Properties

Label 3800.2.a.bd.1.5
Level $3800$
Weight $2$
Character 3800.1
Self dual yes
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 12x^{4} + 16x^{3} + 33x^{2} - 4x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.70452\) 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)
 
\(f(q)\) \(=\) \(q+2.70452 q^{3} +3.77934 q^{7} +4.31442 q^{9} +O(q^{10})\) \(q+2.70452 q^{3} +3.77934 q^{7} +4.31442 q^{9} +4.71724 q^{11} -5.44195 q^{13} -2.40944 q^{17} +1.00000 q^{19} +10.2213 q^{21} +4.38924 q^{23} +3.55488 q^{27} +9.05146 q^{29} -9.92581 q^{31} +12.7579 q^{33} +10.0429 q^{37} -14.7179 q^{39} +1.11712 q^{41} -12.1169 q^{43} +6.72432 q^{47} +7.28339 q^{49} -6.51637 q^{51} +4.34694 q^{53} +2.70452 q^{57} -4.03914 q^{59} -9.14562 q^{61} +16.3057 q^{63} +4.39672 q^{67} +11.8708 q^{69} +7.90222 q^{71} +3.07165 q^{73} +17.8280 q^{77} +11.3996 q^{79} -3.32902 q^{81} -4.37652 q^{83} +24.4798 q^{87} +2.53933 q^{89} -20.5670 q^{91} -26.8445 q^{93} -0.302105 q^{97} +20.3522 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} + 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{3} + 2 q^{7} + 10 q^{9} + 3 q^{11} - 3 q^{13} - 2 q^{17} + 6 q^{19} + 11 q^{21} + 4 q^{23} + 20 q^{27} + 7 q^{29} + 5 q^{31} - 16 q^{33} + 8 q^{39} + 11 q^{41} - 7 q^{43} + 20 q^{47} - 2 q^{49} + 13 q^{51} - 7 q^{53} + 2 q^{57} - 4 q^{59} + 13 q^{61} - q^{63} + 25 q^{67} + 7 q^{69} + 29 q^{71} - 19 q^{73} + 24 q^{77} + 28 q^{79} + 38 q^{81} - 15 q^{83} + 57 q^{87} - 12 q^{89} - 27 q^{93} - 13 q^{97} + 10 q^{99}+O(q^{100}) \) 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
\(3\) 2.70452 1.56145 0.780727 0.624872i \(-0.214849\pi\)
0.780727 + 0.624872i \(0.214849\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.77934 1.42846 0.714228 0.699914i \(-0.246778\pi\)
0.714228 + 0.699914i \(0.246778\pi\)
\(8\) 0 0
\(9\) 4.31442 1.43814
\(10\) 0 0
\(11\) 4.71724 1.42230 0.711150 0.703040i \(-0.248175\pi\)
0.711150 + 0.703040i \(0.248175\pi\)
\(12\) 0 0
\(13\) −5.44195 −1.50933 −0.754663 0.656113i \(-0.772200\pi\)
−0.754663 + 0.656113i \(0.772200\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.40944 −0.584375 −0.292187 0.956361i \(-0.594383\pi\)
−0.292187 + 0.956361i \(0.594383\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 10.2213 2.23047
\(22\) 0 0
\(23\) 4.38924 0.915220 0.457610 0.889153i \(-0.348706\pi\)
0.457610 + 0.889153i \(0.348706\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.55488 0.684137
\(28\) 0 0
\(29\) 9.05146 1.68081 0.840407 0.541956i \(-0.182316\pi\)
0.840407 + 0.541956i \(0.182316\pi\)
\(30\) 0 0
\(31\) −9.92581 −1.78273 −0.891364 0.453288i \(-0.850251\pi\)
−0.891364 + 0.453288i \(0.850251\pi\)
\(32\) 0 0
\(33\) 12.7579 2.22086
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0429 1.65105 0.825524 0.564367i \(-0.190880\pi\)
0.825524 + 0.564367i \(0.190880\pi\)
\(38\) 0 0
\(39\) −14.7179 −2.35674
\(40\) 0 0
\(41\) 1.11712 0.174465 0.0872326 0.996188i \(-0.472198\pi\)
0.0872326 + 0.996188i \(0.472198\pi\)
\(42\) 0 0
\(43\) −12.1169 −1.84781 −0.923904 0.382625i \(-0.875020\pi\)
−0.923904 + 0.382625i \(0.875020\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.72432 0.980842 0.490421 0.871486i \(-0.336843\pi\)
0.490421 + 0.871486i \(0.336843\pi\)
\(48\) 0 0
\(49\) 7.28339 1.04048
\(50\) 0 0
\(51\) −6.51637 −0.912474
\(52\) 0 0
\(53\) 4.34694 0.597098 0.298549 0.954394i \(-0.403497\pi\)
0.298549 + 0.954394i \(0.403497\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.70452 0.358222
\(58\) 0 0
\(59\) −4.03914 −0.525851 −0.262926 0.964816i \(-0.584687\pi\)
−0.262926 + 0.964816i \(0.584687\pi\)
\(60\) 0 0
\(61\) −9.14562 −1.17098 −0.585488 0.810681i \(-0.699097\pi\)
−0.585488 + 0.810681i \(0.699097\pi\)
\(62\) 0 0
\(63\) 16.3057 2.05432
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.39672 0.537145 0.268572 0.963259i \(-0.413448\pi\)
0.268572 + 0.963259i \(0.413448\pi\)
\(68\) 0 0
\(69\) 11.8708 1.42907
\(70\) 0 0
\(71\) 7.90222 0.937821 0.468910 0.883246i \(-0.344647\pi\)
0.468910 + 0.883246i \(0.344647\pi\)
\(72\) 0 0
\(73\) 3.07165 0.359510 0.179755 0.983711i \(-0.442470\pi\)
0.179755 + 0.983711i \(0.442470\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 17.8280 2.03169
\(78\) 0 0
\(79\) 11.3996 1.28256 0.641280 0.767307i \(-0.278404\pi\)
0.641280 + 0.767307i \(0.278404\pi\)
\(80\) 0 0
\(81\) −3.32902 −0.369891
\(82\) 0 0
\(83\) −4.37652 −0.480386 −0.240193 0.970725i \(-0.577211\pi\)
−0.240193 + 0.970725i \(0.577211\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 24.4798 2.62451
\(88\) 0 0
\(89\) 2.53933 0.269169 0.134584 0.990902i \(-0.457030\pi\)
0.134584 + 0.990902i \(0.457030\pi\)
\(90\) 0 0
\(91\) −20.5670 −2.15600
\(92\) 0 0
\(93\) −26.8445 −2.78365
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.302105 −0.0306741 −0.0153371 0.999882i \(-0.504882\pi\)
−0.0153371 + 0.999882i \(0.504882\pi\)
\(98\) 0 0
\(99\) 20.3522 2.04547
\(100\) 0 0
\(101\) 4.77080 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(102\) 0 0
\(103\) −17.9014 −1.76388 −0.881939 0.471364i \(-0.843762\pi\)
−0.881939 + 0.471364i \(0.843762\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −13.1673 −1.27293 −0.636465 0.771305i \(-0.719604\pi\)
−0.636465 + 0.771305i \(0.719604\pi\)
\(108\) 0 0
\(109\) 0.756212 0.0724320 0.0362160 0.999344i \(-0.488470\pi\)
0.0362160 + 0.999344i \(0.488470\pi\)
\(110\) 0 0
\(111\) 27.1613 2.57804
\(112\) 0 0
\(113\) −20.6082 −1.93866 −0.969329 0.245766i \(-0.920960\pi\)
−0.969329 + 0.245766i \(0.920960\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −23.4789 −2.17062
\(118\) 0 0
\(119\) −9.10608 −0.834753
\(120\) 0 0
\(121\) 11.2523 1.02294
\(122\) 0 0
\(123\) 3.02128 0.272419
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 19.3178 1.71418 0.857089 0.515168i \(-0.172270\pi\)
0.857089 + 0.515168i \(0.172270\pi\)
\(128\) 0 0
\(129\) −32.7703 −2.88527
\(130\) 0 0
\(131\) −4.29192 −0.374986 −0.187493 0.982266i \(-0.560036\pi\)
−0.187493 + 0.982266i \(0.560036\pi\)
\(132\) 0 0
\(133\) 3.77934 0.327710
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −21.2413 −1.81476 −0.907382 0.420306i \(-0.861923\pi\)
−0.907382 + 0.420306i \(0.861923\pi\)
\(138\) 0 0
\(139\) 1.94814 0.165239 0.0826197 0.996581i \(-0.473671\pi\)
0.0826197 + 0.996581i \(0.473671\pi\)
\(140\) 0 0
\(141\) 18.1860 1.53154
\(142\) 0 0
\(143\) −25.6710 −2.14671
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 19.6981 1.62467
\(148\) 0 0
\(149\) −4.56655 −0.374107 −0.187053 0.982350i \(-0.559894\pi\)
−0.187053 + 0.982350i \(0.559894\pi\)
\(150\) 0 0
\(151\) −8.84246 −0.719590 −0.359795 0.933031i \(-0.617153\pi\)
−0.359795 + 0.933031i \(0.617153\pi\)
\(152\) 0 0
\(153\) −10.3953 −0.840413
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.63494 0.449717 0.224859 0.974391i \(-0.427808\pi\)
0.224859 + 0.974391i \(0.427808\pi\)
\(158\) 0 0
\(159\) 11.7564 0.932341
\(160\) 0 0
\(161\) 16.5884 1.30735
\(162\) 0 0
\(163\) 3.71114 0.290679 0.145340 0.989382i \(-0.453573\pi\)
0.145340 + 0.989382i \(0.453573\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.403668 0.0312368 0.0156184 0.999878i \(-0.495028\pi\)
0.0156184 + 0.999878i \(0.495028\pi\)
\(168\) 0 0
\(169\) 16.6148 1.27806
\(170\) 0 0
\(171\) 4.31442 0.329932
\(172\) 0 0
\(173\) 13.9048 1.05716 0.528580 0.848884i \(-0.322725\pi\)
0.528580 + 0.848884i \(0.322725\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −10.9239 −0.821093
\(178\) 0 0
\(179\) 3.27673 0.244914 0.122457 0.992474i \(-0.460923\pi\)
0.122457 + 0.992474i \(0.460923\pi\)
\(180\) 0 0
\(181\) 13.0254 0.968173 0.484086 0.875020i \(-0.339152\pi\)
0.484086 + 0.875020i \(0.339152\pi\)
\(182\) 0 0
\(183\) −24.7345 −1.82843
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −11.3659 −0.831156
\(188\) 0 0
\(189\) 13.4351 0.977260
\(190\) 0 0
\(191\) −17.6403 −1.27641 −0.638203 0.769868i \(-0.720322\pi\)
−0.638203 + 0.769868i \(0.720322\pi\)
\(192\) 0 0
\(193\) 16.8445 1.21250 0.606248 0.795276i \(-0.292674\pi\)
0.606248 + 0.795276i \(0.292674\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −19.8742 −1.41598 −0.707988 0.706224i \(-0.750397\pi\)
−0.707988 + 0.706224i \(0.750397\pi\)
\(198\) 0 0
\(199\) −8.54003 −0.605386 −0.302693 0.953088i \(-0.597886\pi\)
−0.302693 + 0.953088i \(0.597886\pi\)
\(200\) 0 0
\(201\) 11.8910 0.838727
\(202\) 0 0
\(203\) 34.2085 2.40097
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 18.9370 1.31622
\(208\) 0 0
\(209\) 4.71724 0.326298
\(210\) 0 0
\(211\) −0.0127177 −0.000875525 0 −0.000437762 1.00000i \(-0.500139\pi\)
−0.000437762 1.00000i \(0.500139\pi\)
\(212\) 0 0
\(213\) 21.3717 1.46437
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −37.5130 −2.54655
\(218\) 0 0
\(219\) 8.30734 0.561358
\(220\) 0 0
\(221\) 13.1120 0.882012
\(222\) 0 0
\(223\) −18.6403 −1.24825 −0.624124 0.781326i \(-0.714544\pi\)
−0.624124 + 0.781326i \(0.714544\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.75187 0.514510 0.257255 0.966344i \(-0.417182\pi\)
0.257255 + 0.966344i \(0.417182\pi\)
\(228\) 0 0
\(229\) −19.3735 −1.28023 −0.640117 0.768277i \(-0.721114\pi\)
−0.640117 + 0.768277i \(0.721114\pi\)
\(230\) 0 0
\(231\) 48.2162 3.17240
\(232\) 0 0
\(233\) −8.63389 −0.565625 −0.282813 0.959175i \(-0.591267\pi\)
−0.282813 + 0.959175i \(0.591267\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 30.8306 2.00266
\(238\) 0 0
\(239\) 9.47644 0.612980 0.306490 0.951874i \(-0.400845\pi\)
0.306490 + 0.951874i \(0.400845\pi\)
\(240\) 0 0
\(241\) 29.3641 1.89151 0.945755 0.324880i \(-0.105324\pi\)
0.945755 + 0.324880i \(0.105324\pi\)
\(242\) 0 0
\(243\) −19.6681 −1.26171
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −5.44195 −0.346263
\(248\) 0 0
\(249\) −11.8364 −0.750101
\(250\) 0 0
\(251\) −7.57377 −0.478052 −0.239026 0.971013i \(-0.576828\pi\)
−0.239026 + 0.971013i \(0.576828\pi\)
\(252\) 0 0
\(253\) 20.7051 1.30172
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −11.7573 −0.733400 −0.366700 0.930339i \(-0.619512\pi\)
−0.366700 + 0.930339i \(0.619512\pi\)
\(258\) 0 0
\(259\) 37.9556 2.35845
\(260\) 0 0
\(261\) 39.0518 2.41725
\(262\) 0 0
\(263\) 2.20334 0.135864 0.0679319 0.997690i \(-0.478360\pi\)
0.0679319 + 0.997690i \(0.478360\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.86767 0.420295
\(268\) 0 0
\(269\) 6.14642 0.374754 0.187377 0.982288i \(-0.440001\pi\)
0.187377 + 0.982288i \(0.440001\pi\)
\(270\) 0 0
\(271\) 25.6620 1.55886 0.779428 0.626491i \(-0.215510\pi\)
0.779428 + 0.626491i \(0.215510\pi\)
\(272\) 0 0
\(273\) −55.6238 −3.36650
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.64202 −0.278912 −0.139456 0.990228i \(-0.544535\pi\)
−0.139456 + 0.990228i \(0.544535\pi\)
\(278\) 0 0
\(279\) −42.8241 −2.56381
\(280\) 0 0
\(281\) −0.369967 −0.0220704 −0.0110352 0.999939i \(-0.503513\pi\)
−0.0110352 + 0.999939i \(0.503513\pi\)
\(282\) 0 0
\(283\) −1.21325 −0.0721203 −0.0360602 0.999350i \(-0.511481\pi\)
−0.0360602 + 0.999350i \(0.511481\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.22198 0.249216
\(288\) 0 0
\(289\) −11.1946 −0.658506
\(290\) 0 0
\(291\) −0.817050 −0.0478963
\(292\) 0 0
\(293\) 8.66265 0.506077 0.253039 0.967456i \(-0.418570\pi\)
0.253039 + 0.967456i \(0.418570\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 16.7692 0.973049
\(298\) 0 0
\(299\) −23.8860 −1.38137
\(300\) 0 0
\(301\) −45.7938 −2.63951
\(302\) 0 0
\(303\) 12.9027 0.741242
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 8.26123 0.471493 0.235747 0.971815i \(-0.424246\pi\)
0.235747 + 0.971815i \(0.424246\pi\)
\(308\) 0 0
\(309\) −48.4147 −2.75421
\(310\) 0 0
\(311\) 7.54408 0.427786 0.213893 0.976857i \(-0.431386\pi\)
0.213893 + 0.976857i \(0.431386\pi\)
\(312\) 0 0
\(313\) −3.43152 −0.193961 −0.0969804 0.995286i \(-0.530918\pi\)
−0.0969804 + 0.995286i \(0.530918\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.50784 0.140854 0.0704270 0.997517i \(-0.477564\pi\)
0.0704270 + 0.997517i \(0.477564\pi\)
\(318\) 0 0
\(319\) 42.6979 2.39062
\(320\) 0 0
\(321\) −35.6112 −1.98762
\(322\) 0 0
\(323\) −2.40944 −0.134065
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.04519 0.113099
\(328\) 0 0
\(329\) 25.4135 1.40109
\(330\) 0 0
\(331\) 29.1167 1.60040 0.800200 0.599734i \(-0.204727\pi\)
0.800200 + 0.599734i \(0.204727\pi\)
\(332\) 0 0
\(333\) 43.3295 2.37444
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 20.4033 1.11144 0.555720 0.831370i \(-0.312443\pi\)
0.555720 + 0.831370i \(0.312443\pi\)
\(338\) 0 0
\(339\) −55.7353 −3.02713
\(340\) 0 0
\(341\) −46.8224 −2.53557
\(342\) 0 0
\(343\) 1.07103 0.0578300
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −25.9967 −1.39557 −0.697787 0.716305i \(-0.745832\pi\)
−0.697787 + 0.716305i \(0.745832\pi\)
\(348\) 0 0
\(349\) 3.32628 0.178052 0.0890258 0.996029i \(-0.471625\pi\)
0.0890258 + 0.996029i \(0.471625\pi\)
\(350\) 0 0
\(351\) −19.3455 −1.03259
\(352\) 0 0
\(353\) −15.0904 −0.803179 −0.401590 0.915820i \(-0.631542\pi\)
−0.401590 + 0.915820i \(0.631542\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −24.6276 −1.30343
\(358\) 0 0
\(359\) −12.8292 −0.677102 −0.338551 0.940948i \(-0.609937\pi\)
−0.338551 + 0.940948i \(0.609937\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 30.4321 1.59727
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4.11198 0.214644 0.107322 0.994224i \(-0.465772\pi\)
0.107322 + 0.994224i \(0.465772\pi\)
\(368\) 0 0
\(369\) 4.81974 0.250905
\(370\) 0 0
\(371\) 16.4285 0.852927
\(372\) 0 0
\(373\) −16.5930 −0.859153 −0.429576 0.903031i \(-0.641337\pi\)
−0.429576 + 0.903031i \(0.641337\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −49.2576 −2.53690
\(378\) 0 0
\(379\) 30.6822 1.57604 0.788021 0.615649i \(-0.211106\pi\)
0.788021 + 0.615649i \(0.211106\pi\)
\(380\) 0 0
\(381\) 52.2454 2.67661
\(382\) 0 0
\(383\) −30.4909 −1.55801 −0.779006 0.627017i \(-0.784276\pi\)
−0.779006 + 0.627017i \(0.784276\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −52.2774 −2.65741
\(388\) 0 0
\(389\) 16.3448 0.828714 0.414357 0.910114i \(-0.364006\pi\)
0.414357 + 0.910114i \(0.364006\pi\)
\(390\) 0 0
\(391\) −10.5756 −0.534831
\(392\) 0 0
\(393\) −11.6076 −0.585524
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −15.5946 −0.782670 −0.391335 0.920248i \(-0.627987\pi\)
−0.391335 + 0.920248i \(0.627987\pi\)
\(398\) 0 0
\(399\) 10.2213 0.511705
\(400\) 0 0
\(401\) −20.3465 −1.01606 −0.508029 0.861340i \(-0.669626\pi\)
−0.508029 + 0.861340i \(0.669626\pi\)
\(402\) 0 0
\(403\) 54.0158 2.69072
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 47.3749 2.34829
\(408\) 0 0
\(409\) −5.65655 −0.279699 −0.139849 0.990173i \(-0.544662\pi\)
−0.139849 + 0.990173i \(0.544662\pi\)
\(410\) 0 0
\(411\) −57.4474 −2.83367
\(412\) 0 0
\(413\) −15.2653 −0.751155
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5.26879 0.258014
\(418\) 0 0
\(419\) −18.1490 −0.886635 −0.443318 0.896365i \(-0.646199\pi\)
−0.443318 + 0.896365i \(0.646199\pi\)
\(420\) 0 0
\(421\) 5.94874 0.289924 0.144962 0.989437i \(-0.453694\pi\)
0.144962 + 0.989437i \(0.453694\pi\)
\(422\) 0 0
\(423\) 29.0115 1.41059
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −34.5644 −1.67269
\(428\) 0 0
\(429\) −69.4276 −3.35200
\(430\) 0 0
\(431\) 1.93242 0.0930816 0.0465408 0.998916i \(-0.485180\pi\)
0.0465408 + 0.998916i \(0.485180\pi\)
\(432\) 0 0
\(433\) −14.7014 −0.706505 −0.353253 0.935528i \(-0.614924\pi\)
−0.353253 + 0.935528i \(0.614924\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.38924 0.209966
\(438\) 0 0
\(439\) 10.5922 0.505540 0.252770 0.967526i \(-0.418658\pi\)
0.252770 + 0.967526i \(0.418658\pi\)
\(440\) 0 0
\(441\) 31.4236 1.49636
\(442\) 0 0
\(443\) −38.7357 −1.84039 −0.920195 0.391459i \(-0.871970\pi\)
−0.920195 + 0.391459i \(0.871970\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −12.3503 −0.584151
\(448\) 0 0
\(449\) −29.2298 −1.37944 −0.689719 0.724078i \(-0.742266\pi\)
−0.689719 + 0.724078i \(0.742266\pi\)
\(450\) 0 0
\(451\) 5.26973 0.248142
\(452\) 0 0
\(453\) −23.9146 −1.12361
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.18004 0.195534 0.0977671 0.995209i \(-0.468830\pi\)
0.0977671 + 0.995209i \(0.468830\pi\)
\(458\) 0 0
\(459\) −8.56527 −0.399793
\(460\) 0 0
\(461\) 35.9590 1.67478 0.837389 0.546607i \(-0.184081\pi\)
0.837389 + 0.546607i \(0.184081\pi\)
\(462\) 0 0
\(463\) −32.5866 −1.51443 −0.757214 0.653167i \(-0.773440\pi\)
−0.757214 + 0.653167i \(0.773440\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5.66769 0.262269 0.131135 0.991365i \(-0.458138\pi\)
0.131135 + 0.991365i \(0.458138\pi\)
\(468\) 0 0
\(469\) 16.6167 0.767287
\(470\) 0 0
\(471\) 15.2398 0.702213
\(472\) 0 0
\(473\) −57.1582 −2.62814
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 18.7545 0.858711
\(478\) 0 0
\(479\) −16.2096 −0.740635 −0.370318 0.928905i \(-0.620751\pi\)
−0.370318 + 0.928905i \(0.620751\pi\)
\(480\) 0 0
\(481\) −54.6531 −2.49197
\(482\) 0 0
\(483\) 44.8637 2.04137
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 19.0329 0.862461 0.431231 0.902242i \(-0.358080\pi\)
0.431231 + 0.902242i \(0.358080\pi\)
\(488\) 0 0
\(489\) 10.0369 0.453882
\(490\) 0 0
\(491\) −16.5658 −0.747602 −0.373801 0.927509i \(-0.621946\pi\)
−0.373801 + 0.927509i \(0.621946\pi\)
\(492\) 0 0
\(493\) −21.8089 −0.982224
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 29.8652 1.33964
\(498\) 0 0
\(499\) 6.36691 0.285022 0.142511 0.989793i \(-0.454482\pi\)
0.142511 + 0.989793i \(0.454482\pi\)
\(500\) 0 0
\(501\) 1.09173 0.0487748
\(502\) 0 0
\(503\) 5.79766 0.258505 0.129252 0.991612i \(-0.458742\pi\)
0.129252 + 0.991612i \(0.458742\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 44.9352 1.99564
\(508\) 0 0
\(509\) 23.0521 1.02177 0.510883 0.859650i \(-0.329319\pi\)
0.510883 + 0.859650i \(0.329319\pi\)
\(510\) 0 0
\(511\) 11.6088 0.513544
\(512\) 0 0
\(513\) 3.55488 0.156952
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 31.7202 1.39505
\(518\) 0 0
\(519\) 37.6057 1.65071
\(520\) 0 0
\(521\) 11.7470 0.514645 0.257323 0.966326i \(-0.417160\pi\)
0.257323 + 0.966326i \(0.417160\pi\)
\(522\) 0 0
\(523\) −24.0715 −1.05257 −0.526286 0.850308i \(-0.676416\pi\)
−0.526286 + 0.850308i \(0.676416\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 23.9156 1.04178
\(528\) 0 0
\(529\) −3.73456 −0.162372
\(530\) 0 0
\(531\) −17.4266 −0.756248
\(532\) 0 0
\(533\) −6.07932 −0.263325
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 8.86198 0.382423
\(538\) 0 0
\(539\) 34.3575 1.47988
\(540\) 0 0
\(541\) −3.93335 −0.169108 −0.0845540 0.996419i \(-0.526947\pi\)
−0.0845540 + 0.996419i \(0.526947\pi\)
\(542\) 0 0
\(543\) 35.2275 1.51176
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −21.2622 −0.909105 −0.454553 0.890720i \(-0.650201\pi\)
−0.454553 + 0.890720i \(0.650201\pi\)
\(548\) 0 0
\(549\) −39.4581 −1.68403
\(550\) 0 0
\(551\) 9.05146 0.385605
\(552\) 0 0
\(553\) 43.0831 1.83208
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −10.2899 −0.435996 −0.217998 0.975949i \(-0.569953\pi\)
−0.217998 + 0.975949i \(0.569953\pi\)
\(558\) 0 0
\(559\) 65.9395 2.78894
\(560\) 0 0
\(561\) −30.7393 −1.29781
\(562\) 0 0
\(563\) 21.8745 0.921901 0.460950 0.887426i \(-0.347509\pi\)
0.460950 + 0.887426i \(0.347509\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −12.5815 −0.528373
\(568\) 0 0
\(569\) 22.8560 0.958172 0.479086 0.877768i \(-0.340968\pi\)
0.479086 + 0.877768i \(0.340968\pi\)
\(570\) 0 0
\(571\) −5.58019 −0.233524 −0.116762 0.993160i \(-0.537251\pi\)
−0.116762 + 0.993160i \(0.537251\pi\)
\(572\) 0 0
\(573\) −47.7085 −1.99305
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −8.48419 −0.353201 −0.176601 0.984283i \(-0.556510\pi\)
−0.176601 + 0.984283i \(0.556510\pi\)
\(578\) 0 0
\(579\) 45.5564 1.89326
\(580\) 0 0
\(581\) −16.5404 −0.686210
\(582\) 0 0
\(583\) 20.5055 0.849252
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 25.8166 1.06557 0.532783 0.846252i \(-0.321146\pi\)
0.532783 + 0.846252i \(0.321146\pi\)
\(588\) 0 0
\(589\) −9.92581 −0.408986
\(590\) 0 0
\(591\) −53.7501 −2.21098
\(592\) 0 0
\(593\) 26.3984 1.08405 0.542025 0.840362i \(-0.317658\pi\)
0.542025 + 0.840362i \(0.317658\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −23.0967 −0.945284
\(598\) 0 0
\(599\) 36.1825 1.47838 0.739188 0.673499i \(-0.235210\pi\)
0.739188 + 0.673499i \(0.235210\pi\)
\(600\) 0 0
\(601\) 3.20839 0.130873 0.0654364 0.997857i \(-0.479156\pi\)
0.0654364 + 0.997857i \(0.479156\pi\)
\(602\) 0 0
\(603\) 18.9693 0.772490
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −20.3390 −0.825534 −0.412767 0.910837i \(-0.635438\pi\)
−0.412767 + 0.910837i \(0.635438\pi\)
\(608\) 0 0
\(609\) 92.5176 3.74900
\(610\) 0 0
\(611\) −36.5934 −1.48041
\(612\) 0 0
\(613\) 18.9062 0.763616 0.381808 0.924242i \(-0.375302\pi\)
0.381808 + 0.924242i \(0.375302\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 35.0702 1.41187 0.705936 0.708275i \(-0.250527\pi\)
0.705936 + 0.708275i \(0.250527\pi\)
\(618\) 0 0
\(619\) −45.1329 −1.81404 −0.907021 0.421085i \(-0.861650\pi\)
−0.907021 + 0.421085i \(0.861650\pi\)
\(620\) 0 0
\(621\) 15.6032 0.626136
\(622\) 0 0
\(623\) 9.59699 0.384495
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 12.7579 0.509500
\(628\) 0 0
\(629\) −24.1978 −0.964830
\(630\) 0 0
\(631\) 35.0467 1.39519 0.697593 0.716494i \(-0.254254\pi\)
0.697593 + 0.716494i \(0.254254\pi\)
\(632\) 0 0
\(633\) −0.0343954 −0.00136709
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −39.6359 −1.57043
\(638\) 0 0
\(639\) 34.0935 1.34872
\(640\) 0 0
\(641\) −10.8470 −0.428431 −0.214215 0.976786i \(-0.568719\pi\)
−0.214215 + 0.976786i \(0.568719\pi\)
\(642\) 0 0
\(643\) 37.6665 1.48542 0.742712 0.669611i \(-0.233539\pi\)
0.742712 + 0.669611i \(0.233539\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −37.2893 −1.46599 −0.732997 0.680231i \(-0.761879\pi\)
−0.732997 + 0.680231i \(0.761879\pi\)
\(648\) 0 0
\(649\) −19.0536 −0.747918
\(650\) 0 0
\(651\) −101.455 −3.97632
\(652\) 0 0
\(653\) −5.10026 −0.199588 −0.0997942 0.995008i \(-0.531818\pi\)
−0.0997942 + 0.995008i \(0.531818\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 13.2524 0.517026
\(658\) 0 0
\(659\) 36.1180 1.40696 0.703479 0.710716i \(-0.251629\pi\)
0.703479 + 0.710716i \(0.251629\pi\)
\(660\) 0 0
\(661\) −15.5014 −0.602934 −0.301467 0.953477i \(-0.597476\pi\)
−0.301467 + 0.953477i \(0.597476\pi\)
\(662\) 0 0
\(663\) 35.4618 1.37722
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 39.7290 1.53831
\(668\) 0 0
\(669\) −50.4131 −1.94908
\(670\) 0 0
\(671\) −43.1420 −1.66548
\(672\) 0 0
\(673\) 20.8897 0.805240 0.402620 0.915367i \(-0.368100\pi\)
0.402620 + 0.915367i \(0.368100\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −39.3686 −1.51306 −0.756529 0.653960i \(-0.773106\pi\)
−0.756529 + 0.653960i \(0.773106\pi\)
\(678\) 0 0
\(679\) −1.14176 −0.0438166
\(680\) 0 0
\(681\) 20.9651 0.803383
\(682\) 0 0
\(683\) −41.6596 −1.59406 −0.797030 0.603940i \(-0.793597\pi\)
−0.797030 + 0.603940i \(0.793597\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −52.3959 −1.99903
\(688\) 0 0
\(689\) −23.6558 −0.901215
\(690\) 0 0
\(691\) 3.90123 0.148410 0.0742050 0.997243i \(-0.476358\pi\)
0.0742050 + 0.997243i \(0.476358\pi\)
\(692\) 0 0
\(693\) 76.9177 2.92186
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −2.69164 −0.101953
\(698\) 0 0
\(699\) −23.3505 −0.883198
\(700\) 0 0
\(701\) −24.6495 −0.931000 −0.465500 0.885048i \(-0.654125\pi\)
−0.465500 + 0.885048i \(0.654125\pi\)
\(702\) 0 0
\(703\) 10.0429 0.378776
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18.0305 0.678106
\(708\) 0 0
\(709\) 9.56624 0.359268 0.179634 0.983734i \(-0.442509\pi\)
0.179634 + 0.983734i \(0.442509\pi\)
\(710\) 0 0
\(711\) 49.1829 1.84450
\(712\) 0 0
\(713\) −43.5668 −1.63159
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 25.6292 0.957141
\(718\) 0 0
\(719\) −26.6273 −0.993030 −0.496515 0.868028i \(-0.665387\pi\)
−0.496515 + 0.868028i \(0.665387\pi\)
\(720\) 0 0
\(721\) −67.6554 −2.51962
\(722\) 0 0
\(723\) 79.4159 2.95351
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −24.3779 −0.904127 −0.452063 0.891986i \(-0.649312\pi\)
−0.452063 + 0.891986i \(0.649312\pi\)
\(728\) 0 0
\(729\) −43.2056 −1.60021
\(730\) 0 0
\(731\) 29.1949 1.07981
\(732\) 0 0
\(733\) −6.53865 −0.241511 −0.120755 0.992682i \(-0.538532\pi\)
−0.120755 + 0.992682i \(0.538532\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 20.7404 0.763981
\(738\) 0 0
\(739\) 11.7605 0.432617 0.216308 0.976325i \(-0.430598\pi\)
0.216308 + 0.976325i \(0.430598\pi\)
\(740\) 0 0
\(741\) −14.7179 −0.540674
\(742\) 0 0
\(743\) −40.7714 −1.49576 −0.747879 0.663835i \(-0.768928\pi\)
−0.747879 + 0.663835i \(0.768928\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −18.8822 −0.690863
\(748\) 0 0
\(749\) −49.7636 −1.81832
\(750\) 0 0
\(751\) −0.578858 −0.0211228 −0.0105614 0.999944i \(-0.503362\pi\)
−0.0105614 + 0.999944i \(0.503362\pi\)
\(752\) 0 0
\(753\) −20.4834 −0.746457
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −14.3710 −0.522325 −0.261162 0.965295i \(-0.584106\pi\)
−0.261162 + 0.965295i \(0.584106\pi\)
\(758\) 0 0
\(759\) 55.9973 2.03257
\(760\) 0 0
\(761\) −16.4753 −0.597230 −0.298615 0.954374i \(-0.596525\pi\)
−0.298615 + 0.954374i \(0.596525\pi\)
\(762\) 0 0
\(763\) 2.85798 0.103466
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 21.9808 0.793681
\(768\) 0 0
\(769\) −18.8410 −0.679425 −0.339712 0.940529i \(-0.610330\pi\)
−0.339712 + 0.940529i \(0.610330\pi\)
\(770\) 0 0
\(771\) −31.7978 −1.14517
\(772\) 0 0
\(773\) 37.5917 1.35208 0.676039 0.736866i \(-0.263695\pi\)
0.676039 + 0.736866i \(0.263695\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 102.652 3.68261
\(778\) 0 0
\(779\) 1.11712 0.0400251
\(780\) 0 0
\(781\) 37.2766 1.33386
\(782\) 0 0
\(783\) 32.1769 1.14991
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −44.5235 −1.58709 −0.793545 0.608511i \(-0.791767\pi\)
−0.793545 + 0.608511i \(0.791767\pi\)
\(788\) 0 0
\(789\) 5.95897 0.212145
\(790\) 0 0
\(791\) −77.8854 −2.76929
\(792\) 0 0
\(793\) 49.7700 1.76738
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.84212 −0.136095 −0.0680475 0.997682i \(-0.521677\pi\)
−0.0680475 + 0.997682i \(0.521677\pi\)
\(798\) 0 0
\(799\) −16.2018 −0.573179
\(800\) 0 0
\(801\) 10.9558 0.387103
\(802\) 0 0
\(803\) 14.4897 0.511331
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 16.6231 0.585161
\(808\) 0 0
\(809\) −1.51537 −0.0532774 −0.0266387 0.999645i \(-0.508480\pi\)
−0.0266387 + 0.999645i \(0.508480\pi\)
\(810\) 0 0
\(811\) −20.1768 −0.708502 −0.354251 0.935150i \(-0.615264\pi\)
−0.354251 + 0.935150i \(0.615264\pi\)
\(812\) 0 0
\(813\) 69.4034 2.43408
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −12.1169 −0.423916
\(818\) 0 0
\(819\) −88.7346 −3.10064
\(820\) 0 0
\(821\) −33.0855 −1.15469 −0.577347 0.816499i \(-0.695912\pi\)
−0.577347 + 0.816499i \(0.695912\pi\)
\(822\) 0 0
\(823\) −22.9779 −0.800958 −0.400479 0.916306i \(-0.631156\pi\)
−0.400479 + 0.916306i \(0.631156\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 54.9730 1.91160 0.955798 0.294023i \(-0.0949942\pi\)
0.955798 + 0.294023i \(0.0949942\pi\)
\(828\) 0 0
\(829\) 36.6692 1.27357 0.636787 0.771039i \(-0.280263\pi\)
0.636787 + 0.771039i \(0.280263\pi\)
\(830\) 0 0
\(831\) −12.5544 −0.435508
\(832\) 0 0
\(833\) −17.5489 −0.608033
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −35.2851 −1.21963
\(838\) 0 0
\(839\) −31.4619 −1.08619 −0.543093 0.839672i \(-0.682747\pi\)
−0.543093 + 0.839672i \(0.682747\pi\)
\(840\) 0 0
\(841\) 52.9289 1.82513
\(842\) 0 0
\(843\) −1.00058 −0.0344619
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 42.5263 1.46122
\(848\) 0 0
\(849\) −3.28126 −0.112613
\(850\) 0 0
\(851\) 44.0808 1.51107
\(852\) 0 0
\(853\) 9.45230 0.323641 0.161820 0.986820i \(-0.448263\pi\)
0.161820 + 0.986820i \(0.448263\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 17.9551 0.613335 0.306668 0.951817i \(-0.400786\pi\)
0.306668 + 0.951817i \(0.400786\pi\)
\(858\) 0 0
\(859\) −24.3628 −0.831249 −0.415624 0.909536i \(-0.636437\pi\)
−0.415624 + 0.909536i \(0.636437\pi\)
\(860\) 0 0
\(861\) 11.4184 0.389139
\(862\) 0 0
\(863\) 45.1946 1.53844 0.769222 0.638982i \(-0.220644\pi\)
0.769222 + 0.638982i \(0.220644\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −30.2760 −1.02823
\(868\) 0 0
\(869\) 53.7748 1.82419
\(870\) 0 0
\(871\) −23.9267 −0.810727
\(872\) 0 0
\(873\) −1.30341 −0.0441138
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7.55736 0.255194 0.127597 0.991826i \(-0.459274\pi\)
0.127597 + 0.991826i \(0.459274\pi\)
\(878\) 0 0
\(879\) 23.4283 0.790217
\(880\) 0 0
\(881\) −50.5760 −1.70395 −0.851974 0.523584i \(-0.824595\pi\)
−0.851974 + 0.523584i \(0.824595\pi\)
\(882\) 0 0
\(883\) 6.40946 0.215696 0.107848 0.994167i \(-0.465604\pi\)
0.107848 + 0.994167i \(0.465604\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.14684 0.0720838 0.0360419 0.999350i \(-0.488525\pi\)
0.0360419 + 0.999350i \(0.488525\pi\)
\(888\) 0 0
\(889\) 73.0085 2.44863
\(890\) 0 0
\(891\) −15.7038 −0.526097
\(892\) 0 0
\(893\) 6.72432 0.225021
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −64.6003 −2.15694
\(898\) 0 0
\(899\) −89.8430 −2.99643
\(900\) 0 0
\(901\) −10.4737 −0.348929
\(902\) 0 0
\(903\) −123.850 −4.12148
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −41.3186 −1.37196 −0.685980 0.727620i \(-0.740626\pi\)
−0.685980 + 0.727620i \(0.740626\pi\)
\(908\) 0 0
\(909\) 20.5833 0.682704
\(910\) 0 0
\(911\) 59.7328 1.97903 0.989517 0.144415i \(-0.0461300\pi\)
0.989517 + 0.144415i \(0.0461300\pi\)
\(912\) 0 0
\(913\) −20.6451 −0.683253
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −16.2206 −0.535651
\(918\) 0 0
\(919\) −34.3526 −1.13319 −0.566594 0.823997i \(-0.691739\pi\)
−0.566594 + 0.823997i \(0.691739\pi\)
\(920\) 0 0
\(921\) 22.3426 0.736215
\(922\) 0 0
\(923\) −43.0035 −1.41548
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −77.2342 −2.53670
\(928\) 0 0
\(929\) −48.0752 −1.57729 −0.788647 0.614846i \(-0.789218\pi\)
−0.788647 + 0.614846i \(0.789218\pi\)
\(930\) 0 0
\(931\) 7.28339 0.238703
\(932\) 0 0
\(933\) 20.4031 0.667968
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −42.9537 −1.40324 −0.701619 0.712552i \(-0.747539\pi\)
−0.701619 + 0.712552i \(0.747539\pi\)
\(938\) 0 0
\(939\) −9.28060 −0.302861
\(940\) 0 0
\(941\) −42.8745 −1.39767 −0.698835 0.715283i \(-0.746298\pi\)
−0.698835 + 0.715283i \(0.746298\pi\)
\(942\) 0 0
\(943\) 4.90332 0.159674
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28.1350 0.914264 0.457132 0.889399i \(-0.348877\pi\)
0.457132 + 0.889399i \(0.348877\pi\)
\(948\) 0 0
\(949\) −16.7158 −0.542617
\(950\) 0 0
\(951\) 6.78249 0.219937
\(952\) 0 0
\(953\) −16.7612 −0.542949 −0.271474 0.962446i \(-0.587511\pi\)
−0.271474 + 0.962446i \(0.587511\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 115.477 3.73285
\(958\) 0 0
\(959\) −80.2780 −2.59231
\(960\) 0 0
\(961\) 67.5217 2.17812
\(962\) 0 0
\(963\) −56.8093 −1.83065
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −35.3277 −1.13606 −0.568030 0.823008i \(-0.692294\pi\)
−0.568030 + 0.823008i \(0.692294\pi\)
\(968\) 0 0
\(969\) −6.51637 −0.209336
\(970\) 0 0
\(971\) 54.3917 1.74551 0.872756 0.488156i \(-0.162330\pi\)
0.872756 + 0.488156i \(0.162330\pi\)
\(972\) 0 0
\(973\) 7.36269 0.236037
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −47.4481 −1.51800 −0.759000 0.651091i \(-0.774312\pi\)
−0.759000 + 0.651091i \(0.774312\pi\)
\(978\) 0 0
\(979\) 11.9786 0.382839
\(980\) 0 0
\(981\) 3.26262 0.104167
\(982\) 0 0
\(983\) 42.8550 1.36686 0.683432 0.730014i \(-0.260487\pi\)
0.683432 + 0.730014i \(0.260487\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 68.7312 2.18774
\(988\) 0 0
\(989\) −53.1839 −1.69115
\(990\) 0 0
\(991\) 23.4118 0.743700 0.371850 0.928293i \(-0.378724\pi\)
0.371850 + 0.928293i \(0.378724\pi\)
\(992\) 0 0
\(993\) 78.7467 2.49895
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 12.9905 0.411412 0.205706 0.978614i \(-0.434051\pi\)
0.205706 + 0.978614i \(0.434051\pi\)
\(998\) 0 0
\(999\) 35.7014 1.12954
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.bd.1.5 yes 6
4.3 odd 2 7600.2.a.ci.1.2 6
5.2 odd 4 3800.2.d.p.3649.3 12
5.3 odd 4 3800.2.d.p.3649.10 12
5.4 even 2 3800.2.a.bb.1.2 6
20.19 odd 2 7600.2.a.cm.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.bb.1.2 6 5.4 even 2
3800.2.a.bd.1.5 yes 6 1.1 even 1 trivial
3800.2.d.p.3649.3 12 5.2 odd 4
3800.2.d.p.3649.10 12 5.3 odd 4
7600.2.a.ci.1.2 6 4.3 odd 2
7600.2.a.cm.1.5 6 20.19 odd 2