i \(-0.483637\pi\)
0.0513828 + 0.998679i \(0.483637\pi\)
\(308\) 0 0
\(309\) 10.2364 0.582326
\(310\) 0 0
\(311\) −9.26554 −0.525401 −0.262700 0.964877i \(-0.584613\pi\)
−0.262700 + 0.964877i \(0.584613\pi\)
\(312\) 0 0
\(313\) 4.29779 0.242925 0.121463 0.992596i \(-0.461242\pi\)
0.121463 + 0.992596i \(0.461242\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4.61065 0.258960 0.129480 0.991582i \(-0.458669\pi\)
0.129480 + 0.991582i \(0.458669\pi\)
\(318\) 0 0
\(319\) 31.2653 1.75052
\(320\) 0 0
\(321\) 43.7132 2.43983
\(322\) 0 0
\(323\) −7.15378 −0.398047
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 6.26734 0.346584
\(328\) 0 0
\(329\) −0.849442 −0.0468312
\(330\) 0 0
\(331\) −0.934366 −0.0513574 −0.0256787 0.999670i \(-0.508175\pi\)
−0.0256787 + 0.999670i \(0.508175\pi\)
\(332\) 0 0
\(333\) 25.7206 1.40948
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −29.8588 −1.62651 −0.813256 0.581905i \(-0.802307\pi\)
−0.813256 + 0.581905i \(0.802307\pi\)
\(338\) 0 0
\(339\) 35.1980 1.91169
\(340\) 0 0
\(341\) 21.5852 1.16891
\(342\) 0 0
\(343\) −2.02743 −0.109471
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −10.6502 −0.571732 −0.285866 0.958270i \(-0.592281\pi\)
−0.285866 + 0.958270i \(0.592281\pi\)
\(348\) 0 0
\(349\) 29.3479 1.57096 0.785480 0.618888i \(-0.212416\pi\)
0.785480 + 0.618888i \(0.212416\pi\)
\(350\) 0 0
\(351\) −31.4705 −1.67977
\(352\) 0 0
\(353\) 14.3847 0.765619 0.382810 0.923827i \(-0.374957\pi\)
0.382810 + 0.923827i \(0.374957\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −2.95433 −0.156360
\(358\) 0 0
\(359\) −2.23773 −0.118103 −0.0590514 0.998255i \(-0.518808\pi\)
−0.0590514 + 0.998255i \(0.518808\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 61.7680 3.24198
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0.636873 0.0332445 0.0166222 0.999862i \(-0.494709\pi\)
0.0166222 + 0.999862i \(0.494709\pi\)
\(368\) 0 0
\(369\) 29.4983 1.53562
\(370\) 0 0
\(371\) 1.01121 0.0524993
\(372\) 0 0
\(373\) −16.2358 −0.840655 −0.420328 0.907372i \(-0.638085\pi\)
−0.420328 + 0.907372i \(0.638085\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 28.6718 1.47667
\(378\) 0 0
\(379\) −17.9624 −0.922667 −0.461333 0.887227i \(-0.652629\pi\)
−0.461333 + 0.887227i \(0.652629\pi\)
\(380\) 0 0
\(381\) −51.5118 −2.63903
\(382\) 0 0
\(383\) −4.06305 −0.207612 −0.103806 0.994598i \(-0.533102\pi\)
−0.103806 + 0.994598i \(0.533102\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 16.9737 0.862822
\(388\) 0 0
\(389\) −28.7507 −1.45772 −0.728859 0.684663i \(-0.759949\pi\)
−0.728859 + 0.684663i \(0.759949\pi\)
\(390\) 0 0
\(391\) −4.45510 −0.225304
\(392\) 0 0
\(393\) 34.5232 1.74147
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 11.6518 0.584786 0.292393 0.956298i \(-0.405548\pi\)
0.292393 + 0.956298i \(0.405548\pi\)
\(398\) 0 0
\(399\) 0.412974 0.0206746
\(400\) 0 0
\(401\) 1.98190 0.0989716 0.0494858 0.998775i \(-0.484242\pi\)
0.0494858 + 0.998775i \(0.484242\pi\)
\(402\) 0 0
\(403\) 19.7947 0.986045
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −28.7919 −1.42716
\(408\) 0 0
\(409\) 26.3984 1.30532 0.652660 0.757651i \(-0.273653\pi\)
0.652660 + 0.757651i \(0.273653\pi\)
\(410\) 0 0
\(411\) 57.0582 2.81447
\(412\) 0 0
\(413\) 1.31849 0.0648788
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 32.4596 1.58955
\(418\) 0 0
\(419\) −16.3134 −0.796963 −0.398481 0.917176i \(-0.630463\pi\)
−0.398481 + 0.917176i \(0.630463\pi\)
\(420\) 0 0
\(421\) −1.90815 −0.0929976 −0.0464988 0.998918i \(-0.514806\pi\)
−0.0464988 + 0.998918i \(0.514806\pi\)
\(422\) 0 0
\(423\) −29.9156 −1.45455
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.18451 −0.0573223
\(428\) 0 0
\(429\) 85.3678 4.12160
\(430\) 0 0
\(431\) 13.9013 0.669600 0.334800 0.942289i \(-0.391331\pi\)
0.334800 + 0.942289i \(0.391331\pi\)
\(432\) 0 0
\(433\) 4.93885 0.237346 0.118673 0.992933i \(-0.462136\pi\)
0.118673 + 0.992933i \(0.462136\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.622762 0.0297907
\(438\) 0 0
\(439\) −18.0918 −0.863473 −0.431736 0.902000i \(-0.642099\pi\)
−0.431736 + 0.902000i \(0.642099\pi\)
\(440\) 0 0
\(441\) −35.6473 −1.69749
\(442\) 0 0
\(443\) −8.13690 −0.386596 −0.193298 0.981140i \(-0.561918\pi\)
−0.193298 + 0.981140i \(0.561918\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −44.3681 −2.09854
\(448\) 0 0
\(449\) 19.9187 0.940020 0.470010 0.882661i \(-0.344250\pi\)
0.470010 + 0.882661i \(0.344250\pi\)
\(450\) 0 0
\(451\) −33.0207 −1.55488
\(452\) 0 0
\(453\) 63.7018 2.99297
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −21.4434 −1.00308 −0.501540 0.865134i \(-0.667233\pi\)
−0.501540 + 0.865134i \(0.667233\pi\)
\(458\) 0 0
\(459\) −42.9360 −2.00408
\(460\) 0 0
\(461\) 11.2325 0.523148 0.261574 0.965183i \(-0.415758\pi\)
0.261574 + 0.965183i \(0.415758\pi\)
\(462\) 0 0
\(463\) −37.8262 −1.75793 −0.878967 0.476882i \(-0.841767\pi\)
−0.878967 + 0.476882i \(0.841767\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 30.3388 1.40391 0.701955 0.712221i \(-0.252311\pi\)
0.701955 + 0.712221i \(0.252311\pi\)
\(468\) 0 0
\(469\) −1.98092 −0.0914705
\(470\) 0 0
\(471\) −65.7888 −3.03139
\(472\) 0 0
\(473\) −19.0005 −0.873645
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 35.6127 1.63059
\(478\) 0 0
\(479\) −35.3121 −1.61345 −0.806725 0.590927i \(-0.798762\pi\)
−0.806725 + 0.590927i \(0.798762\pi\)
\(480\) 0 0
\(481\) −26.4035 −1.20390
\(482\) 0 0
\(483\) 0.257185 0.0117023
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −6.88482 −0.311981 −0.155991 0.987759i \(-0.549857\pi\)
−0.155991 + 0.987759i \(0.549857\pi\)
\(488\) 0 0
\(489\) −32.3250 −1.46179
\(490\) 0 0
\(491\) −10.1116 −0.456328 −0.228164 0.973623i \(-0.573272\pi\)
−0.228164 + 0.973623i \(0.573272\pi\)
\(492\) 0 0
\(493\) 39.1177 1.76177
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.350234 0.0157102
\(498\) 0 0
\(499\) −36.4594 −1.63215 −0.816074 0.577947i \(-0.803854\pi\)
−0.816074 + 0.577947i \(0.803854\pi\)
\(500\) 0 0
\(501\) −17.3690 −0.775989
\(502\) 0 0
\(503\) −9.00269 −0.401410 −0.200705 0.979652i \(-0.564323\pi\)
−0.200705 + 0.979652i \(0.564323\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 41.2700 1.83286
\(508\) 0 0
\(509\) 10.3542 0.458943 0.229471 0.973315i \(-0.426300\pi\)
0.229471 + 0.973315i \(0.426300\pi\)
\(510\) 0 0
\(511\) −1.07961 −0.0477594
\(512\) 0 0
\(513\) 6.00186 0.264989
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 33.4878 1.47279
\(518\) 0 0
\(519\) 31.0229 1.36175
\(520\) 0 0
\(521\) 11.5209 0.504738 0.252369 0.967631i \(-0.418790\pi\)
0.252369 + 0.967631i \(0.418790\pi\)
\(522\) 0 0
\(523\) −24.0131 −1.05002 −0.525009 0.851097i \(-0.675938\pi\)
−0.525009 + 0.851097i \(0.675938\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 27.0065 1.17642
\(528\) 0 0
\(529\) −22.6122 −0.983138
\(530\) 0 0
\(531\) 46.4347 2.01509
\(532\) 0 0
\(533\) −30.2816 −1.31164
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 12.7030 0.548175
\(538\) 0 0
\(539\) 39.9039 1.71878
\(540\) 0 0
\(541\) −34.5131 −1.48383 −0.741916 0.670492i \(-0.766083\pi\)
−0.741916 + 0.670492i \(0.766083\pi\)
\(542\) 0 0
\(543\) 10.2161 0.438415
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.89383 0.166488 0.0832441 0.996529i \(-0.473472\pi\)
0.0832441 + 0.996529i \(0.473472\pi\)
\(548\) 0 0
\(549\) −41.7159 −1.78039
\(550\) 0 0
\(551\) −5.46811 −0.232949
\(552\) 0 0
\(553\) −1.40609 −0.0597929
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11.1816 0.473779 0.236889 0.971537i \(-0.423872\pi\)
0.236889 + 0.971537i \(0.423872\pi\)
\(558\) 0 0
\(559\) −17.4244 −0.736974
\(560\) 0 0
\(561\) 116.470 4.91735
\(562\) 0 0
\(563\) −2.54455 −0.107240 −0.0536200 0.998561i \(-0.517076\pi\)
−0.0536200 + 0.998561i \(0.517076\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.256186 0.0107588
\(568\) 0 0
\(569\) 2.09511 0.0878314 0.0439157 0.999035i \(-0.486017\pi\)
0.0439157 + 0.999035i \(0.486017\pi\)
\(570\) 0 0
\(571\) 36.2358 1.51642 0.758211 0.652009i \(-0.226074\pi\)
0.758211 + 0.652009i \(0.226074\pi\)
\(572\) 0 0
\(573\) 22.8015 0.952547
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 15.5756 0.648419 0.324209 0.945985i \(-0.394902\pi\)
0.324209 + 0.945985i \(0.394902\pi\)
\(578\) 0 0
\(579\) 4.14702 0.172344
\(580\) 0 0
\(581\) 0.314893 0.0130639
\(582\) 0 0
\(583\) −39.8652 −1.65105
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.54894 0.352853 0.176426 0.984314i \(-0.443546\pi\)
0.176426 + 0.984314i \(0.443546\pi\)
\(588\) 0 0
\(589\) −3.77513 −0.155551
\(590\) 0 0
\(591\) −75.0325 −3.08642
\(592\) 0 0
\(593\) 25.1005 1.03075 0.515377 0.856964i \(-0.327652\pi\)
0.515377 + 0.856964i \(0.327652\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 34.6642 1.41871
\(598\) 0 0
\(599\) −24.9528 −1.01954 −0.509772 0.860310i \(-0.670270\pi\)
−0.509772 + 0.860310i \(0.670270\pi\)
\(600\) 0 0
\(601\) 11.9744 0.488445 0.244223 0.969719i \(-0.421467\pi\)
0.244223 + 0.969719i \(0.421467\pi\)
\(602\) 0 0
\(603\) −69.7641 −2.84101
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 6.89412 0.279824 0.139912 0.990164i \(-0.455318\pi\)
0.139912 + 0.990164i \(0.455318\pi\)
\(608\) 0 0
\(609\) −2.25819 −0.0915065
\(610\) 0 0
\(611\) 30.7100 1.24239
\(612\) 0 0
\(613\) −25.8193 −1.04283 −0.521416 0.853303i \(-0.674596\pi\)
−0.521416 + 0.853303i \(0.674596\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 15.7584 0.634408 0.317204 0.948357i \(-0.397256\pi\)
0.317204 + 0.948357i \(0.397256\pi\)
\(618\) 0 0
\(619\) −25.9111 −1.04145 −0.520727 0.853723i \(-0.674339\pi\)
−0.520727 + 0.853723i \(0.674339\pi\)
\(620\) 0 0
\(621\) 3.73773 0.149990
\(622\) 0 0
\(623\) 0.566768 0.0227071
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −16.2808 −0.650194
\(628\) 0 0
\(629\) −36.0231 −1.43633
\(630\) 0 0
\(631\) −6.86287 −0.273207 −0.136603 0.990626i \(-0.543619\pi\)
−0.136603 + 0.990626i \(0.543619\pi\)
\(632\) 0 0
\(633\) −55.5499 −2.20791
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 36.5939 1.44990
\(638\) 0 0
\(639\) 12.3346 0.487947
\(640\) 0 0
\(641\) −35.4136 −1.39875 −0.699376 0.714754i \(-0.746539\pi\)
−0.699376 + 0.714754i \(0.746539\pi\)
\(642\) 0 0
\(643\) −27.6618 −1.09088 −0.545438 0.838151i \(-0.683637\pi\)
−0.545438 + 0.838151i \(0.683637\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −29.0871 −1.14353 −0.571765 0.820417i \(-0.693741\pi\)
−0.571765 + 0.820417i \(0.693741\pi\)
\(648\) 0 0
\(649\) −51.9794 −2.04037
\(650\) 0 0
\(651\) −1.55903 −0.0611033
\(652\) 0 0
\(653\) 22.7798 0.891444 0.445722 0.895171i \(-0.352947\pi\)
0.445722 + 0.895171i \(0.352947\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −38.0219 −1.48337
\(658\) 0 0
\(659\) −13.5698 −0.528605 −0.264302 0.964440i \(-0.585142\pi\)
−0.264302 + 0.964440i \(0.585142\pi\)
\(660\) 0 0
\(661\) 47.4318 1.84489 0.922443 0.386134i \(-0.126190\pi\)
0.922443 + 0.386134i \(0.126190\pi\)
\(662\) 0 0
\(663\) 106.808 4.14809
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3.40533 −0.131855
\(668\) 0 0
\(669\) −42.4857 −1.64259
\(670\) 0 0
\(671\) 46.6972 1.80273
\(672\) 0 0
\(673\) 4.71398 0.181711 0.0908553 0.995864i \(-0.471040\pi\)
0.0908553 + 0.995864i \(0.471040\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −42.5524 −1.63542 −0.817711 0.575629i \(-0.804757\pi\)
−0.817711 + 0.575629i \(0.804757\pi\)
\(678\) 0 0
\(679\) −0.722747 −0.0277365
\(680\) 0 0
\(681\) −38.2757 −1.46673
\(682\) 0 0
\(683\) 34.6186 1.32464 0.662321 0.749220i \(-0.269571\pi\)
0.662321 + 0.749220i \(0.269571\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −20.9747 −0.800235
\(688\) 0 0
\(689\) −36.5584 −1.39276
\(690\) 0 0
\(691\) 22.4131 0.852636 0.426318 0.904573i \(-0.359811\pi\)
0.426318 + 0.904573i \(0.359811\pi\)
\(692\) 0 0
\(693\) −4.23576 −0.160903
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −41.3140 −1.56488
\(698\) 0 0
\(699\) −12.0846 −0.457082
\(700\) 0 0
\(701\) −30.3599 −1.14668 −0.573339 0.819318i \(-0.694352\pi\)
−0.573339 + 0.819318i \(0.694352\pi\)
\(702\) 0 0
\(703\) 5.03553 0.189919
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −0.776149 −0.0291901
\(708\) 0 0
\(709\) −11.2281 −0.421679 −0.210839 0.977521i \(-0.567620\pi\)
−0.210839 + 0.977521i \(0.567620\pi\)
\(710\) 0 0
\(711\) −49.5196 −1.85713
\(712\) 0 0
\(713\) −2.35101 −0.0880459
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −43.6757 −1.63110
\(718\) 0 0
\(719\) −27.4485 −1.02366 −0.511829 0.859088i \(-0.671032\pi\)
−0.511829 + 0.859088i \(0.671032\pi\)
\(720\) 0 0
\(721\) 0.521392 0.0194177
\(722\) 0 0
\(723\) 76.0336 2.82772
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 17.2650 0.640323 0.320162 0.947363i \(-0.396263\pi\)
0.320162 + 0.947363i \(0.396263\pi\)
\(728\) 0 0
\(729\) −42.2472 −1.56471
\(730\) 0 0
\(731\) −23.7726 −0.879261
\(732\) 0 0
\(733\) 15.6789 0.579112 0.289556 0.957161i \(-0.406492\pi\)
0.289556 + 0.957161i \(0.406492\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 78.0946 2.87665
\(738\) 0 0
\(739\) 47.7761 1.75747 0.878737 0.477307i \(-0.158387\pi\)
0.878737 + 0.477307i \(0.158387\pi\)
\(740\) 0 0
\(741\) −14.9303 −0.548479
\(742\) 0 0
\(743\) −37.5789 −1.37864 −0.689318 0.724459i \(-0.742090\pi\)
−0.689318 + 0.724459i \(0.742090\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 11.0899 0.405758
\(748\) 0 0
\(749\) 2.22654 0.0813561
\(750\) 0 0
\(751\) 50.4411 1.84062 0.920311 0.391188i \(-0.127936\pi\)
0.920311 + 0.391188i \(0.127936\pi\)
\(752\) 0 0
\(753\) 10.3778 0.378188
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −10.8756 −0.395280 −0.197640 0.980275i \(-0.563328\pi\)
−0.197640 + 0.980275i \(0.563328\pi\)
\(758\) 0 0
\(759\) −10.1391 −0.368026
\(760\) 0 0
\(761\) 10.7675 0.390322 0.195161 0.980771i \(-0.437477\pi\)
0.195161 + 0.980771i \(0.437477\pi\)
\(762\) 0 0
\(763\) 0.319229 0.0115569
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −47.6677 −1.72118
\(768\) 0 0
\(769\) −1.76320 −0.0635827 −0.0317914 0.999495i \(-0.510121\pi\)
−0.0317914 + 0.999495i \(0.510121\pi\)
\(770\) 0 0
\(771\) −27.6514 −0.995841
\(772\) 0 0
\(773\) −26.5661 −0.955517 −0.477759 0.878491i \(-0.658551\pi\)
−0.477759 + 0.878491i \(0.658551\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 2.07954 0.0746032
\(778\) 0 0
\(779\) 5.77513 0.206915
\(780\) 0 0
\(781\) −13.8074 −0.494068
\(782\) 0 0
\(783\) −32.8189 −1.17285
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −34.4557 −1.22821 −0.614106 0.789223i \(-0.710483\pi\)
−0.614106 + 0.789223i \(0.710483\pi\)
\(788\) 0 0
\(789\) 5.18379 0.184548
\(790\) 0 0
\(791\) 1.79282 0.0637453
\(792\) 0 0
\(793\) 42.8236 1.52071
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11.5121 0.407779 0.203890 0.978994i \(-0.434642\pi\)
0.203890 + 0.978994i \(0.434642\pi\)
\(798\) 0 0
\(799\) 41.8985 1.48226
\(800\) 0 0
\(801\) 19.9604 0.705267
\(802\) 0 0
\(803\) 42.5621 1.50198
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 61.6048 2.16859
\(808\) 0 0
\(809\) 35.7367 1.25643 0.628217 0.778038i \(-0.283785\pi\)
0.628217 + 0.778038i \(0.283785\pi\)
\(810\) 0 0
\(811\) −47.4425 −1.66593 −0.832966 0.553324i \(-0.813359\pi\)
−0.832966 + 0.553324i \(0.813359\pi\)
\(812\) 0 0
\(813\) −21.4820 −0.753406
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 3.32308 0.116260
\(818\) 0 0
\(819\) −3.88440 −0.135732
\(820\) 0 0
\(821\) −5.70237 −0.199014 −0.0995070 0.995037i \(-0.531727\pi\)
−0.0995070 + 0.995037i \(0.531727\pi\)
\(822\) 0 0
\(823\) 3.37125 0.117514 0.0587572 0.998272i \(-0.481286\pi\)
0.0587572 + 0.998272i \(0.481286\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.22704 0.112215 0.0561075 0.998425i \(-0.482131\pi\)
0.0561075 + 0.998425i \(0.482131\pi\)
\(828\) 0 0
\(829\) 10.5746 0.367273 0.183636 0.982994i \(-0.441213\pi\)
0.183636 + 0.982994i \(0.441213\pi\)
\(830\) 0 0
\(831\) 35.7519 1.24022
\(832\) 0 0
\(833\) 49.9260 1.72983
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −22.6578 −0.783168
\(838\) 0 0
\(839\) 51.5214 1.77872 0.889359 0.457210i \(-0.151151\pi\)
0.889359 + 0.457210i \(0.151151\pi\)
\(840\) 0 0
\(841\) 0.900274 0.0310439
\(842\) 0 0
\(843\) 63.1977 2.17665
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3.14617 0.108104
\(848\) 0 0
\(849\) −58.7055 −2.01477
\(850\) 0 0
\(851\) 3.13593 0.107498
\(852\) 0 0
\(853\) 14.3074 0.489874 0.244937 0.969539i \(-0.421233\pi\)
0.244937 + 0.969539i \(0.421233\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.55632 −0.121482 −0.0607408 0.998154i \(-0.519346\pi\)
−0.0607408 + 0.998154i \(0.519346\pi\)
\(858\) 0 0
\(859\) 13.8084 0.471135 0.235568 0.971858i \(-0.424305\pi\)
0.235568 + 0.971858i \(0.424305\pi\)
\(860\) 0 0
\(861\) 2.38498 0.0812799
\(862\) 0 0
\(863\) −11.3476 −0.386277 −0.193139 0.981171i \(-0.561867\pi\)
−0.193139 + 0.981171i \(0.561867\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 97.3154 3.30500
\(868\) 0 0
\(869\) 55.4327 1.88042
\(870\) 0 0
\(871\) 71.6166 2.42664
\(872\) 0 0
\(873\) −25.4537 −0.861478
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 44.6155 1.50656 0.753279 0.657701i \(-0.228471\pi\)
0.753279 + 0.657701i \(0.228471\pi\)
\(878\) 0 0
\(879\) −63.0479 −2.12655
\(880\) 0 0
\(881\) 34.3973 1.15887 0.579437 0.815017i \(-0.303272\pi\)
0.579437 + 0.815017i \(0.303272\pi\)
\(882\) 0 0
\(883\) −54.2057 −1.82417 −0.912084 0.410004i \(-0.865527\pi\)
−0.912084 + 0.410004i \(0.865527\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.69998 0.0570798 0.0285399 0.999593i \(-0.490914\pi\)
0.0285399 + 0.999593i \(0.490914\pi\)
\(888\) 0 0
\(889\) −2.62377 −0.0879984
\(890\) 0 0
\(891\) −10.0997 −0.338353
\(892\) 0 0
\(893\) −5.85683 −0.195991
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −9.29804 −0.310453
\(898\) 0 0
\(899\) 20.6428 0.688477
\(900\) 0 0
\(901\) −49.8776 −1.66166
\(902\) 0 0
\(903\) 1.37235 0.0456689
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −13.1338 −0.436101 −0.218050 0.975938i \(-0.569970\pi\)
−0.218050 + 0.975938i \(0.569970\pi\)
\(908\) 0 0
\(909\) −27.3344 −0.906626
\(910\) 0 0
\(911\) 24.7511 0.820040 0.410020 0.912076i \(-0.365522\pi\)
0.410020 + 0.912076i \(0.365522\pi\)
\(912\) 0 0
\(913\) −12.4141 −0.410848
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.75845 0.0580692
\(918\) 0 0
\(919\) −14.2789 −0.471019 −0.235510 0.971872i \(-0.575676\pi\)
−0.235510 + 0.971872i \(0.575676\pi\)
\(920\) 0 0
\(921\) 5.12707 0.168943
\(922\) 0 0
\(923\) −12.6621 −0.416777
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 18.3624 0.603100
\(928\) 0 0
\(929\) 18.6484 0.611834 0.305917 0.952058i \(-0.401037\pi\)
0.305917 + 0.952058i \(0.401037\pi\)
\(930\) 0 0
\(931\) −6.97897 −0.228726
\(932\) 0 0
\(933\) −26.3829 −0.863738
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 17.9120 0.585159 0.292580 0.956241i \(-0.405486\pi\)
0.292580 + 0.956241i \(0.405486\pi\)
\(938\) 0 0
\(939\) 12.2376 0.399360
\(940\) 0 0
\(941\) 10.7412 0.350152 0.175076 0.984555i \(-0.443983\pi\)
0.175076 + 0.984555i \(0.443983\pi\)
\(942\) 0 0
\(943\) 3.59653 0.117119
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −58.5220 −1.90171 −0.950855 0.309636i \(-0.899793\pi\)
−0.950855 + 0.309636i \(0.899793\pi\)
\(948\) 0 0
\(949\) 39.0315 1.26702
\(950\) 0 0
\(951\) 13.1285 0.425720
\(952\) 0 0
\(953\) −2.93656 −0.0951244 −0.0475622 0.998868i \(-0.515145\pi\)
−0.0475622 + 0.998868i \(0.515145\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 89.0255 2.87779
\(958\) 0 0
\(959\) 2.90628 0.0938486
\(960\) 0 0
\(961\) −16.7484 −0.540271
\(962\) 0 0
\(963\) 78.4144 2.52687
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −20.5192 −0.659852 −0.329926 0.944007i \(-0.607024\pi\)
−0.329926 + 0.944007i \(0.607024\pi\)
\(968\) 0 0
\(969\) −20.3699 −0.654374
\(970\) 0 0
\(971\) −2.37837 −0.0763255 −0.0381628 0.999272i \(-0.512151\pi\)
−0.0381628 + 0.999272i \(0.512151\pi\)
\(972\) 0 0
\(973\) 1.65334 0.0530037
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 46.2822 1.48070 0.740349 0.672222i \(-0.234660\pi\)
0.740349 + 0.672222i \(0.234660\pi\)
\(978\) 0 0
\(979\) −22.3439 −0.714114
\(980\) 0 0
\(981\) 11.2426 0.358948
\(982\) 0 0
\(983\) 49.2913 1.57215 0.786074 0.618133i \(-0.212111\pi\)
0.786074 + 0.618133i \(0.212111\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −2.41872 −0.0769887
\(988\) 0 0
\(989\) 2.06949 0.0658059
\(990\) 0 0
\(991\) 6.92186 0.219880 0.109940 0.993938i \(-0.464934\pi\)
0.109940 + 0.993938i \(0.464934\pi\)
\(992\) 0 0
\(993\) −2.66054 −0.0844296
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 28.1533 0.891624 0.445812 0.895127i \(-0.352915\pi\)
0.445812 + 0.895127i \(0.352915\pi\)
\(998\) 0 0
\(999\) 30.2225 0.956199
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 3800.2.a.z.1.6 6
4.3 odd 2 7600.2.a.cn.1.1 6
5.2 odd 4 760.2.d.e.609.2 12
5.3 odd 4 760.2.d.e.609.11 yes 12
5.4 even 2 3800.2.a.be.1.1 6
20.3 even 4 1520.2.d.k.609.2 12
20.7 even 4 1520.2.d.k.609.11 12
20.19 odd 2 7600.2.a.cg.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.d.e.609.2 12 5.2 odd 4
760.2.d.e.609.11 yes 12 5.3 odd 4
1520.2.d.k.609.2 12 20.3 even 4
1520.2.d.k.609.11 12 20.7 even 4
3800.2.a.z.1.6 6 1.1 even 1 trivial
3800.2.a.be.1.1 6 5.4 even 2
7600.2.a.cg.1.6 6 20.19 odd 2
7600.2.a.cn.1.1 6 4.3 odd 2