Properties

Label 8281.2.a.bj.1.1
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8281,2,Mod(1,8281)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8281.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.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: \(66.1241179138\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2]\)
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 1.1
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.801938 q^{2} +2.24698 q^{3} -1.35690 q^{4} +0.246980 q^{5} -1.80194 q^{6} +2.69202 q^{8} +2.04892 q^{9} +O(q^{10})\) \(q-0.801938 q^{2} +2.24698 q^{3} -1.35690 q^{4} +0.246980 q^{5} -1.80194 q^{6} +2.69202 q^{8} +2.04892 q^{9} -0.198062 q^{10} +4.24698 q^{11} -3.04892 q^{12} +0.554958 q^{15} +0.554958 q^{16} -2.15883 q^{17} -1.64310 q^{18} -0.0881460 q^{19} -0.335126 q^{20} -3.40581 q^{22} +1.49396 q^{23} +6.04892 q^{24} -4.93900 q^{25} -2.13706 q^{27} +4.63102 q^{29} -0.445042 q^{30} -6.63102 q^{31} -5.82908 q^{32} +9.54288 q^{33} +1.73125 q^{34} -2.78017 q^{36} -5.69202 q^{37} +0.0706876 q^{38} +0.664874 q^{40} -11.5918 q^{41} -0.295897 q^{43} -5.76271 q^{44} +0.506041 q^{45} -1.19806 q^{46} -7.35690 q^{47} +1.24698 q^{48} +3.96077 q^{50} -4.85086 q^{51} -10.3937 q^{53} +1.71379 q^{54} +1.04892 q^{55} -0.198062 q^{57} -3.71379 q^{58} -6.78017 q^{59} -0.753020 q^{60} -3.47219 q^{61} +5.31767 q^{62} +3.56465 q^{64} -7.65279 q^{66} -7.67994 q^{67} +2.92931 q^{68} +3.35690 q^{69} +8.66487 q^{71} +5.51573 q^{72} +6.73556 q^{73} +4.56465 q^{74} -11.0978 q^{75} +0.119605 q^{76} +9.97046 q^{79} +0.137063 q^{80} -10.9487 q^{81} +9.29590 q^{82} +1.60925 q^{83} -0.533188 q^{85} +0.237291 q^{86} +10.4058 q^{87} +11.4330 q^{88} -2.88471 q^{89} -0.405813 q^{90} -2.02715 q^{92} -14.8998 q^{93} +5.89977 q^{94} -0.0217703 q^{95} -13.0978 q^{96} -8.05861 q^{97} +8.70171 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 2 q^{3} - 4 q^{5} - q^{6} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 2 q^{3} - 4 q^{5} - q^{6} + 3 q^{8} - 3 q^{9} - 5 q^{10} + 8 q^{11} + 2 q^{15} + 2 q^{16} + 2 q^{17} - 9 q^{18} - 4 q^{19} + 3 q^{22} - 5 q^{23} + 9 q^{24} - 5 q^{25} - q^{27} - q^{29} - q^{30} - 5 q^{31} - 7 q^{32} + 10 q^{33} + 13 q^{34} - 7 q^{36} - 12 q^{37} - 12 q^{38} + 3 q^{40} - 7 q^{41} + 13 q^{43} + 11 q^{45} - 8 q^{46} - 18 q^{47} - q^{48} - q^{50} - q^{51} + q^{53} - 3 q^{54} - 6 q^{55} - 5 q^{57} - 3 q^{58} - 19 q^{59} - 7 q^{60} - 4 q^{61} - q^{62} - 11 q^{64} - 5 q^{66} + q^{67} + 21 q^{68} + 6 q^{69} + 27 q^{71} + 4 q^{72} + 9 q^{73} - 8 q^{74} - 15 q^{75} - 21 q^{76} - 5 q^{79} - 5 q^{80} - q^{81} + 14 q^{82} - 7 q^{83} - 5 q^{85} + 18 q^{86} + 18 q^{87} + 15 q^{88} - 11 q^{89} + 12 q^{90} - 22 q^{93} - 5 q^{94} + 3 q^{95} - 21 q^{96} + 7 q^{97} - q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.801938 −0.567056 −0.283528 0.958964i \(-0.591505\pi\)
−0.283528 + 0.958964i \(0.591505\pi\)
\(3\) 2.24698 1.29729 0.648647 0.761089i \(-0.275335\pi\)
0.648647 + 0.761089i \(0.275335\pi\)
\(4\) −1.35690 −0.678448
\(5\) 0.246980 0.110453 0.0552263 0.998474i \(-0.482412\pi\)
0.0552263 + 0.998474i \(0.482412\pi\)
\(6\) −1.80194 −0.735638
\(7\) 0 0
\(8\) 2.69202 0.951773
\(9\) 2.04892 0.682972
\(10\) −0.198062 −0.0626328
\(11\) 4.24698 1.28051 0.640256 0.768161i \(-0.278828\pi\)
0.640256 + 0.768161i \(0.278828\pi\)
\(12\) −3.04892 −0.880147
\(13\) 0 0
\(14\) 0 0
\(15\) 0.554958 0.143290
\(16\) 0.554958 0.138740
\(17\) −2.15883 −0.523594 −0.261797 0.965123i \(-0.584315\pi\)
−0.261797 + 0.965123i \(0.584315\pi\)
\(18\) −1.64310 −0.387283
\(19\) −0.0881460 −0.0202221 −0.0101110 0.999949i \(-0.503218\pi\)
−0.0101110 + 0.999949i \(0.503218\pi\)
\(20\) −0.335126 −0.0749364
\(21\) 0 0
\(22\) −3.40581 −0.726122
\(23\) 1.49396 0.311512 0.155756 0.987796i \(-0.450219\pi\)
0.155756 + 0.987796i \(0.450219\pi\)
\(24\) 6.04892 1.23473
\(25\) −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 \(0.358521\pi\)
\(30\) −0.445042 −0.0812532
\(31\) −6.63102 −1.19097 −0.595483 0.803368i \(-0.703039\pi\)
−0.595483 + 0.803368i \(0.703039\pi\)
\(32\) −5.82908 −1.03045
\(33\) 9.54288 1.66120
\(34\) 1.73125 0.296907
\(35\) 0 0
\(36\) −2.78017 −0.463361
\(37\) −5.69202 −0.935763 −0.467881 0.883791i \(-0.654983\pi\)
−0.467881 + 0.883791i \(0.654983\pi\)
\(38\) 0.0706876 0.0114670
\(39\) 0 0
\(40\) 0.664874 0.105126
\(41\) −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 \(-0.507182\pi\)
−0.0225619 + 0.999745i \(0.507182\pi\)
\(44\) −5.76271 −0.868761
\(45\) 0.506041 0.0754361
\(46\) −1.19806 −0.176645
\(47\) −7.35690 −1.07311 −0.536557 0.843864i \(-0.680275\pi\)
−0.536557 + 0.843864i \(0.680275\pi\)
\(48\) 1.24698 0.179986
\(49\) 0 0
\(50\) 3.96077 0.560138
\(51\) −4.85086 −0.679256
\(52\) 0 0
\(53\) −10.3937 −1.42769 −0.713844 0.700304i \(-0.753048\pi\)
−0.713844 + 0.700304i \(0.753048\pi\)
\(54\) 1.71379 0.233218
\(55\) 1.04892 0.141436
\(56\) 0 0
\(57\) −0.198062 −0.0262340
\(58\) −3.71379 −0.487645
\(59\) −6.78017 −0.882703 −0.441351 0.897334i \(-0.645501\pi\)
−0.441351 + 0.897334i \(0.645501\pi\)
\(60\) −0.753020 −0.0972145
\(61\) −3.47219 −0.444568 −0.222284 0.974982i \(-0.571351\pi\)
−0.222284 + 0.974982i \(0.571351\pi\)
\(62\) 5.31767 0.675344
\(63\) 0 0
\(64\) 3.56465 0.445581
\(65\) 0 0
\(66\) −7.65279 −0.941994
\(67\) −7.67994 −0.938254 −0.469127 0.883131i \(-0.655431\pi\)
−0.469127 + 0.883131i \(0.655431\pi\)
\(68\) 2.92931 0.355231
\(69\) 3.35690 0.404123
\(70\) 0 0
\(71\) 8.66487 1.02833 0.514166 0.857691i \(-0.328102\pi\)
0.514166 + 0.857691i \(0.328102\pi\)
\(72\) 5.51573 0.650035
\(73\) 6.73556 0.788338 0.394169 0.919038i \(-0.371032\pi\)
0.394169 + 0.919038i \(0.371032\pi\)
\(74\) 4.56465 0.530629
\(75\) −11.0978 −1.28147
\(76\) 0.119605 0.0137196
\(77\) 0 0
\(78\) 0 0
\(79\) 9.97046 1.12176 0.560882 0.827896i \(-0.310462\pi\)
0.560882 + 0.827896i \(0.310462\pi\)
\(80\) 0.137063 0.0153241
\(81\) −10.9487 −1.21652
\(82\) 9.29590 1.02656
\(83\) 1.60925 0.176638 0.0883192 0.996092i \(-0.471850\pi\)
0.0883192 + 0.996092i \(0.471850\pi\)
\(84\) 0 0
\(85\) −0.533188 −0.0578323
\(86\) 0.237291 0.0255877
\(87\) 10.4058 1.11562
\(88\) 11.4330 1.21876
\(89\) −2.88471 −0.305778 −0.152889 0.988243i \(-0.548858\pi\)
−0.152889 + 0.988243i \(0.548858\pi\)
\(90\) −0.405813 −0.0427765
\(91\) 0 0
\(92\) −2.02715 −0.211345
\(93\) −14.8998 −1.54503
\(94\) 5.89977 0.608515
\(95\) −0.0217703 −0.00223358
\(96\) −13.0978 −1.33679
\(97\) −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
\(100\) 6.70171 0.670171
\(101\) 13.3545 1.32882 0.664411 0.747367i \(-0.268682\pi\)
0.664411 + 0.747367i \(0.268682\pi\)
\(102\) 3.89008 0.385176
\(103\) −1.36227 −0.134229 −0.0671144 0.997745i \(-0.521379\pi\)
−0.0671144 + 0.997745i \(0.521379\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 8.33513 0.809579
\(107\) 3.26875 0.316002 0.158001 0.987439i \(-0.449495\pi\)
0.158001 + 0.987439i \(0.449495\pi\)
\(108\) 2.89977 0.279031
\(109\) −15.7017 −1.50395 −0.751976 0.659191i \(-0.770899\pi\)
−0.751976 + 0.659191i \(0.770899\pi\)
\(110\) −0.841166 −0.0802021
\(111\) −12.7899 −1.21396
\(112\) 0 0
\(113\) 12.0489 1.13347 0.566733 0.823901i \(-0.308207\pi\)
0.566733 + 0.823901i \(0.308207\pi\)
\(114\) 0.158834 0.0148761
\(115\) 0.368977 0.0344073
\(116\) −6.28382 −0.583438
\(117\) 0 0
\(118\) 5.43727 0.500541
\(119\) 0 0
\(120\) 1.49396 0.136379
\(121\) 7.03684 0.639712
\(122\) 2.78448 0.252095
\(123\) −26.0465 −2.34854
\(124\) 8.99761 0.808009
\(125\) −2.45473 −0.219558
\(126\) 0 0
\(127\) −9.80731 −0.870258 −0.435129 0.900368i \(-0.643297\pi\)
−0.435129 + 0.900368i \(0.643297\pi\)
\(128\) 8.79954 0.777777
\(129\) −0.664874 −0.0585389
\(130\) 0 0
\(131\) 6.57673 0.574611 0.287306 0.957839i \(-0.407240\pi\)
0.287306 + 0.957839i \(0.407240\pi\)
\(132\) −12.9487 −1.12704
\(133\) 0 0
\(134\) 6.15883 0.532042
\(135\) −0.527811 −0.0454267
\(136\) −5.81163 −0.498343
\(137\) −6.21983 −0.531396 −0.265698 0.964056i \(-0.585602\pi\)
−0.265698 + 0.964056i \(0.585602\pi\)
\(138\) −2.69202 −0.229160
\(139\) 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
\(142\) −6.94869 −0.583121
\(143\) 0 0
\(144\) 1.13706 0.0947553
\(145\) 1.14377 0.0949848
\(146\) −5.40150 −0.447031
\(147\) 0 0
\(148\) 7.72348 0.634866
\(149\) −4.33513 −0.355147 −0.177574 0.984108i \(-0.556825\pi\)
−0.177574 + 0.984108i \(0.556825\pi\)
\(150\) 8.89977 0.726663
\(151\) −3.94438 −0.320989 −0.160494 0.987037i \(-0.551309\pi\)
−0.160494 + 0.987037i \(0.551309\pi\)
\(152\) −0.237291 −0.0192468
\(153\) −4.42327 −0.357600
\(154\) 0 0
\(155\) −1.63773 −0.131545
\(156\) 0 0
\(157\) −4.45473 −0.355526 −0.177763 0.984073i \(-0.556886\pi\)
−0.177763 + 0.984073i \(0.556886\pi\)
\(158\) −7.99569 −0.636103
\(159\) −23.3545 −1.85213
\(160\) −1.43967 −0.113816
\(161\) 0 0
\(162\) 8.78017 0.689835
\(163\) −16.1588 −1.26566 −0.632829 0.774292i \(-0.718106\pi\)
−0.632829 + 0.774292i \(0.718106\pi\)
\(164\) 15.7289 1.22822
\(165\) 2.35690 0.183484
\(166\) −1.29052 −0.100164
\(167\) 16.1172 1.24719 0.623594 0.781749i \(-0.285672\pi\)
0.623594 + 0.781749i \(0.285672\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0.427583 0.0327942
\(171\) −0.180604 −0.0138111
\(172\) 0.401501 0.0306142
\(173\) 21.5362 1.63736 0.818682 0.574247i \(-0.194705\pi\)
0.818682 + 0.574247i \(0.194705\pi\)
\(174\) −8.34481 −0.632619
\(175\) 0 0
\(176\) 2.35690 0.177658
\(177\) −15.2349 −1.14513
\(178\) 2.31336 0.173393
\(179\) 11.4330 0.854540 0.427270 0.904124i \(-0.359475\pi\)
0.427270 + 0.904124i \(0.359475\pi\)
\(180\) −0.686645 −0.0511795
\(181\) −20.9705 −1.55872 −0.779361 0.626575i \(-0.784456\pi\)
−0.779361 + 0.626575i \(0.784456\pi\)
\(182\) 0 0
\(183\) −7.80194 −0.576736
\(184\) 4.02177 0.296489
\(185\) −1.40581 −0.103357
\(186\) 11.9487 0.876120
\(187\) −9.16852 −0.670469
\(188\) 9.98254 0.728052
\(189\) 0 0
\(190\) 0.0174584 0.00126657
\(191\) −14.4373 −1.04464 −0.522322 0.852748i \(-0.674934\pi\)
−0.522322 + 0.852748i \(0.674934\pi\)
\(192\) 8.00969 0.578049
\(193\) 13.5797 0.977489 0.488745 0.872427i \(-0.337455\pi\)
0.488745 + 0.872427i \(0.337455\pi\)
\(194\) 6.46250 0.463980
\(195\) 0 0
\(196\) 0 0
\(197\) 0.560335 0.0399222 0.0199611 0.999801i \(-0.493646\pi\)
0.0199611 + 0.999801i \(0.493646\pi\)
\(198\) −6.97823 −0.495921
\(199\) −11.4916 −0.814616 −0.407308 0.913291i \(-0.633532\pi\)
−0.407308 + 0.913291i \(0.633532\pi\)
\(200\) −13.2959 −0.940162
\(201\) −17.2567 −1.21719
\(202\) −10.7095 −0.753516
\(203\) 0 0
\(204\) 6.58211 0.460840
\(205\) −2.86294 −0.199956
\(206\) 1.09246 0.0761151
\(207\) 3.06100 0.212754
\(208\) 0 0
\(209\) −0.374354 −0.0258946
\(210\) 0 0
\(211\) 8.78448 0.604748 0.302374 0.953189i \(-0.402221\pi\)
0.302374 + 0.953189i \(0.402221\pi\)
\(212\) 14.1032 0.968613
\(213\) 19.4698 1.33405
\(214\) −2.62133 −0.179191
\(215\) −0.0730805 −0.00498405
\(216\) −5.75302 −0.391443
\(217\) 0 0
\(218\) 12.5918 0.852824
\(219\) 15.1347 1.02271
\(220\) −1.42327 −0.0959570
\(221\) 0 0
\(222\) 10.2567 0.688383
\(223\) −2.25906 −0.151278 −0.0756390 0.997135i \(-0.524100\pi\)
−0.0756390 + 0.997135i \(0.524100\pi\)
\(224\) 0 0
\(225\) −10.1196 −0.674640
\(226\) −9.66248 −0.642739
\(227\) −6.96615 −0.462359 −0.231180 0.972911i \(-0.574259\pi\)
−0.231180 + 0.972911i \(0.574259\pi\)
\(228\) 0.268750 0.0177984
\(229\) −24.1739 −1.59746 −0.798728 0.601692i \(-0.794493\pi\)
−0.798728 + 0.601692i \(0.794493\pi\)
\(230\) −0.295897 −0.0195109
\(231\) 0 0
\(232\) 12.4668 0.818486
\(233\) −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
\(236\) 9.19998 0.598868
\(237\) 22.4034 1.45526
\(238\) 0 0
\(239\) 25.1468 1.62661 0.813304 0.581839i \(-0.197667\pi\)
0.813304 + 0.581839i \(0.197667\pi\)
\(240\) 0.307979 0.0198799
\(241\) −20.2664 −1.30547 −0.652735 0.757586i \(-0.726379\pi\)
−0.652735 + 0.757586i \(0.726379\pi\)
\(242\) −5.64310 −0.362752
\(243\) −18.1903 −1.16691
\(244\) 4.71140 0.301616
\(245\) 0 0
\(246\) 20.8877 1.33175
\(247\) 0 0
\(248\) −17.8509 −1.13353
\(249\) 3.61596 0.229152
\(250\) 1.96854 0.124501
\(251\) 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
\(254\) 7.86486 0.493485
\(255\) −1.19806 −0.0750256
\(256\) −14.1860 −0.886624
\(257\) −14.2241 −0.887278 −0.443639 0.896206i \(-0.646313\pi\)
−0.443639 + 0.896206i \(0.646313\pi\)
\(258\) 0.533188 0.0331948
\(259\) 0 0
\(260\) 0 0
\(261\) 9.48858 0.587329
\(262\) −5.27413 −0.325837
\(263\) −17.0954 −1.05415 −0.527075 0.849819i \(-0.676711\pi\)
−0.527075 + 0.849819i \(0.676711\pi\)
\(264\) 25.6896 1.58109
\(265\) −2.56704 −0.157692
\(266\) 0 0
\(267\) −6.48188 −0.396684
\(268\) 10.4209 0.636556
\(269\) 6.46681 0.394288 0.197144 0.980374i \(-0.436833\pi\)
0.197144 + 0.980374i \(0.436833\pi\)
\(270\) 0.423272 0.0257595
\(271\) 6.44803 0.391690 0.195845 0.980635i \(-0.437255\pi\)
0.195845 + 0.980635i \(0.437255\pi\)
\(272\) −1.19806 −0.0726432
\(273\) 0 0
\(274\) 4.98792 0.301331
\(275\) −20.9758 −1.26489
\(276\) −4.55496 −0.274176
\(277\) 13.4601 0.808739 0.404370 0.914596i \(-0.367491\pi\)
0.404370 + 0.914596i \(0.367491\pi\)
\(278\) −11.7942 −0.707367
\(279\) −13.5864 −0.813398
\(280\) 0 0
\(281\) −5.03684 −0.300472 −0.150236 0.988650i \(-0.548003\pi\)
−0.150236 + 0.988650i \(0.548003\pi\)
\(282\) 13.2567 0.789423
\(283\) −22.1280 −1.31537 −0.657686 0.753293i \(-0.728464\pi\)
−0.657686 + 0.753293i \(0.728464\pi\)
\(284\) −11.7573 −0.697669
\(285\) −0.0489173 −0.00289761
\(286\) 0 0
\(287\) 0 0
\(288\) −11.9433 −0.703766
\(289\) −12.3394 −0.725849
\(290\) −0.917231 −0.0538616
\(291\) −18.1075 −1.06148
\(292\) −9.13946 −0.534846
\(293\) 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
\(296\) −15.3230 −0.890634
\(297\) −9.07606 −0.526647
\(298\) 3.47650 0.201388
\(299\) 0 0
\(300\) 15.0586 0.869409
\(301\) 0 0
\(302\) 3.16315 0.182019
\(303\) 30.0073 1.72387
\(304\) −0.0489173 −0.00280560
\(305\) −0.857560 −0.0491037
\(306\) 3.54719 0.202779
\(307\) 19.1293 1.09177 0.545883 0.837861i \(-0.316194\pi\)
0.545883 + 0.837861i \(0.316194\pi\)
\(308\) 0 0
\(309\) −3.06100 −0.174134
\(310\) 1.31336 0.0745936
\(311\) 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 \(-0.270070\pi\)
0.661146 + 0.750257i \(0.270070\pi\)
\(314\) 3.57242 0.201603
\(315\) 0 0
\(316\) −13.5289 −0.761059
\(317\) 13.9952 0.786050 0.393025 0.919528i \(-0.371429\pi\)
0.393025 + 0.919528i \(0.371429\pi\)
\(318\) 18.7289 1.05026
\(319\) 19.6679 1.10119
\(320\) 0.880395 0.0492156
\(321\) 7.34481 0.409948
\(322\) 0 0
\(323\) 0.190293 0.0105882
\(324\) 14.8562 0.825346
\(325\) 0 0
\(326\) 12.9584 0.717698
\(327\) −35.2814 −1.95107
\(328\) −31.2054 −1.72303
\(329\) 0 0
\(330\) −1.89008 −0.104046
\(331\) −17.8213 −0.979548 −0.489774 0.871849i \(-0.662921\pi\)
−0.489774 + 0.871849i \(0.662921\pi\)
\(332\) −2.18359 −0.119840
\(333\) −11.6625 −0.639100
\(334\) −12.9250 −0.707225
\(335\) −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
\(340\) 0.723480 0.0392362
\(341\) −28.1618 −1.52505
\(342\) 0.144833 0.00783167
\(343\) 0 0
\(344\) −0.796561 −0.0429477
\(345\) 0.829085 0.0446364
\(346\) −17.2707 −0.928477
\(347\) 1.50365 0.0807200 0.0403600 0.999185i \(-0.487150\pi\)
0.0403600 + 0.999185i \(0.487150\pi\)
\(348\) −14.1196 −0.756890
\(349\) −14.1860 −0.759358 −0.379679 0.925118i \(-0.623966\pi\)
−0.379679 + 0.925118i \(0.623966\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −24.7560 −1.31950
\(353\) −7.16852 −0.381542 −0.190771 0.981635i \(-0.561099\pi\)
−0.190771 + 0.981635i \(0.561099\pi\)
\(354\) 12.2174 0.649350
\(355\) 2.14005 0.113582
\(356\) 3.91425 0.207455
\(357\) 0 0
\(358\) −9.16852 −0.484571
\(359\) −19.8853 −1.04951 −0.524753 0.851255i \(-0.675842\pi\)
−0.524753 + 0.851255i \(0.675842\pi\)
\(360\) 1.36227 0.0717981
\(361\) −18.9922 −0.999591
\(362\) 16.8170 0.883882
\(363\) 15.8116 0.829895
\(364\) 0 0
\(365\) 1.66355 0.0870740
\(366\) 6.25667 0.327041
\(367\) −1.08383 −0.0565757 −0.0282878 0.999600i \(-0.509006\pi\)
−0.0282878 + 0.999600i \(0.509006\pi\)
\(368\) 0.829085 0.0432190
\(369\) −23.7506 −1.23641
\(370\) 1.12737 0.0586094
\(371\) 0 0
\(372\) 20.2174 1.04823
\(373\) −6.13036 −0.317418 −0.158709 0.987325i \(-0.550733\pi\)
−0.158709 + 0.987325i \(0.550733\pi\)
\(374\) 7.35258 0.380193
\(375\) −5.51573 −0.284831
\(376\) −19.8049 −1.02136
\(377\) 0 0
\(378\) 0 0
\(379\) 2.40880 0.123732 0.0618658 0.998084i \(-0.480295\pi\)
0.0618658 + 0.998084i \(0.480295\pi\)
\(380\) 0.0295400 0.00151537
\(381\) −22.0368 −1.12898
\(382\) 11.5778 0.592371
\(383\) −30.3913 −1.55292 −0.776462 0.630164i \(-0.782988\pi\)
−0.776462 + 0.630164i \(0.782988\pi\)
\(384\) 19.7724 1.00901
\(385\) 0 0
\(386\) −10.8901 −0.554291
\(387\) −0.606268 −0.0308184
\(388\) 10.9347 0.555125
\(389\) −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
\(394\) −0.449354 −0.0226381
\(395\) 2.46250 0.123902
\(396\) −11.8073 −0.593340
\(397\) 16.9148 0.848931 0.424466 0.905444i \(-0.360462\pi\)
0.424466 + 0.905444i \(0.360462\pi\)
\(398\) 9.21552 0.461932
\(399\) 0 0
\(400\) −2.74094 −0.137047
\(401\) −26.6625 −1.33146 −0.665730 0.746192i \(-0.731880\pi\)
−0.665730 + 0.746192i \(0.731880\pi\)
\(402\) 13.8388 0.690215
\(403\) 0 0
\(404\) −18.1207 −0.901537
\(405\) −2.70410 −0.134368
\(406\) 0 0
\(407\) −24.1739 −1.19826
\(408\) −13.0586 −0.646497
\(409\) 28.5163 1.41004 0.705021 0.709187i \(-0.250938\pi\)
0.705021 + 0.709187i \(0.250938\pi\)
\(410\) 2.29590 0.113386
\(411\) −13.9758 −0.689377
\(412\) 1.84846 0.0910672
\(413\) 0 0
\(414\) −2.45473 −0.120643
\(415\) 0.397452 0.0195102
\(416\) 0 0
\(417\) 33.0465 1.61830
\(418\) 0.300209 0.0146837
\(419\) 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 \(-0.408288\pi\)
0.284151 + 0.958780i \(0.408288\pi\)
\(422\) −7.04461 −0.342926
\(423\) −15.0737 −0.732907
\(424\) −27.9801 −1.35884
\(425\) 10.6625 0.517206
\(426\) −15.6136 −0.756480
\(427\) 0 0
\(428\) −4.43535 −0.214391
\(429\) 0 0
\(430\) 0.0586060 0.00282623
\(431\) −4.34913 −0.209490 −0.104745 0.994499i \(-0.533403\pi\)
−0.104745 + 0.994499i \(0.533403\pi\)
\(432\) −1.18598 −0.0570605
\(433\) 14.3884 0.691460 0.345730 0.938334i \(-0.387631\pi\)
0.345730 + 0.938334i \(0.387631\pi\)
\(434\) 0 0
\(435\) 2.57002 0.123223
\(436\) 21.3056 1.02035
\(437\) −0.131687 −0.00629942
\(438\) −12.1371 −0.579931
\(439\) 20.2325 0.965645 0.482822 0.875718i \(-0.339612\pi\)
0.482822 + 0.875718i \(0.339612\pi\)
\(440\) 2.82371 0.134615
\(441\) 0 0
\(442\) 0 0
\(443\) 8.12200 0.385888 0.192944 0.981210i \(-0.438196\pi\)
0.192944 + 0.981210i \(0.438196\pi\)
\(444\) 17.3545 0.823608
\(445\) −0.712464 −0.0337740
\(446\) 1.81163 0.0857830
\(447\) −9.74094 −0.460731
\(448\) 0 0
\(449\) −12.4916 −0.589513 −0.294757 0.955572i \(-0.595239\pi\)
−0.294757 + 0.955572i \(0.595239\pi\)
\(450\) 8.11529 0.382559
\(451\) −49.2301 −2.31816
\(452\) −16.3491 −0.768998
\(453\) −8.86294 −0.416417
\(454\) 5.58642 0.262184
\(455\) 0 0
\(456\) −0.533188 −0.0249688
\(457\) −5.98121 −0.279789 −0.139895 0.990166i \(-0.544676\pi\)
−0.139895 + 0.990166i \(0.544676\pi\)
\(458\) 19.3860 0.905847
\(459\) 4.61356 0.215343
\(460\) −0.500664 −0.0233436
\(461\) −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 \(0.437095\pi\)
\(464\) 2.57002 0.119310
\(465\) −3.67994 −0.170653
\(466\) 2.45473 0.113713
\(467\) −33.5139 −1.55084 −0.775420 0.631446i \(-0.782462\pi\)
−0.775420 + 0.631446i \(0.782462\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 1.45712 0.0672121
\(471\) −10.0097 −0.461222
\(472\) −18.2524 −0.840133
\(473\) −1.25667 −0.0577817
\(474\) −17.9661 −0.825213
\(475\) 0.435353 0.0199754
\(476\) 0 0
\(477\) −21.2959 −0.975072
\(478\) −20.1661 −0.922377
\(479\) −24.7313 −1.13000 −0.565000 0.825091i \(-0.691124\pi\)
−0.565000 + 0.825091i \(0.691124\pi\)
\(480\) −3.23490 −0.147652
\(481\) 0 0
\(482\) 16.2524 0.740275
\(483\) 0 0
\(484\) −9.54825 −0.434012
\(485\) −1.99031 −0.0903754
\(486\) 14.5875 0.661702
\(487\) 37.7555 1.71087 0.855433 0.517913i \(-0.173291\pi\)
0.855433 + 0.517913i \(0.173291\pi\)
\(488\) −9.34721 −0.423128
\(489\) −36.3086 −1.64193
\(490\) 0 0
\(491\) 31.3110 1.41304 0.706522 0.707691i \(-0.250263\pi\)
0.706522 + 0.707691i \(0.250263\pi\)
\(492\) 35.3424 1.59336
\(493\) −9.99761 −0.450270
\(494\) 0 0
\(495\) 2.14914 0.0965969
\(496\) −3.67994 −0.165234
\(497\) 0 0
\(498\) −2.89977 −0.129942
\(499\) −21.4873 −0.961902 −0.480951 0.876748i \(-0.659708\pi\)
−0.480951 + 0.876748i \(0.659708\pi\)
\(500\) 3.33081 0.148959
\(501\) 36.2150 1.61797
\(502\) −19.0228 −0.849031
\(503\) −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
\(506\) −5.08815 −0.226196
\(507\) 0 0
\(508\) 13.3075 0.590425
\(509\) −17.1075 −0.758278 −0.379139 0.925340i \(-0.623780\pi\)
−0.379139 + 0.925340i \(0.623780\pi\)
\(510\) 0.960771 0.0425437
\(511\) 0 0
\(512\) −6.22282 −0.275012
\(513\) 0.188374 0.00831690
\(514\) 11.4069 0.503136
\(515\) −0.336454 −0.0148259
\(516\) 0.902165 0.0397156
\(517\) −31.2446 −1.37414
\(518\) 0 0
\(519\) 48.3913 2.12414
\(520\) 0 0
\(521\) 19.8465 0.869493 0.434746 0.900553i \(-0.356838\pi\)
0.434746 + 0.900553i \(0.356838\pi\)
\(522\) −7.60925 −0.333048
\(523\) 11.4300 0.499798 0.249899 0.968272i \(-0.419603\pi\)
0.249899 + 0.968272i \(0.419603\pi\)
\(524\) −8.92394 −0.389844
\(525\) 0 0
\(526\) 13.7095 0.597762
\(527\) 14.3153 0.623583
\(528\) 5.29590 0.230474
\(529\) −20.7681 −0.902960
\(530\) 2.05861 0.0894201
\(531\) −13.8920 −0.602862
\(532\) 0 0
\(533\) 0 0
\(534\) 5.19806 0.224942
\(535\) 0.807315 0.0349033
\(536\) −20.6746 −0.893005
\(537\) 25.6896 1.10859
\(538\) −5.18598 −0.223584
\(539\) 0 0
\(540\) 0.716185 0.0308197
\(541\) −16.1884 −0.695993 −0.347996 0.937496i \(-0.613138\pi\)
−0.347996 + 0.937496i \(0.613138\pi\)
\(542\) −5.17092 −0.222110
\(543\) −47.1202 −2.02212
\(544\) 12.5840 0.539536
\(545\) −3.87800 −0.166115
\(546\) 0 0
\(547\) 5.33081 0.227929 0.113965 0.993485i \(-0.463645\pi\)
0.113965 + 0.993485i \(0.463645\pi\)
\(548\) 8.43967 0.360525
\(549\) −7.11423 −0.303628
\(550\) 16.8213 0.717263
\(551\) −0.408206 −0.0173902
\(552\) 9.03684 0.384633
\(553\) 0 0
\(554\) −10.7942 −0.458600
\(555\) −3.15883 −0.134085
\(556\) −19.9560 −0.846322
\(557\) −7.39075 −0.313156 −0.156578 0.987666i \(-0.550046\pi\)
−0.156578 + 0.987666i \(0.550046\pi\)
\(558\) 10.8955 0.461242
\(559\) 0 0
\(560\) 0 0
\(561\) −20.6015 −0.869795
\(562\) 4.03923 0.170385
\(563\) 9.47889 0.399488 0.199744 0.979848i \(-0.435989\pi\)
0.199744 + 0.979848i \(0.435989\pi\)
\(564\) 22.4306 0.944497
\(565\) 2.97584 0.125194
\(566\) 17.7453 0.745889
\(567\) 0 0
\(568\) 23.3260 0.978738
\(569\) −10.1438 −0.425249 −0.212624 0.977134i \(-0.568201\pi\)
−0.212624 + 0.977134i \(0.568201\pi\)
\(570\) 0.0392287 0.00164311
\(571\) −14.0925 −0.589751 −0.294876 0.955536i \(-0.595278\pi\)
−0.294876 + 0.955536i \(0.595278\pi\)
\(572\) 0 0
\(573\) −32.4403 −1.35521
\(574\) 0 0
\(575\) −7.37867 −0.307712
\(576\) 7.30367 0.304319
\(577\) 25.1545 1.04720 0.523598 0.851965i \(-0.324589\pi\)
0.523598 + 0.851965i \(0.324589\pi\)
\(578\) 9.89546 0.411597
\(579\) 30.5133 1.26809
\(580\) −1.55197 −0.0644422
\(581\) 0 0
\(582\) 14.5211 0.601919
\(583\) −44.1420 −1.82817
\(584\) 18.1323 0.750319
\(585\) 0 0
\(586\) −11.9860 −0.495137
\(587\) 43.8353 1.80928 0.904639 0.426180i \(-0.140141\pi\)
0.904639 + 0.426180i \(0.140141\pi\)
\(588\) 0 0
\(589\) 0.584498 0.0240838
\(590\) 1.34290 0.0552861
\(591\) 1.25906 0.0517909
\(592\) −3.15883 −0.129827
\(593\) −24.9965 −1.02648 −0.513242 0.858244i \(-0.671556\pi\)
−0.513242 + 0.858244i \(0.671556\pi\)
\(594\) 7.27844 0.298638
\(595\) 0 0
\(596\) 5.88231 0.240949
\(597\) −25.8213 −1.05680
\(598\) 0 0
\(599\) −6.24027 −0.254971 −0.127485 0.991840i \(-0.540691\pi\)
−0.127485 + 0.991840i \(0.540691\pi\)
\(600\) −29.8756 −1.21967
\(601\) −6.32975 −0.258196 −0.129098 0.991632i \(-0.541208\pi\)
−0.129098 + 0.991632i \(0.541208\pi\)
\(602\) 0 0
\(603\) −15.7356 −0.640802
\(604\) 5.35211 0.217774
\(605\) 1.73795 0.0706579
\(606\) −24.0640 −0.977532
\(607\) 43.6480 1.77162 0.885809 0.464050i \(-0.153604\pi\)
0.885809 + 0.464050i \(0.153604\pi\)
\(608\) 0.513811 0.0208378
\(609\) 0 0
\(610\) 0.687710 0.0278445
\(611\) 0 0
\(612\) 6.00192 0.242613
\(613\) 25.9541 1.04827 0.524137 0.851634i \(-0.324388\pi\)
0.524137 + 0.851634i \(0.324388\pi\)
\(614\) −15.3405 −0.619092
\(615\) −6.43296 −0.259402
\(616\) 0 0
\(617\) −45.9396 −1.84946 −0.924729 0.380626i \(-0.875709\pi\)
−0.924729 + 0.380626i \(0.875709\pi\)
\(618\) 2.45473 0.0987437
\(619\) 6.73556 0.270725 0.135363 0.990796i \(-0.456780\pi\)
0.135363 + 0.990796i \(0.456780\pi\)
\(620\) 2.22223 0.0892467
\(621\) −3.19269 −0.128118
\(622\) −0.216375 −0.00867583
\(623\) 0 0
\(624\) 0 0
\(625\) 24.0887 0.963549
\(626\) −18.7603 −0.749813
\(627\) −0.841166 −0.0335930
\(628\) 6.04461 0.241206
\(629\) 12.2881 0.489960
\(630\) 0 0
\(631\) −45.0998 −1.79539 −0.897696 0.440614i \(-0.854761\pi\)
−0.897696 + 0.440614i \(0.854761\pi\)
\(632\) 26.8407 1.06767
\(633\) 19.7385 0.784537
\(634\) −11.2233 −0.445734
\(635\) −2.42221 −0.0961223
\(636\) 31.6896 1.25658
\(637\) 0 0
\(638\) −15.7724 −0.624435
\(639\) 17.7536 0.702322
\(640\) 2.17331 0.0859075
\(641\) 32.5821 1.28692 0.643458 0.765482i \(-0.277499\pi\)
0.643458 + 0.765482i \(0.277499\pi\)
\(642\) −5.89008 −0.232463
\(643\) 25.5754 1.00860 0.504298 0.863530i \(-0.331751\pi\)
0.504298 + 0.863530i \(0.331751\pi\)
\(644\) 0 0
\(645\) −0.164210 −0.00646578
\(646\) −0.152603 −0.00600408
\(647\) 30.1715 1.18616 0.593082 0.805142i \(-0.297911\pi\)
0.593082 + 0.805142i \(0.297911\pi\)
\(648\) −29.4741 −1.15785
\(649\) −28.7952 −1.13031
\(650\) 0 0
\(651\) 0 0
\(652\) 21.9259 0.858683
\(653\) 36.9028 1.44412 0.722058 0.691832i \(-0.243196\pi\)
0.722058 + 0.691832i \(0.243196\pi\)
\(654\) 28.2935 1.10636
\(655\) 1.62432 0.0634673
\(656\) −6.43296 −0.251165
\(657\) 13.8006 0.538413
\(658\) 0 0
\(659\) 23.6866 0.922701 0.461350 0.887218i \(-0.347365\pi\)
0.461350 + 0.887218i \(0.347365\pi\)
\(660\) −3.19806 −0.124484
\(661\) −31.7590 −1.23528 −0.617641 0.786460i \(-0.711911\pi\)
−0.617641 + 0.786460i \(0.711911\pi\)
\(662\) 14.2916 0.555458
\(663\) 0 0
\(664\) 4.33214 0.168120
\(665\) 0 0
\(666\) 9.35258 0.362405
\(667\) 6.91856 0.267888
\(668\) −21.8694 −0.846152
\(669\) −5.07606 −0.196252
\(670\) 1.52111 0.0587655
\(671\) −14.7463 −0.569275
\(672\) 0 0
\(673\) −7.50232 −0.289193 −0.144597 0.989491i \(-0.546188\pi\)
−0.144597 + 0.989491i \(0.546188\pi\)
\(674\) 22.3327 0.860225
\(675\) 10.5550 0.406261
\(676\) 0 0
\(677\) 35.0315 1.34637 0.673184 0.739475i \(-0.264926\pi\)
0.673184 + 0.739475i \(0.264926\pi\)
\(678\) −21.7114 −0.833821
\(679\) 0 0
\(680\) −1.43535 −0.0550433
\(681\) −15.6528 −0.599816
\(682\) 22.5840 0.864787
\(683\) 24.0834 0.921524 0.460762 0.887524i \(-0.347576\pi\)
0.460762 + 0.887524i \(0.347576\pi\)
\(684\) 0.245061 0.00937013
\(685\) −1.53617 −0.0586941
\(686\) 0 0
\(687\) −54.3183 −2.07237
\(688\) −0.164210 −0.00626046
\(689\) 0 0
\(690\) −0.664874 −0.0253113
\(691\) 2.01447 0.0766342 0.0383171 0.999266i \(-0.487800\pi\)
0.0383171 + 0.999266i \(0.487800\pi\)
\(692\) −29.2223 −1.11087
\(693\) 0 0
\(694\) −1.20583 −0.0457728
\(695\) 3.63235 0.137783
\(696\) 28.0127 1.06182
\(697\) 25.0248 0.947880
\(698\) 11.3763 0.430598
\(699\) −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
\(704\) 15.1390 0.570572
\(705\) −4.08277 −0.153766
\(706\) 5.74871 0.216355
\(707\) 0 0
\(708\) 20.6722 0.776908
\(709\) 20.8060 0.781385 0.390693 0.920521i \(-0.372236\pi\)
0.390693 + 0.920521i \(0.372236\pi\)
\(710\) −1.71618 −0.0644073
\(711\) 20.4286 0.766134
\(712\) −7.76569 −0.291032
\(713\) −9.90648 −0.371000
\(714\) 0 0
\(715\) 0 0
\(716\) −15.5133 −0.579761
\(717\) 56.5042 2.11019
\(718\) 15.9468 0.595128
\(719\) −21.4306 −0.799225 −0.399613 0.916684i \(-0.630855\pi\)
−0.399613 + 0.916684i \(0.630855\pi\)
\(720\) 0.280831 0.0104660
\(721\) 0 0
\(722\) 15.2306 0.566824
\(723\) −45.5381 −1.69358
\(724\) 28.4547 1.05751
\(725\) −22.8726 −0.849468
\(726\) −12.6799 −0.470597
\(727\) −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
\(730\) −1.33406 −0.0493758
\(731\) 0.638792 0.0236266
\(732\) 10.5864 0.391285
\(733\) −43.5424 −1.60828 −0.804138 0.594443i \(-0.797373\pi\)
−0.804138 + 0.594443i \(0.797373\pi\)
\(734\) 0.869167 0.0320816
\(735\) 0 0
\(736\) −8.70841 −0.320996
\(737\) −32.6165 −1.20145
\(738\) 19.0465 0.701112
\(739\) −20.0543 −0.737709 −0.368855 0.929487i \(-0.620250\pi\)
−0.368855 + 0.929487i \(0.620250\pi\)
\(740\) 1.90754 0.0701226
\(741\) 0 0
\(742\) 0 0
\(743\) 33.1685 1.21684 0.608418 0.793617i \(-0.291805\pi\)
0.608418 + 0.793617i \(0.291805\pi\)
\(744\) −40.1105 −1.47052
\(745\) −1.07069 −0.0392270
\(746\) 4.91617 0.179994
\(747\) 3.29722 0.120639
\(748\) 12.4407 0.454878
\(749\) 0 0
\(750\) 4.42327 0.161515
\(751\) 39.2814 1.43340 0.716700 0.697382i \(-0.245652\pi\)
0.716700 + 0.697382i \(0.245652\pi\)
\(752\) −4.08277 −0.148883
\(753\) 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 \(0.821970\pi\)
\(758\) −1.93171 −0.0701627
\(759\) 14.2567 0.517484
\(760\) −0.0586060 −0.00212586
\(761\) −21.8984 −0.793818 −0.396909 0.917858i \(-0.629917\pi\)
−0.396909 + 0.917858i \(0.629917\pi\)
\(762\) 17.6722 0.640195
\(763\) 0 0
\(764\) 19.5899 0.708737
\(765\) −1.09246 −0.0394979
\(766\) 24.3720 0.880595
\(767\) 0 0
\(768\) −31.8756 −1.15021
\(769\) 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
\(772\) −18.4263 −0.663175
\(773\) −30.2416 −1.08771 −0.543857 0.839178i \(-0.683037\pi\)
−0.543857 + 0.839178i \(0.683037\pi\)
\(774\) 0.486189 0.0174757
\(775\) 32.7506 1.17644
\(776\) −21.6939 −0.778767
\(777\) 0 0
\(778\) 12.7836 0.458315
\(779\) 1.02177 0.0366087
\(780\) 0 0
\(781\) 36.7995 1.31679
\(782\) 2.58642 0.0924901
\(783\) −9.89679 −0.353682
\(784\) 0 0
\(785\) −1.10023 −0.0392688
\(786\) −11.8509 −0.422706
\(787\) 28.7023 1.02313 0.511563 0.859246i \(-0.329067\pi\)
0.511563 + 0.859246i \(0.329067\pi\)
\(788\) −0.760316 −0.0270851
\(789\) −38.4131 −1.36754
\(790\) −1.97477 −0.0702592
\(791\) 0 0
\(792\) 23.4252 0.832378
\(793\) 0 0
\(794\) −13.5646 −0.481391
\(795\) −5.76809 −0.204573
\(796\) 15.5929 0.552674
\(797\) 18.5418 0.656785 0.328392 0.944541i \(-0.393493\pi\)
0.328392 + 0.944541i \(0.393493\pi\)
\(798\) 0 0
\(799\) 15.8823 0.561876
\(800\) 28.7899 1.01788
\(801\) −5.91053 −0.208838
\(802\) 21.3817 0.755012
\(803\) 28.6058 1.00948
\(804\) 23.4155 0.825801
\(805\) 0 0
\(806\) 0 0
\(807\) 14.5308 0.511508
\(808\) 35.9506 1.26474
\(809\) −10.0677 −0.353962 −0.176981 0.984214i \(-0.556633\pi\)
−0.176981 + 0.984214i \(0.556633\pi\)
\(810\) 2.16852 0.0761941
\(811\) −10.0285 −0.352147 −0.176074 0.984377i \(-0.556340\pi\)
−0.176074 + 0.984377i \(0.556340\pi\)
\(812\) 0 0
\(813\) 14.4886 0.508137
\(814\) 19.3860 0.679478
\(815\) −3.99090 −0.139795
\(816\) −2.69202 −0.0942396
\(817\) 0.0260821 0.000912498 0
\(818\) −22.8683 −0.799572
\(819\) 0 0
\(820\) 3.88471 0.135660
\(821\) 26.1704 0.913355 0.456677 0.889632i \(-0.349039\pi\)
0.456677 + 0.889632i \(0.349039\pi\)
\(822\) 11.2078 0.390915
\(823\) 1.82238 0.0635242 0.0317621 0.999495i \(-0.489888\pi\)
0.0317621 + 0.999495i \(0.489888\pi\)
\(824\) −3.66727 −0.127755
\(825\) −47.1323 −1.64094
\(826\) 0 0
\(827\) 32.2941 1.12298 0.561488 0.827485i \(-0.310229\pi\)
0.561488 + 0.827485i \(0.310229\pi\)
\(828\) −4.15346 −0.144343
\(829\) −15.1002 −0.524453 −0.262226 0.965006i \(-0.584457\pi\)
−0.262226 + 0.965006i \(0.584457\pi\)
\(830\) −0.318732 −0.0110634
\(831\) 30.2446 1.04917
\(832\) 0 0
\(833\) 0 0
\(834\) −26.5013 −0.917663
\(835\) 3.98062 0.137755
\(836\) 0.507960 0.0175682
\(837\) 14.1709 0.489818
\(838\) −23.7448 −0.820250
\(839\) −32.9965 −1.13917 −0.569584 0.821933i \(-0.692896\pi\)
−0.569584 + 0.821933i \(0.692896\pi\)
\(840\) 0 0
\(841\) −7.55363 −0.260470
\(842\) −9.35105 −0.322258
\(843\) −11.3177 −0.389801
\(844\) −11.9196 −0.410290
\(845\) 0 0
\(846\) 12.0881 0.415599
\(847\) 0 0
\(848\) −5.76809 −0.198077
\(849\) −49.7211 −1.70642
\(850\) −8.55065 −0.293285
\(851\) −8.50365 −0.291501
\(852\) −26.4185 −0.905082
\(853\) −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
\(856\) 8.79954 0.300762
\(857\) −27.3623 −0.934677 −0.467339 0.884078i \(-0.654787\pi\)
−0.467339 + 0.884078i \(0.654787\pi\)
\(858\) 0 0
\(859\) 20.0629 0.684538 0.342269 0.939602i \(-0.388805\pi\)
0.342269 + 0.939602i \(0.388805\pi\)
\(860\) 0.0991626 0.00338142
\(861\) 0 0
\(862\) 3.48773 0.118792
\(863\) 6.14483 0.209173 0.104586 0.994516i \(-0.466648\pi\)
0.104586 + 0.994516i \(0.466648\pi\)
\(864\) 12.4571 0.423800
\(865\) 5.31900 0.180851
\(866\) −11.5386 −0.392096
\(867\) −27.7265 −0.941640
\(868\) 0 0
\(869\) 42.3443 1.43643
\(870\) −2.06100 −0.0698744
\(871\) 0 0
\(872\) −42.2693 −1.43142
\(873\) −16.5114 −0.558827
\(874\) 0.105604 0.00357212
\(875\) 0 0
\(876\) −20.5362 −0.693853
\(877\) 13.5077 0.456123 0.228061 0.973647i \(-0.426761\pi\)
0.228061 + 0.973647i \(0.426761\pi\)
\(878\) −16.2252 −0.547574
\(879\) 33.5840 1.13276
\(880\) 0.582105 0.0196228
\(881\) 5.23431 0.176348 0.0881741 0.996105i \(-0.471897\pi\)
0.0881741 + 0.996105i \(0.471897\pi\)
\(882\) 0 0
\(883\) −4.57301 −0.153894 −0.0769470 0.997035i \(-0.524517\pi\)
−0.0769470 + 0.997035i \(0.524517\pi\)
\(884\) 0 0
\(885\) −3.76271 −0.126482
\(886\) −6.51334 −0.218820
\(887\) 1.64071 0.0550897 0.0275448 0.999621i \(-0.491231\pi\)
0.0275448 + 0.999621i \(0.491231\pi\)
\(888\) −34.4306 −1.15541
\(889\) 0 0
\(890\) 0.571352 0.0191517
\(891\) −46.4989 −1.55777
\(892\) 3.06531 0.102634
\(893\) 0.648481 0.0217006
\(894\) 7.81163 0.261260
\(895\) 2.82371 0.0943861
\(896\) 0 0
\(897\) 0 0
\(898\) 10.0175 0.334287
\(899\) −30.7084 −1.02418
\(900\) 13.7313 0.457708
\(901\) 22.4383 0.747529
\(902\) 39.4795 1.31452
\(903\) 0 0
\(904\) 32.4359 1.07880
\(905\) −5.17928 −0.172165
\(906\) 7.10752 0.236132
\(907\) 8.10215 0.269027 0.134514 0.990912i \(-0.457053\pi\)
0.134514 + 0.990912i \(0.457053\pi\)
\(908\) 9.45234 0.313687
\(909\) 27.3623 0.907549
\(910\) 0 0
\(911\) −9.18119 −0.304187 −0.152093 0.988366i \(-0.548601\pi\)
−0.152093 + 0.988366i \(0.548601\pi\)
\(912\) −0.109916 −0.00363969
\(913\) 6.83446 0.226188
\(914\) 4.79656 0.158656
\(915\) −1.92692 −0.0637020
\(916\) 32.8015 1.08379
\(917\) 0 0
\(918\) −3.69979 −0.122111
\(919\) 27.5036 0.907262 0.453631 0.891190i \(-0.350128\pi\)
0.453631 + 0.891190i \(0.350128\pi\)
\(920\) 0.993295 0.0327480
\(921\) 42.9831 1.41634
\(922\) 1.64933 0.0543180
\(923\) 0 0
\(924\) 0 0
\(925\) 28.1129 0.924346
\(926\) −6.77586 −0.222668
\(927\) −2.79118 −0.0916745
\(928\) −26.9946 −0.886142
\(929\) 24.2131 0.794407 0.397203 0.917731i \(-0.369981\pi\)
0.397203 + 0.917731i \(0.369981\pi\)
\(930\) 2.95108 0.0967698
\(931\) 0 0
\(932\) 4.15346 0.136051
\(933\) 0.606268 0.0198483
\(934\) 26.8761 0.879412
\(935\) −2.26444 −0.0740550
\(936\) 0 0
\(937\) −11.1830 −0.365333 −0.182666 0.983175i \(-0.558473\pi\)
−0.182666 + 0.983175i \(0.558473\pi\)
\(938\) 0 0
\(939\) 52.5652 1.71540
\(940\) 2.46548 0.0804152
\(941\) −15.9638 −0.520404 −0.260202 0.965554i \(-0.583789\pi\)
−0.260202 + 0.965554i \(0.583789\pi\)
\(942\) 8.02715 0.261539
\(943\) −17.3177 −0.563941
\(944\) −3.76271 −0.122466
\(945\) 0 0
\(946\) 1.00777 0.0327654
\(947\) −6.51466 −0.211698 −0.105849 0.994382i \(-0.533756\pi\)
−0.105849 + 0.994382i \(0.533756\pi\)
\(948\) −30.3991 −0.987317
\(949\) 0 0
\(950\) −0.349126 −0.0113271
\(951\) 31.4470 1.01974
\(952\) 0 0
\(953\) 47.6469 1.54344 0.771718 0.635965i \(-0.219398\pi\)
0.771718 + 0.635965i \(0.219398\pi\)
\(954\) 17.0780 0.552920
\(955\) −3.56571 −0.115384
\(956\) −34.1215 −1.10357
\(957\) 44.1933 1.42857
\(958\) 19.8329 0.640773
\(959\) 0 0
\(960\) 1.97823 0.0638471
\(961\) 12.9705 0.418402
\(962\) 0 0
\(963\) 6.69740 0.215821
\(964\) 27.4993 0.885694
\(965\) 3.35391 0.107966
\(966\) 0 0
\(967\) 43.8122 1.40891 0.704453 0.709751i \(-0.251192\pi\)
0.704453 + 0.709751i \(0.251192\pi\)
\(968\) 18.9433 0.608861
\(969\) 0.427583 0.0137360
\(970\) 1.59611 0.0512479
\(971\) −4.29483 −0.137828 −0.0689139 0.997623i \(-0.521953\pi\)
−0.0689139 + 0.997623i \(0.521953\pi\)
\(972\) 24.6823 0.791686
\(973\) 0 0
\(974\) −30.2776 −0.970156
\(975\) 0 0
\(976\) −1.92692 −0.0616792
\(977\) −26.8019 −0.857470 −0.428735 0.903430i \(-0.641041\pi\)
−0.428735 + 0.903430i \(0.641041\pi\)
\(978\) 29.1172 0.931066
\(979\) −12.2513 −0.391553
\(980\) 0 0
\(981\) −32.1715 −1.02716
\(982\) −25.1094 −0.801275
\(983\) 27.2495 0.869124 0.434562 0.900642i \(-0.356903\pi\)
0.434562 + 0.900642i \(0.356903\pi\)
\(984\) −70.1178 −2.23527
\(985\) 0.138391 0.00440951
\(986\) 8.01746 0.255328
\(987\) 0 0
\(988\) 0 0
\(989\) −0.442058 −0.0140566
\(990\) −1.72348 −0.0547758
\(991\) 24.3889 0.774740 0.387370 0.921924i \(-0.373384\pi\)
0.387370 + 0.921924i \(0.373384\pi\)
\(992\) 38.6528 1.22723
\(993\) −40.0441 −1.27076
\(994\) 0 0
\(995\) −2.83818 −0.0899764
\(996\) −4.90648 −0.155468
\(997\) −31.3207 −0.991935 −0.495967 0.868341i \(-0.665186\pi\)
−0.495967 + 0.868341i \(0.665186\pi\)
\(998\) 17.2314 0.545452
\(999\) 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 8281.2.a.bj.1.1 3
7.6 odd 2 169.2.a.c.1.1 yes 3
13.12 even 2 8281.2.a.bf.1.3 3
21.20 even 2 1521.2.a.o.1.3 3
28.27 even 2 2704.2.a.ba.1.3 3
35.34 odd 2 4225.2.a.bb.1.3 3
91.6 even 12 169.2.e.b.23.5 12
91.20 even 12 169.2.e.b.23.2 12
91.34 even 4 169.2.b.b.168.2 6
91.41 even 12 169.2.e.b.147.2 12
91.48 odd 6 169.2.c.b.146.3 6
91.55 odd 6 169.2.c.b.22.3 6
91.62 odd 6 169.2.c.c.22.1 6
91.69 odd 6 169.2.c.c.146.1 6
91.76 even 12 169.2.e.b.147.5 12
91.83 even 4 169.2.b.b.168.5 6
91.90 odd 2 169.2.a.b.1.3 3
273.83 odd 4 1521.2.b.l.1351.2 6
273.125 odd 4 1521.2.b.l.1351.5 6
273.272 even 2 1521.2.a.r.1.1 3
364.83 odd 4 2704.2.f.o.337.6 6
364.307 odd 4 2704.2.f.o.337.5 6
364.363 even 2 2704.2.a.z.1.3 3
455.454 odd 2 4225.2.a.bg.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.3 3 91.90 odd 2
169.2.a.c.1.1 yes 3 7.6 odd 2
169.2.b.b.168.2 6 91.34 even 4
169.2.b.b.168.5 6 91.83 even 4
169.2.c.b.22.3 6 91.55 odd 6
169.2.c.b.146.3 6 91.48 odd 6
169.2.c.c.22.1 6 91.62 odd 6
169.2.c.c.146.1 6 91.69 odd 6
169.2.e.b.23.2 12 91.20 even 12
169.2.e.b.23.5 12 91.6 even 12
169.2.e.b.147.2 12 91.41 even 12
169.2.e.b.147.5 12 91.76 even 12
1521.2.a.o.1.3 3 21.20 even 2
1521.2.a.r.1.1 3 273.272 even 2
1521.2.b.l.1351.2 6 273.83 odd 4
1521.2.b.l.1351.5 6 273.125 odd 4
2704.2.a.z.1.3 3 364.363 even 2
2704.2.a.ba.1.3 3 28.27 even 2
2704.2.f.o.337.5 6 364.307 odd 4
2704.2.f.o.337.6 6 364.83 odd 4
4225.2.a.bb.1.3 3 35.34 odd 2
4225.2.a.bg.1.1 3 455.454 odd 2
8281.2.a.bf.1.3 3 13.12 even 2
8281.2.a.bj.1.1 3 1.1 even 1 trivial