Properties

Label 2704.2.a.z.1.3
Level $2704$
Weight $2$
Character 2704.1
Self dual yes
Analytic conductor $21.592$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2704,2,Mod(1,2704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2704, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2704.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2704 = 2^{4} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2704.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: \(21.5915487066\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) 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 169)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 2704.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.24698 q^{3} +0.246980 q^{5} +2.35690 q^{7} +2.04892 q^{9} +O(q^{10})\) \(q+2.24698 q^{3} +0.246980 q^{5} +2.35690 q^{7} +2.04892 q^{9} +4.24698 q^{11} +0.554958 q^{15} +2.15883 q^{17} +0.0881460 q^{19} +5.29590 q^{21} -1.49396 q^{23} -4.93900 q^{25} -2.13706 q^{27} +4.63102 q^{29} +6.63102 q^{31} +9.54288 q^{33} +0.582105 q^{35} +5.69202 q^{37} -11.5918 q^{41} +0.295897 q^{43} +0.506041 q^{45} +7.35690 q^{47} -1.44504 q^{49} +4.85086 q^{51} -10.3937 q^{53} +1.04892 q^{55} +0.198062 q^{57} +6.78017 q^{59} +3.47219 q^{61} +4.82908 q^{63} -7.67994 q^{67} -3.35690 q^{69} +8.66487 q^{71} +6.73556 q^{73} -11.0978 q^{75} +10.0097 q^{77} -9.97046 q^{79} -10.9487 q^{81} -1.60925 q^{83} +0.533188 q^{85} +10.4058 q^{87} -2.88471 q^{89} +14.8998 q^{93} +0.0217703 q^{95} -8.05861 q^{97} +8.70171 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} - 4 q^{5} + 3 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} - 4 q^{5} + 3 q^{7} - 3 q^{9} + 8 q^{11} + 2 q^{15} - 2 q^{17} + 4 q^{19} + 2 q^{21} + 5 q^{23} - 5 q^{25} - q^{27} - q^{29} + 5 q^{31} + 10 q^{33} - 4 q^{35} + 12 q^{37} - 7 q^{41} - 13 q^{43} + 11 q^{45} + 18 q^{47} - 4 q^{49} + q^{51} + q^{53} - 6 q^{55} + 5 q^{57} + 19 q^{59} + 4 q^{61} + 4 q^{63} + q^{67} - 6 q^{69} + 27 q^{71} + 9 q^{73} - 15 q^{75} + 8 q^{77} + 5 q^{79} - q^{81} + 7 q^{83} + 5 q^{85} + 18 q^{87} - 11 q^{89} + 22 q^{93} - 3 q^{95} + 7 q^{97} - q^{99}+O(q^{100}) \) 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
\(3\) 2.24698 1.29729 0.648647 0.761089i \(-0.275335\pi\)
0.648647 + 0.761089i \(0.275335\pi\)
\(4\) 0 0
\(5\) 0.246980 0.110453 0.0552263 0.998474i \(-0.482412\pi\)
0.0552263 + 0.998474i \(0.482412\pi\)
\(6\) 0 0
\(7\) 2.35690 0.890823 0.445411 0.895326i \(-0.353057\pi\)
0.445411 + 0.895326i \(0.353057\pi\)
\(8\) 0 0
\(9\) 2.04892 0.682972
\(10\) 0 0
\(11\) 4.24698 1.28051 0.640256 0.768161i \(-0.278828\pi\)
0.640256 + 0.768161i \(0.278828\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 0.554958 0.143290
\(16\) 0 0
\(17\) 2.15883 0.523594 0.261797 0.965123i \(-0.415685\pi\)
0.261797 + 0.965123i \(0.415685\pi\)
\(18\) 0 0
\(19\) 0.0881460 0.0202221 0.0101110 0.999949i \(-0.496782\pi\)
0.0101110 + 0.999949i \(0.496782\pi\)
\(20\) 0 0
\(21\) 5.29590 1.15566
\(22\) 0 0
\(23\) −1.49396 −0.311512 −0.155756 0.987796i \(-0.549781\pi\)
−0.155756 + 0.987796i \(0.549781\pi\)
\(24\) 0 0
\(25\) −4.93900 −0.987800
\(26\) 0 0
\(27\) −2.13706 −0.411278
\(28\) 0 0
\(29\) 4.63102 0.859959 0.429980 0.902839i \(-0.358521\pi\)
0.429980 + 0.902839i \(0.358521\pi\)
\(30\) 0 0
\(31\) 6.63102 1.19097 0.595483 0.803368i \(-0.296961\pi\)
0.595483 + 0.803368i \(0.296961\pi\)
\(32\) 0 0
\(33\) 9.54288 1.66120
\(34\) 0 0
\(35\) 0.582105 0.0983937
\(36\) 0 0
\(37\) 5.69202 0.935763 0.467881 0.883791i \(-0.345017\pi\)
0.467881 + 0.883791i \(0.345017\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −11.5918 −1.81033 −0.905167 0.425056i \(-0.860254\pi\)
−0.905167 + 0.425056i \(0.860254\pi\)
\(42\) 0 0
\(43\) 0.295897 0.0451239 0.0225619 0.999745i \(-0.492818\pi\)
0.0225619 + 0.999745i \(0.492818\pi\)
\(44\) 0 0
\(45\) 0.506041 0.0754361
\(46\) 0 0
\(47\) 7.35690 1.07311 0.536557 0.843864i \(-0.319725\pi\)
0.536557 + 0.843864i \(0.319725\pi\)
\(48\) 0 0
\(49\) −1.44504 −0.206435
\(50\) 0 0
\(51\) 4.85086 0.679256
\(52\) 0 0
\(53\) −10.3937 −1.42769 −0.713844 0.700304i \(-0.753048\pi\)
−0.713844 + 0.700304i \(0.753048\pi\)
\(54\) 0 0
\(55\) 1.04892 0.141436
\(56\) 0 0
\(57\) 0.198062 0.0262340
\(58\) 0 0
\(59\) 6.78017 0.882703 0.441351 0.897334i \(-0.354499\pi\)
0.441351 + 0.897334i \(0.354499\pi\)
\(60\) 0 0
\(61\) 3.47219 0.444568 0.222284 0.974982i \(-0.428649\pi\)
0.222284 + 0.974982i \(0.428649\pi\)
\(62\) 0 0
\(63\) 4.82908 0.608407
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −7.67994 −0.938254 −0.469127 0.883131i \(-0.655431\pi\)
−0.469127 + 0.883131i \(0.655431\pi\)
\(68\) 0 0
\(69\) −3.35690 −0.404123
\(70\) 0 0
\(71\) 8.66487 1.02833 0.514166 0.857691i \(-0.328102\pi\)
0.514166 + 0.857691i \(0.328102\pi\)
\(72\) 0 0
\(73\) 6.73556 0.788338 0.394169 0.919038i \(-0.371032\pi\)
0.394169 + 0.919038i \(0.371032\pi\)
\(74\) 0 0
\(75\) −11.0978 −1.28147
\(76\) 0 0
\(77\) 10.0097 1.14071
\(78\) 0 0
\(79\) −9.97046 −1.12176 −0.560882 0.827896i \(-0.689538\pi\)
−0.560882 + 0.827896i \(0.689538\pi\)
\(80\) 0 0
\(81\) −10.9487 −1.21652
\(82\) 0 0
\(83\) −1.60925 −0.176638 −0.0883192 0.996092i \(-0.528150\pi\)
−0.0883192 + 0.996092i \(0.528150\pi\)
\(84\) 0 0
\(85\) 0.533188 0.0578323
\(86\) 0 0
\(87\) 10.4058 1.11562
\(88\) 0 0
\(89\) −2.88471 −0.305778 −0.152889 0.988243i \(-0.548858\pi\)
−0.152889 + 0.988243i \(0.548858\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 14.8998 1.54503
\(94\) 0 0
\(95\) 0.0217703 0.00223358
\(96\) 0 0
\(97\) −8.05861 −0.818227 −0.409114 0.912483i \(-0.634162\pi\)
−0.409114 + 0.912483i \(0.634162\pi\)
\(98\) 0 0
\(99\) 8.70171 0.874555
\(100\) 0 0
\(101\) −13.3545 −1.32882 −0.664411 0.747367i \(-0.731318\pi\)
−0.664411 + 0.747367i \(0.731318\pi\)
\(102\) 0 0
\(103\) −1.36227 −0.134229 −0.0671144 0.997745i \(-0.521379\pi\)
−0.0671144 + 0.997745i \(0.521379\pi\)
\(104\) 0 0
\(105\) 1.30798 0.127646
\(106\) 0 0
\(107\) −3.26875 −0.316002 −0.158001 0.987439i \(-0.550505\pi\)
−0.158001 + 0.987439i \(0.550505\pi\)
\(108\) 0 0
\(109\) 15.7017 1.50395 0.751976 0.659191i \(-0.229101\pi\)
0.751976 + 0.659191i \(0.229101\pi\)
\(110\) 0 0
\(111\) 12.7899 1.21396
\(112\) 0 0
\(113\) 12.0489 1.13347 0.566733 0.823901i \(-0.308207\pi\)
0.566733 + 0.823901i \(0.308207\pi\)
\(114\) 0 0
\(115\) −0.368977 −0.0344073
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.08815 0.466430
\(120\) 0 0
\(121\) 7.03684 0.639712
\(122\) 0 0
\(123\) −26.0465 −2.34854
\(124\) 0 0
\(125\) −2.45473 −0.219558
\(126\) 0 0
\(127\) 9.80731 0.870258 0.435129 0.900368i \(-0.356703\pi\)
0.435129 + 0.900368i \(0.356703\pi\)
\(128\) 0 0
\(129\) 0.664874 0.0585389
\(130\) 0 0
\(131\) 6.57673 0.574611 0.287306 0.957839i \(-0.407240\pi\)
0.287306 + 0.957839i \(0.407240\pi\)
\(132\) 0 0
\(133\) 0.207751 0.0180143
\(134\) 0 0
\(135\) −0.527811 −0.0454267
\(136\) 0 0
\(137\) 6.21983 0.531396 0.265698 0.964056i \(-0.414398\pi\)
0.265698 + 0.964056i \(0.414398\pi\)
\(138\) 0 0
\(139\) 14.7071 1.24744 0.623719 0.781648i \(-0.285621\pi\)
0.623719 + 0.781648i \(0.285621\pi\)
\(140\) 0 0
\(141\) 16.5308 1.39214
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 1.14377 0.0949848
\(146\) 0 0
\(147\) −3.24698 −0.267806
\(148\) 0 0
\(149\) 4.33513 0.355147 0.177574 0.984108i \(-0.443175\pi\)
0.177574 + 0.984108i \(0.443175\pi\)
\(150\) 0 0
\(151\) −3.94438 −0.320989 −0.160494 0.987037i \(-0.551309\pi\)
−0.160494 + 0.987037i \(0.551309\pi\)
\(152\) 0 0
\(153\) 4.42327 0.357600
\(154\) 0 0
\(155\) 1.63773 0.131545
\(156\) 0 0
\(157\) 4.45473 0.355526 0.177763 0.984073i \(-0.443114\pi\)
0.177763 + 0.984073i \(0.443114\pi\)
\(158\) 0 0
\(159\) −23.3545 −1.85213
\(160\) 0 0
\(161\) −3.52111 −0.277502
\(162\) 0 0
\(163\) −16.1588 −1.26566 −0.632829 0.774292i \(-0.718106\pi\)
−0.632829 + 0.774292i \(0.718106\pi\)
\(164\) 0 0
\(165\) 2.35690 0.183484
\(166\) 0 0
\(167\) −16.1172 −1.24719 −0.623594 0.781749i \(-0.714328\pi\)
−0.623594 + 0.781749i \(0.714328\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0.180604 0.0138111
\(172\) 0 0
\(173\) −21.5362 −1.63736 −0.818682 0.574247i \(-0.805295\pi\)
−0.818682 + 0.574247i \(0.805295\pi\)
\(174\) 0 0
\(175\) −11.6407 −0.879955
\(176\) 0 0
\(177\) 15.2349 1.14513
\(178\) 0 0
\(179\) −11.4330 −0.854540 −0.427270 0.904124i \(-0.640525\pi\)
−0.427270 + 0.904124i \(0.640525\pi\)
\(180\) 0 0
\(181\) 20.9705 1.55872 0.779361 0.626575i \(-0.215544\pi\)
0.779361 + 0.626575i \(0.215544\pi\)
\(182\) 0 0
\(183\) 7.80194 0.576736
\(184\) 0 0
\(185\) 1.40581 0.103357
\(186\) 0 0
\(187\) 9.16852 0.670469
\(188\) 0 0
\(189\) −5.03684 −0.366376
\(190\) 0 0
\(191\) 14.4373 1.04464 0.522322 0.852748i \(-0.325066\pi\)
0.522322 + 0.852748i \(0.325066\pi\)
\(192\) 0 0
\(193\) −13.5797 −0.977489 −0.488745 0.872427i \(-0.662545\pi\)
−0.488745 + 0.872427i \(0.662545\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −0.560335 −0.0399222 −0.0199611 0.999801i \(-0.506354\pi\)
−0.0199611 + 0.999801i \(0.506354\pi\)
\(198\) 0 0
\(199\) −11.4916 −0.814616 −0.407308 0.913291i \(-0.633532\pi\)
−0.407308 + 0.913291i \(0.633532\pi\)
\(200\) 0 0
\(201\) −17.2567 −1.21719
\(202\) 0 0
\(203\) 10.9148 0.766071
\(204\) 0 0
\(205\) −2.86294 −0.199956
\(206\) 0 0
\(207\) −3.06100 −0.212754
\(208\) 0 0
\(209\) 0.374354 0.0258946
\(210\) 0 0
\(211\) −8.78448 −0.604748 −0.302374 0.953189i \(-0.597779\pi\)
−0.302374 + 0.953189i \(0.597779\pi\)
\(212\) 0 0
\(213\) 19.4698 1.33405
\(214\) 0 0
\(215\) 0.0730805 0.00498405
\(216\) 0 0
\(217\) 15.6286 1.06094
\(218\) 0 0
\(219\) 15.1347 1.02271
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 2.25906 0.151278 0.0756390 0.997135i \(-0.475900\pi\)
0.0756390 + 0.997135i \(0.475900\pi\)
\(224\) 0 0
\(225\) −10.1196 −0.674640
\(226\) 0 0
\(227\) 6.96615 0.462359 0.231180 0.972911i \(-0.425741\pi\)
0.231180 + 0.972911i \(0.425741\pi\)
\(228\) 0 0
\(229\) −24.1739 −1.59746 −0.798728 0.601692i \(-0.794493\pi\)
−0.798728 + 0.601692i \(0.794493\pi\)
\(230\) 0 0
\(231\) 22.4916 1.47984
\(232\) 0 0
\(233\) −3.06100 −0.200533 −0.100266 0.994961i \(-0.531969\pi\)
−0.100266 + 0.994961i \(0.531969\pi\)
\(234\) 0 0
\(235\) 1.81700 0.118528
\(236\) 0 0
\(237\) −22.4034 −1.45526
\(238\) 0 0
\(239\) 25.1468 1.62661 0.813304 0.581839i \(-0.197667\pi\)
0.813304 + 0.581839i \(0.197667\pi\)
\(240\) 0 0
\(241\) −20.2664 −1.30547 −0.652735 0.757586i \(-0.726379\pi\)
−0.652735 + 0.757586i \(0.726379\pi\)
\(242\) 0 0
\(243\) −18.1903 −1.16691
\(244\) 0 0
\(245\) −0.356896 −0.0228012
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −3.61596 −0.229152
\(250\) 0 0
\(251\) 23.7211 1.49726 0.748631 0.662987i \(-0.230712\pi\)
0.748631 + 0.662987i \(0.230712\pi\)
\(252\) 0 0
\(253\) −6.34481 −0.398895
\(254\) 0 0
\(255\) 1.19806 0.0750256
\(256\) 0 0
\(257\) 14.2241 0.887278 0.443639 0.896206i \(-0.353687\pi\)
0.443639 + 0.896206i \(0.353687\pi\)
\(258\) 0 0
\(259\) 13.4155 0.833599
\(260\) 0 0
\(261\) 9.48858 0.587329
\(262\) 0 0
\(263\) 17.0954 1.05415 0.527075 0.849819i \(-0.323289\pi\)
0.527075 + 0.849819i \(0.323289\pi\)
\(264\) 0 0
\(265\) −2.56704 −0.157692
\(266\) 0 0
\(267\) −6.48188 −0.396684
\(268\) 0 0
\(269\) −6.46681 −0.394288 −0.197144 0.980374i \(-0.563167\pi\)
−0.197144 + 0.980374i \(0.563167\pi\)
\(270\) 0 0
\(271\) −6.44803 −0.391690 −0.195845 0.980635i \(-0.562745\pi\)
−0.195845 + 0.980635i \(0.562745\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −20.9758 −1.26489
\(276\) 0 0
\(277\) 13.4601 0.808739 0.404370 0.914596i \(-0.367491\pi\)
0.404370 + 0.914596i \(0.367491\pi\)
\(278\) 0 0
\(279\) 13.5864 0.813398
\(280\) 0 0
\(281\) 5.03684 0.300472 0.150236 0.988650i \(-0.451997\pi\)
0.150236 + 0.988650i \(0.451997\pi\)
\(282\) 0 0
\(283\) −22.1280 −1.31537 −0.657686 0.753293i \(-0.728464\pi\)
−0.657686 + 0.753293i \(0.728464\pi\)
\(284\) 0 0
\(285\) 0.0489173 0.00289761
\(286\) 0 0
\(287\) −27.3207 −1.61269
\(288\) 0 0
\(289\) −12.3394 −0.725849
\(290\) 0 0
\(291\) −18.1075 −1.06148
\(292\) 0 0
\(293\) 14.9463 0.873172 0.436586 0.899663i \(-0.356187\pi\)
0.436586 + 0.899663i \(0.356187\pi\)
\(294\) 0 0
\(295\) 1.67456 0.0974968
\(296\) 0 0
\(297\) −9.07606 −0.526647
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0.697398 0.0401974
\(302\) 0 0
\(303\) −30.0073 −1.72387
\(304\) 0 0
\(305\) 0.857560 0.0491037
\(306\) 0 0
\(307\) −19.1293 −1.09177 −0.545883 0.837861i \(-0.683806\pi\)
−0.545883 + 0.837861i \(0.683806\pi\)
\(308\) 0 0
\(309\) −3.06100 −0.174134
\(310\) 0 0
\(311\) 0.269815 0.0152998 0.00764990 0.999971i \(-0.497565\pi\)
0.00764990 + 0.999971i \(0.497565\pi\)
\(312\) 0 0
\(313\) −23.3937 −1.32229 −0.661146 0.750257i \(-0.729930\pi\)
−0.661146 + 0.750257i \(0.729930\pi\)
\(314\) 0 0
\(315\) 1.19269 0.0672002
\(316\) 0 0
\(317\) −13.9952 −0.786050 −0.393025 0.919528i \(-0.628571\pi\)
−0.393025 + 0.919528i \(0.628571\pi\)
\(318\) 0 0
\(319\) 19.6679 1.10119
\(320\) 0 0
\(321\) −7.34481 −0.409948
\(322\) 0 0
\(323\) 0.190293 0.0105882
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 35.2814 1.95107
\(328\) 0 0
\(329\) 17.3394 0.955954
\(330\) 0 0
\(331\) −17.8213 −0.979548 −0.489774 0.871849i \(-0.662921\pi\)
−0.489774 + 0.871849i \(0.662921\pi\)
\(332\) 0 0
\(333\) 11.6625 0.639100
\(334\) 0 0
\(335\) −1.89679 −0.103633
\(336\) 0 0
\(337\) −27.8485 −1.51700 −0.758501 0.651672i \(-0.774068\pi\)
−0.758501 + 0.651672i \(0.774068\pi\)
\(338\) 0 0
\(339\) 27.0737 1.47044
\(340\) 0 0
\(341\) 28.1618 1.52505
\(342\) 0 0
\(343\) −19.9041 −1.07472
\(344\) 0 0
\(345\) −0.829085 −0.0446364
\(346\) 0 0
\(347\) −1.50365 −0.0807200 −0.0403600 0.999185i \(-0.512850\pi\)
−0.0403600 + 0.999185i \(0.512850\pi\)
\(348\) 0 0
\(349\) −14.1860 −0.759358 −0.379679 0.925118i \(-0.623966\pi\)
−0.379679 + 0.925118i \(0.623966\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −7.16852 −0.381542 −0.190771 0.981635i \(-0.561099\pi\)
−0.190771 + 0.981635i \(0.561099\pi\)
\(354\) 0 0
\(355\) 2.14005 0.113582
\(356\) 0 0
\(357\) 11.4330 0.605096
\(358\) 0 0
\(359\) −19.8853 −1.04951 −0.524753 0.851255i \(-0.675842\pi\)
−0.524753 + 0.851255i \(0.675842\pi\)
\(360\) 0 0
\(361\) −18.9922 −0.999591
\(362\) 0 0
\(363\) 15.8116 0.829895
\(364\) 0 0
\(365\) 1.66355 0.0870740
\(366\) 0 0
\(367\) −1.08383 −0.0565757 −0.0282878 0.999600i \(-0.509006\pi\)
−0.0282878 + 0.999600i \(0.509006\pi\)
\(368\) 0 0
\(369\) −23.7506 −1.23641
\(370\) 0 0
\(371\) −24.4969 −1.27182
\(372\) 0 0
\(373\) −6.13036 −0.317418 −0.158709 0.987325i \(-0.550733\pi\)
−0.158709 + 0.987325i \(0.550733\pi\)
\(374\) 0 0
\(375\) −5.51573 −0.284831
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.40880 0.123732 0.0618658 0.998084i \(-0.480295\pi\)
0.0618658 + 0.998084i \(0.480295\pi\)
\(380\) 0 0
\(381\) 22.0368 1.12898
\(382\) 0 0
\(383\) 30.3913 1.55292 0.776462 0.630164i \(-0.217012\pi\)
0.776462 + 0.630164i \(0.217012\pi\)
\(384\) 0 0
\(385\) 2.47219 0.125994
\(386\) 0 0
\(387\) 0.606268 0.0308184
\(388\) 0 0
\(389\) −15.9409 −0.808237 −0.404118 0.914707i \(-0.632422\pi\)
−0.404118 + 0.914707i \(0.632422\pi\)
\(390\) 0 0
\(391\) −3.22521 −0.163106
\(392\) 0 0
\(393\) 14.7778 0.745440
\(394\) 0 0
\(395\) −2.46250 −0.123902
\(396\) 0 0
\(397\) 16.9148 0.848931 0.424466 0.905444i \(-0.360462\pi\)
0.424466 + 0.905444i \(0.360462\pi\)
\(398\) 0 0
\(399\) 0.466812 0.0233698
\(400\) 0 0
\(401\) 26.6625 1.33146 0.665730 0.746192i \(-0.268120\pi\)
0.665730 + 0.746192i \(0.268120\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −2.70410 −0.134368
\(406\) 0 0
\(407\) 24.1739 1.19826
\(408\) 0 0
\(409\) 28.5163 1.41004 0.705021 0.709187i \(-0.250938\pi\)
0.705021 + 0.709187i \(0.250938\pi\)
\(410\) 0 0
\(411\) 13.9758 0.689377
\(412\) 0 0
\(413\) 15.9801 0.786332
\(414\) 0 0
\(415\) −0.397452 −0.0195102
\(416\) 0 0
\(417\) 33.0465 1.61830
\(418\) 0 0
\(419\) 29.6093 1.44651 0.723253 0.690583i \(-0.242646\pi\)
0.723253 + 0.690583i \(0.242646\pi\)
\(420\) 0 0
\(421\) −11.6606 −0.568301 −0.284151 0.958780i \(-0.591712\pi\)
−0.284151 + 0.958780i \(0.591712\pi\)
\(422\) 0 0
\(423\) 15.0737 0.732907
\(424\) 0 0
\(425\) −10.6625 −0.517206
\(426\) 0 0
\(427\) 8.18359 0.396032
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4.34913 −0.209490 −0.104745 0.994499i \(-0.533403\pi\)
−0.104745 + 0.994499i \(0.533403\pi\)
\(432\) 0 0
\(433\) −14.3884 −0.691460 −0.345730 0.938334i \(-0.612369\pi\)
−0.345730 + 0.938334i \(0.612369\pi\)
\(434\) 0 0
\(435\) 2.57002 0.123223
\(436\) 0 0
\(437\) −0.131687 −0.00629942
\(438\) 0 0
\(439\) 20.2325 0.965645 0.482822 0.875718i \(-0.339612\pi\)
0.482822 + 0.875718i \(0.339612\pi\)
\(440\) 0 0
\(441\) −2.96077 −0.140989
\(442\) 0 0
\(443\) −8.12200 −0.385888 −0.192944 0.981210i \(-0.561804\pi\)
−0.192944 + 0.981210i \(0.561804\pi\)
\(444\) 0 0
\(445\) −0.712464 −0.0337740
\(446\) 0 0
\(447\) 9.74094 0.460731
\(448\) 0 0
\(449\) 12.4916 0.589513 0.294757 0.955572i \(-0.404761\pi\)
0.294757 + 0.955572i \(0.404761\pi\)
\(450\) 0 0
\(451\) −49.2301 −2.31816
\(452\) 0 0
\(453\) −8.86294 −0.416417
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.98121 0.279789 0.139895 0.990166i \(-0.455324\pi\)
0.139895 + 0.990166i \(0.455324\pi\)
\(458\) 0 0
\(459\) −4.61356 −0.215343
\(460\) 0 0
\(461\) −2.05669 −0.0957895 −0.0478947 0.998852i \(-0.515251\pi\)
−0.0478947 + 0.998852i \(0.515251\pi\)
\(462\) 0 0
\(463\) 8.44935 0.392675 0.196337 0.980536i \(-0.437095\pi\)
0.196337 + 0.980536i \(0.437095\pi\)
\(464\) 0 0
\(465\) 3.67994 0.170653
\(466\) 0 0
\(467\) −33.5139 −1.55084 −0.775420 0.631446i \(-0.782462\pi\)
−0.775420 + 0.631446i \(0.782462\pi\)
\(468\) 0 0
\(469\) −18.1008 −0.835818
\(470\) 0 0
\(471\) 10.0097 0.461222
\(472\) 0 0
\(473\) 1.25667 0.0577817
\(474\) 0 0
\(475\) −0.435353 −0.0199754
\(476\) 0 0
\(477\) −21.2959 −0.975072
\(478\) 0 0
\(479\) 24.7313 1.13000 0.565000 0.825091i \(-0.308876\pi\)
0.565000 + 0.825091i \(0.308876\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −7.91185 −0.360002
\(484\) 0 0
\(485\) −1.99031 −0.0903754
\(486\) 0 0
\(487\) 37.7555 1.71087 0.855433 0.517913i \(-0.173291\pi\)
0.855433 + 0.517913i \(0.173291\pi\)
\(488\) 0 0
\(489\) −36.3086 −1.64193
\(490\) 0 0
\(491\) −31.3110 −1.41304 −0.706522 0.707691i \(-0.749737\pi\)
−0.706522 + 0.707691i \(0.749737\pi\)
\(492\) 0 0
\(493\) 9.99761 0.450270
\(494\) 0 0
\(495\) 2.14914 0.0965969
\(496\) 0 0
\(497\) 20.4222 0.916061
\(498\) 0 0
\(499\) −21.4873 −0.961902 −0.480951 0.876748i \(-0.659708\pi\)
−0.480951 + 0.876748i \(0.659708\pi\)
\(500\) 0 0
\(501\) −36.2150 −1.61797
\(502\) 0 0
\(503\) −37.5924 −1.67616 −0.838081 0.545546i \(-0.816322\pi\)
−0.838081 + 0.545546i \(0.816322\pi\)
\(504\) 0 0
\(505\) −3.29829 −0.146772
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −17.1075 −0.758278 −0.379139 0.925340i \(-0.623780\pi\)
−0.379139 + 0.925340i \(0.623780\pi\)
\(510\) 0 0
\(511\) 15.8750 0.702269
\(512\) 0 0
\(513\) −0.188374 −0.00831690
\(514\) 0 0
\(515\) −0.336454 −0.0148259
\(516\) 0 0
\(517\) 31.2446 1.37414
\(518\) 0 0
\(519\) −48.3913 −2.12414
\(520\) 0 0
\(521\) −19.8465 −0.869493 −0.434746 0.900553i \(-0.643162\pi\)
−0.434746 + 0.900553i \(0.643162\pi\)
\(522\) 0 0
\(523\) 11.4300 0.499798 0.249899 0.968272i \(-0.419603\pi\)
0.249899 + 0.968272i \(0.419603\pi\)
\(524\) 0 0
\(525\) −26.1564 −1.14156
\(526\) 0 0
\(527\) 14.3153 0.623583
\(528\) 0 0
\(529\) −20.7681 −0.902960
\(530\) 0 0
\(531\) 13.8920 0.602862
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −0.807315 −0.0349033
\(536\) 0 0
\(537\) −25.6896 −1.10859
\(538\) 0 0
\(539\) −6.13706 −0.264342
\(540\) 0 0
\(541\) 16.1884 0.695993 0.347996 0.937496i \(-0.386862\pi\)
0.347996 + 0.937496i \(0.386862\pi\)
\(542\) 0 0
\(543\) 47.1202 2.02212
\(544\) 0 0
\(545\) 3.87800 0.166115
\(546\) 0 0
\(547\) −5.33081 −0.227929 −0.113965 0.993485i \(-0.536355\pi\)
−0.113965 + 0.993485i \(0.536355\pi\)
\(548\) 0 0
\(549\) 7.11423 0.303628
\(550\) 0 0
\(551\) 0.408206 0.0173902
\(552\) 0 0
\(553\) −23.4993 −0.999293
\(554\) 0 0
\(555\) 3.15883 0.134085
\(556\) 0 0
\(557\) 7.39075 0.313156 0.156578 0.987666i \(-0.449954\pi\)
0.156578 + 0.987666i \(0.449954\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 20.6015 0.869795
\(562\) 0 0
\(563\) 9.47889 0.399488 0.199744 0.979848i \(-0.435989\pi\)
0.199744 + 0.979848i \(0.435989\pi\)
\(564\) 0 0
\(565\) 2.97584 0.125194
\(566\) 0 0
\(567\) −25.8049 −1.08370
\(568\) 0 0
\(569\) −10.1438 −0.425249 −0.212624 0.977134i \(-0.568201\pi\)
−0.212624 + 0.977134i \(0.568201\pi\)
\(570\) 0 0
\(571\) 14.0925 0.589751 0.294876 0.955536i \(-0.404722\pi\)
0.294876 + 0.955536i \(0.404722\pi\)
\(572\) 0 0
\(573\) 32.4403 1.35521
\(574\) 0 0
\(575\) 7.37867 0.307712
\(576\) 0 0
\(577\) 25.1545 1.04720 0.523598 0.851965i \(-0.324589\pi\)
0.523598 + 0.851965i \(0.324589\pi\)
\(578\) 0 0
\(579\) −30.5133 −1.26809
\(580\) 0 0
\(581\) −3.79284 −0.157354
\(582\) 0 0
\(583\) −44.1420 −1.82817
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −43.8353 −1.80928 −0.904639 0.426180i \(-0.859859\pi\)
−0.904639 + 0.426180i \(0.859859\pi\)
\(588\) 0 0
\(589\) 0.584498 0.0240838
\(590\) 0 0
\(591\) −1.25906 −0.0517909
\(592\) 0 0
\(593\) −24.9965 −1.02648 −0.513242 0.858244i \(-0.671556\pi\)
−0.513242 + 0.858244i \(0.671556\pi\)
\(594\) 0 0
\(595\) 1.25667 0.0515184
\(596\) 0 0
\(597\) −25.8213 −1.05680
\(598\) 0 0
\(599\) 6.24027 0.254971 0.127485 0.991840i \(-0.459309\pi\)
0.127485 + 0.991840i \(0.459309\pi\)
\(600\) 0 0
\(601\) 6.32975 0.258196 0.129098 0.991632i \(-0.458792\pi\)
0.129098 + 0.991632i \(0.458792\pi\)
\(602\) 0 0
\(603\) −15.7356 −0.640802
\(604\) 0 0
\(605\) 1.73795 0.0706579
\(606\) 0 0
\(607\) 43.6480 1.77162 0.885809 0.464050i \(-0.153604\pi\)
0.885809 + 0.464050i \(0.153604\pi\)
\(608\) 0 0
\(609\) 24.5254 0.993820
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −25.9541 −1.04827 −0.524137 0.851634i \(-0.675612\pi\)
−0.524137 + 0.851634i \(0.675612\pi\)
\(614\) 0 0
\(615\) −6.43296 −0.259402
\(616\) 0 0
\(617\) 45.9396 1.84946 0.924729 0.380626i \(-0.124291\pi\)
0.924729 + 0.380626i \(0.124291\pi\)
\(618\) 0 0
\(619\) −6.73556 −0.270725 −0.135363 0.990796i \(-0.543220\pi\)
−0.135363 + 0.990796i \(0.543220\pi\)
\(620\) 0 0
\(621\) 3.19269 0.128118
\(622\) 0 0
\(623\) −6.79895 −0.272394
\(624\) 0 0
\(625\) 24.0887 0.963549
\(626\) 0 0
\(627\) 0.841166 0.0335930
\(628\) 0 0
\(629\) 12.2881 0.489960
\(630\) 0 0
\(631\) −45.0998 −1.79539 −0.897696 0.440614i \(-0.854761\pi\)
−0.897696 + 0.440614i \(0.854761\pi\)
\(632\) 0 0
\(633\) −19.7385 −0.784537
\(634\) 0 0
\(635\) 2.42221 0.0961223
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 17.7536 0.702322
\(640\) 0 0
\(641\) 32.5821 1.28692 0.643458 0.765482i \(-0.277499\pi\)
0.643458 + 0.765482i \(0.277499\pi\)
\(642\) 0 0
\(643\) −25.5754 −1.00860 −0.504298 0.863530i \(-0.668249\pi\)
−0.504298 + 0.863530i \(0.668249\pi\)
\(644\) 0 0
\(645\) 0.164210 0.00646578
\(646\) 0 0
\(647\) 30.1715 1.18616 0.593082 0.805142i \(-0.297911\pi\)
0.593082 + 0.805142i \(0.297911\pi\)
\(648\) 0 0
\(649\) 28.7952 1.13031
\(650\) 0 0
\(651\) 35.1172 1.37635
\(652\) 0 0
\(653\) 36.9028 1.44412 0.722058 0.691832i \(-0.243196\pi\)
0.722058 + 0.691832i \(0.243196\pi\)
\(654\) 0 0
\(655\) 1.62432 0.0634673
\(656\) 0 0
\(657\) 13.8006 0.538413
\(658\) 0 0
\(659\) −23.6866 −0.922701 −0.461350 0.887218i \(-0.652635\pi\)
−0.461350 + 0.887218i \(0.652635\pi\)
\(660\) 0 0
\(661\) −31.7590 −1.23528 −0.617641 0.786460i \(-0.711911\pi\)
−0.617641 + 0.786460i \(0.711911\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.0513102 0.00198973
\(666\) 0 0
\(667\) −6.91856 −0.267888
\(668\) 0 0
\(669\) 5.07606 0.196252
\(670\) 0 0
\(671\) 14.7463 0.569275
\(672\) 0 0
\(673\) −7.50232 −0.289193 −0.144597 0.989491i \(-0.546188\pi\)
−0.144597 + 0.989491i \(0.546188\pi\)
\(674\) 0 0
\(675\) 10.5550 0.406261
\(676\) 0 0
\(677\) −35.0315 −1.34637 −0.673184 0.739475i \(-0.735074\pi\)
−0.673184 + 0.739475i \(0.735074\pi\)
\(678\) 0 0
\(679\) −18.9933 −0.728896
\(680\) 0 0
\(681\) 15.6528 0.599816
\(682\) 0 0
\(683\) 24.0834 0.921524 0.460762 0.887524i \(-0.347576\pi\)
0.460762 + 0.887524i \(0.347576\pi\)
\(684\) 0 0
\(685\) 1.53617 0.0586941
\(686\) 0 0
\(687\) −54.3183 −2.07237
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −2.01447 −0.0766342 −0.0383171 0.999266i \(-0.512200\pi\)
−0.0383171 + 0.999266i \(0.512200\pi\)
\(692\) 0 0
\(693\) 20.5090 0.779073
\(694\) 0 0
\(695\) 3.63235 0.137783
\(696\) 0 0
\(697\) −25.0248 −0.947880
\(698\) 0 0
\(699\) −6.87800 −0.260150
\(700\) 0 0
\(701\) −48.8189 −1.84387 −0.921933 0.387350i \(-0.873390\pi\)
−0.921933 + 0.387350i \(0.873390\pi\)
\(702\) 0 0
\(703\) 0.501729 0.0189231
\(704\) 0 0
\(705\) 4.08277 0.153766
\(706\) 0 0
\(707\) −31.4752 −1.18375
\(708\) 0 0
\(709\) −20.8060 −0.781385 −0.390693 0.920521i \(-0.627764\pi\)
−0.390693 + 0.920521i \(0.627764\pi\)
\(710\) 0 0
\(711\) −20.4286 −0.766134
\(712\) 0 0
\(713\) −9.90648 −0.371000
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 56.5042 2.11019
\(718\) 0 0
\(719\) −21.4306 −0.799225 −0.399613 0.916684i \(-0.630855\pi\)
−0.399613 + 0.916684i \(0.630855\pi\)
\(720\) 0 0
\(721\) −3.21073 −0.119574
\(722\) 0 0
\(723\) −45.5381 −1.69358
\(724\) 0 0
\(725\) −22.8726 −0.849468
\(726\) 0 0
\(727\) −13.4862 −0.500175 −0.250088 0.968223i \(-0.580459\pi\)
−0.250088 + 0.968223i \(0.580459\pi\)
\(728\) 0 0
\(729\) −8.02715 −0.297302
\(730\) 0 0
\(731\) 0.638792 0.0236266
\(732\) 0 0
\(733\) −43.5424 −1.60828 −0.804138 0.594443i \(-0.797373\pi\)
−0.804138 + 0.594443i \(0.797373\pi\)
\(734\) 0 0
\(735\) −0.801938 −0.0295799
\(736\) 0 0
\(737\) −32.6165 −1.20145
\(738\) 0 0
\(739\) −20.0543 −0.737709 −0.368855 0.929487i \(-0.620250\pi\)
−0.368855 + 0.929487i \(0.620250\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 33.1685 1.21684 0.608418 0.793617i \(-0.291805\pi\)
0.608418 + 0.793617i \(0.291805\pi\)
\(744\) 0 0
\(745\) 1.07069 0.0392270
\(746\) 0 0
\(747\) −3.29722 −0.120639
\(748\) 0 0
\(749\) −7.70410 −0.281502
\(750\) 0 0
\(751\) −39.2814 −1.43340 −0.716700 0.697382i \(-0.754348\pi\)
−0.716700 + 0.697382i \(0.754348\pi\)
\(752\) 0 0
\(753\) 53.3008 1.94239
\(754\) 0 0
\(755\) −0.974181 −0.0354541
\(756\) 0 0
\(757\) −46.6426 −1.69526 −0.847628 0.530592i \(-0.821970\pi\)
−0.847628 + 0.530592i \(0.821970\pi\)
\(758\) 0 0
\(759\) −14.2567 −0.517484
\(760\) 0 0
\(761\) −21.8984 −0.793818 −0.396909 0.917858i \(-0.629917\pi\)
−0.396909 + 0.917858i \(0.629917\pi\)
\(762\) 0 0
\(763\) 37.0073 1.33975
\(764\) 0 0
\(765\) 1.09246 0.0394979
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 46.7096 1.68439 0.842196 0.539172i \(-0.181263\pi\)
0.842196 + 0.539172i \(0.181263\pi\)
\(770\) 0 0
\(771\) 31.9614 1.15106
\(772\) 0 0
\(773\) −30.2416 −1.08771 −0.543857 0.839178i \(-0.683037\pi\)
−0.543857 + 0.839178i \(0.683037\pi\)
\(774\) 0 0
\(775\) −32.7506 −1.17644
\(776\) 0 0
\(777\) 30.1444 1.08142
\(778\) 0 0
\(779\) −1.02177 −0.0366087
\(780\) 0 0
\(781\) 36.7995 1.31679
\(782\) 0 0
\(783\) −9.89679 −0.353682
\(784\) 0 0
\(785\) 1.10023 0.0392688
\(786\) 0 0
\(787\) −28.7023 −1.02313 −0.511563 0.859246i \(-0.670933\pi\)
−0.511563 + 0.859246i \(0.670933\pi\)
\(788\) 0 0
\(789\) 38.4131 1.36754
\(790\) 0 0
\(791\) 28.3980 1.00972
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −5.76809 −0.204573
\(796\) 0 0
\(797\) −18.5418 −0.656785 −0.328392 0.944541i \(-0.606507\pi\)
−0.328392 + 0.944541i \(0.606507\pi\)
\(798\) 0 0
\(799\) 15.8823 0.561876
\(800\) 0 0
\(801\) −5.91053 −0.208838
\(802\) 0 0
\(803\) 28.6058 1.00948
\(804\) 0 0
\(805\) −0.869641 −0.0306508
\(806\) 0 0
\(807\) −14.5308 −0.511508
\(808\) 0 0
\(809\) −10.0677 −0.353962 −0.176981 0.984214i \(-0.556633\pi\)
−0.176981 + 0.984214i \(0.556633\pi\)
\(810\) 0 0
\(811\) 10.0285 0.352147 0.176074 0.984377i \(-0.443660\pi\)
0.176074 + 0.984377i \(0.443660\pi\)
\(812\) 0 0
\(813\) −14.4886 −0.508137
\(814\) 0 0
\(815\) −3.99090 −0.139795
\(816\) 0 0
\(817\) 0.0260821 0.000912498 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −26.1704 −0.913355 −0.456677 0.889632i \(-0.650961\pi\)
−0.456677 + 0.889632i \(0.650961\pi\)
\(822\) 0 0
\(823\) −1.82238 −0.0635242 −0.0317621 0.999495i \(-0.510112\pi\)
−0.0317621 + 0.999495i \(0.510112\pi\)
\(824\) 0 0
\(825\) −47.1323 −1.64094
\(826\) 0 0
\(827\) 32.2941 1.12298 0.561488 0.827485i \(-0.310229\pi\)
0.561488 + 0.827485i \(0.310229\pi\)
\(828\) 0 0
\(829\) 15.1002 0.524453 0.262226 0.965006i \(-0.415543\pi\)
0.262226 + 0.965006i \(0.415543\pi\)
\(830\) 0 0
\(831\) 30.2446 1.04917
\(832\) 0 0
\(833\) −3.11960 −0.108088
\(834\) 0 0
\(835\) −3.98062 −0.137755
\(836\) 0 0
\(837\) −14.1709 −0.489818
\(838\) 0 0
\(839\) 32.9965 1.13917 0.569584 0.821933i \(-0.307104\pi\)
0.569584 + 0.821933i \(0.307104\pi\)
\(840\) 0 0
\(841\) −7.55363 −0.260470
\(842\) 0 0
\(843\) 11.3177 0.389801
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 16.5851 0.569870
\(848\) 0 0
\(849\) −49.7211 −1.70642
\(850\) 0 0
\(851\) −8.50365 −0.291501
\(852\) 0 0
\(853\) −37.7802 −1.29357 −0.646784 0.762673i \(-0.723887\pi\)
−0.646784 + 0.762673i \(0.723887\pi\)
\(854\) 0 0
\(855\) 0.0446055 0.00152547
\(856\) 0 0
\(857\) 27.3623 0.934677 0.467339 0.884078i \(-0.345213\pi\)
0.467339 + 0.884078i \(0.345213\pi\)
\(858\) 0 0
\(859\) 20.0629 0.684538 0.342269 0.939602i \(-0.388805\pi\)
0.342269 + 0.939602i \(0.388805\pi\)
\(860\) 0 0
\(861\) −61.3889 −2.09213
\(862\) 0 0
\(863\) 6.14483 0.209173 0.104586 0.994516i \(-0.466648\pi\)
0.104586 + 0.994516i \(0.466648\pi\)
\(864\) 0 0
\(865\) −5.31900 −0.180851
\(866\) 0 0
\(867\) −27.7265 −0.941640
\(868\) 0 0
\(869\) −42.3443 −1.43643
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −16.5114 −0.558827
\(874\) 0 0
\(875\) −5.78554 −0.195587
\(876\) 0 0
\(877\) −13.5077 −0.456123 −0.228061 0.973647i \(-0.573239\pi\)
−0.228061 + 0.973647i \(0.573239\pi\)
\(878\) 0 0
\(879\) 33.5840 1.13276
\(880\) 0 0
\(881\) −5.23431 −0.176348 −0.0881741 0.996105i \(-0.528103\pi\)
−0.0881741 + 0.996105i \(0.528103\pi\)
\(882\) 0 0
\(883\) 4.57301 0.153894 0.0769470 0.997035i \(-0.475483\pi\)
0.0769470 + 0.997035i \(0.475483\pi\)
\(884\) 0 0
\(885\) 3.76271 0.126482
\(886\) 0 0
\(887\) 1.64071 0.0550897 0.0275448 0.999621i \(-0.491231\pi\)
0.0275448 + 0.999621i \(0.491231\pi\)
\(888\) 0 0
\(889\) 23.1148 0.775246
\(890\) 0 0
\(891\) −46.4989 −1.55777
\(892\) 0 0
\(893\) 0.648481 0.0217006
\(894\) 0 0
\(895\) −2.82371 −0.0943861
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 30.7084 1.02418
\(900\) 0 0
\(901\) −22.4383 −0.747529
\(902\) 0 0
\(903\) 1.56704 0.0521478
\(904\) 0 0
\(905\) 5.17928 0.172165
\(906\) 0 0
\(907\) −8.10215 −0.269027 −0.134514 0.990912i \(-0.542947\pi\)
−0.134514 + 0.990912i \(0.542947\pi\)
\(908\) 0 0
\(909\) −27.3623 −0.907549
\(910\) 0 0
\(911\) 9.18119 0.304187 0.152093 0.988366i \(-0.451399\pi\)
0.152093 + 0.988366i \(0.451399\pi\)
\(912\) 0 0
\(913\) −6.83446 −0.226188
\(914\) 0 0
\(915\) 1.92692 0.0637020
\(916\) 0 0
\(917\) 15.5007 0.511877
\(918\) 0 0
\(919\) −27.5036 −0.907262 −0.453631 0.891190i \(-0.649872\pi\)
−0.453631 + 0.891190i \(0.649872\pi\)
\(920\) 0 0
\(921\) −42.9831 −1.41634
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −28.1129 −0.924346
\(926\) 0 0
\(927\) −2.79118 −0.0916745
\(928\) 0 0
\(929\) 24.2131 0.794407 0.397203 0.917731i \(-0.369981\pi\)
0.397203 + 0.917731i \(0.369981\pi\)
\(930\) 0 0
\(931\) −0.127375 −0.00417454
\(932\) 0 0
\(933\) 0.606268 0.0198483
\(934\) 0 0
\(935\) 2.26444 0.0740550
\(936\) 0 0
\(937\) 11.1830 0.365333 0.182666 0.983175i \(-0.441527\pi\)
0.182666 + 0.983175i \(0.441527\pi\)
\(938\) 0 0
\(939\) −52.5652 −1.71540
\(940\) 0 0
\(941\) −15.9638 −0.520404 −0.260202 0.965554i \(-0.583789\pi\)
−0.260202 + 0.965554i \(0.583789\pi\)
\(942\) 0 0
\(943\) 17.3177 0.563941
\(944\) 0 0
\(945\) −1.24400 −0.0404672
\(946\) 0 0
\(947\) −6.51466 −0.211698 −0.105849 0.994382i \(-0.533756\pi\)
−0.105849 + 0.994382i \(0.533756\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −31.4470 −1.01974
\(952\) 0 0
\(953\) 47.6469 1.54344 0.771718 0.635965i \(-0.219398\pi\)
0.771718 + 0.635965i \(0.219398\pi\)
\(954\) 0 0
\(955\) 3.56571 0.115384
\(956\) 0 0
\(957\) 44.1933 1.42857
\(958\) 0 0
\(959\) 14.6595 0.473380
\(960\) 0 0
\(961\) 12.9705 0.418402
\(962\) 0 0
\(963\) −6.69740 −0.215821
\(964\) 0 0
\(965\) −3.35391 −0.107966
\(966\) 0 0
\(967\) 43.8122 1.40891 0.704453 0.709751i \(-0.251192\pi\)
0.704453 + 0.709751i \(0.251192\pi\)
\(968\) 0 0
\(969\) 0.427583 0.0137360
\(970\) 0 0
\(971\) −4.29483 −0.137828 −0.0689139 0.997623i \(-0.521953\pi\)
−0.0689139 + 0.997623i \(0.521953\pi\)
\(972\) 0 0
\(973\) 34.6631 1.11125
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 26.8019 0.857470 0.428735 0.903430i \(-0.358959\pi\)
0.428735 + 0.903430i \(0.358959\pi\)
\(978\) 0 0
\(979\) −12.2513 −0.391553
\(980\) 0 0
\(981\) 32.1715 1.02716
\(982\) 0 0
\(983\) −27.2495 −0.869124 −0.434562 0.900642i \(-0.643097\pi\)
−0.434562 + 0.900642i \(0.643097\pi\)
\(984\) 0 0
\(985\) −0.138391 −0.00440951
\(986\) 0 0
\(987\) 38.9614 1.24015
\(988\) 0 0
\(989\) −0.442058 −0.0140566
\(990\) 0 0
\(991\) −24.3889 −0.774740 −0.387370 0.921924i \(-0.626616\pi\)
−0.387370 + 0.921924i \(0.626616\pi\)
\(992\) 0 0
\(993\) −40.0441 −1.27076
\(994\) 0 0
\(995\) −2.83818 −0.0899764
\(996\) 0 0
\(997\) 31.3207 0.991935 0.495967 0.868341i \(-0.334814\pi\)
0.495967 + 0.868341i \(0.334814\pi\)
\(998\) 0 0
\(999\) −12.1642 −0.384859
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 2704.2.a.z.1.3 3
4.3 odd 2 169.2.a.b.1.3 3
12.11 even 2 1521.2.a.r.1.1 3
13.5 odd 4 2704.2.f.o.337.5 6
13.8 odd 4 2704.2.f.o.337.6 6
13.12 even 2 2704.2.a.ba.1.3 3
20.19 odd 2 4225.2.a.bg.1.1 3
28.27 even 2 8281.2.a.bf.1.3 3
52.3 odd 6 169.2.c.c.22.1 6
52.7 even 12 169.2.e.b.23.5 12
52.11 even 12 169.2.e.b.147.2 12
52.15 even 12 169.2.e.b.147.5 12
52.19 even 12 169.2.e.b.23.2 12
52.23 odd 6 169.2.c.b.22.3 6
52.31 even 4 169.2.b.b.168.2 6
52.35 odd 6 169.2.c.c.146.1 6
52.43 odd 6 169.2.c.b.146.3 6
52.47 even 4 169.2.b.b.168.5 6
52.51 odd 2 169.2.a.c.1.1 yes 3
156.47 odd 4 1521.2.b.l.1351.2 6
156.83 odd 4 1521.2.b.l.1351.5 6
156.155 even 2 1521.2.a.o.1.3 3
260.259 odd 2 4225.2.a.bb.1.3 3
364.363 even 2 8281.2.a.bj.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.3 3 4.3 odd 2
169.2.a.c.1.1 yes 3 52.51 odd 2
169.2.b.b.168.2 6 52.31 even 4
169.2.b.b.168.5 6 52.47 even 4
169.2.c.b.22.3 6 52.23 odd 6
169.2.c.b.146.3 6 52.43 odd 6
169.2.c.c.22.1 6 52.3 odd 6
169.2.c.c.146.1 6 52.35 odd 6
169.2.e.b.23.2 12 52.19 even 12
169.2.e.b.23.5 12 52.7 even 12
169.2.e.b.147.2 12 52.11 even 12
169.2.e.b.147.5 12 52.15 even 12
1521.2.a.o.1.3 3 156.155 even 2
1521.2.a.r.1.1 3 12.11 even 2
1521.2.b.l.1351.2 6 156.47 odd 4
1521.2.b.l.1351.5 6 156.83 odd 4
2704.2.a.z.1.3 3 1.1 even 1 trivial
2704.2.a.ba.1.3 3 13.12 even 2
2704.2.f.o.337.5 6 13.5 odd 4
2704.2.f.o.337.6 6 13.8 odd 4
4225.2.a.bb.1.3 3 260.259 odd 2
4225.2.a.bg.1.1 3 20.19 odd 2
8281.2.a.bf.1.3 3 28.27 even 2
8281.2.a.bj.1.1 3 364.363 even 2