Properties

Label 3328.2.a.m.1.1
Level $3328$
Weight $2$
Character 3328.1
Self dual yes
Analytic conductor $26.574$
Analytic rank $1$
Dimension $2$
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] = [3328,2,Mod(1,3328)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3328, 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("3328.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3328 = 2^{8} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3328.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: \(26.5742137927\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3328.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} -3.46410 q^{5} -1.26795 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} -3.46410 q^{5} -1.26795 q^{7} +1.00000 q^{9} -4.73205 q^{11} +1.00000 q^{13} +6.92820 q^{15} +5.46410 q^{17} -0.732051 q^{19} +2.53590 q^{21} +4.00000 q^{23} +7.00000 q^{25} +4.00000 q^{27} +2.00000 q^{29} +6.73205 q^{31} +9.46410 q^{33} +4.39230 q^{35} +8.92820 q^{37} -2.00000 q^{39} -8.92820 q^{41} -0.535898 q^{43} -3.46410 q^{45} -6.73205 q^{47} -5.39230 q^{49} -10.9282 q^{51} +2.92820 q^{53} +16.3923 q^{55} +1.46410 q^{57} -10.1962 q^{59} +2.92820 q^{61} -1.26795 q^{63} -3.46410 q^{65} -0.732051 q^{67} -8.00000 q^{69} -8.19615 q^{71} +7.46410 q^{73} -14.0000 q^{75} +6.00000 q^{77} -5.46410 q^{79} -11.0000 q^{81} -3.26795 q^{83} -18.9282 q^{85} -4.00000 q^{87} +17.3205 q^{89} -1.26795 q^{91} -13.4641 q^{93} +2.53590 q^{95} +6.39230 q^{97} -4.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} - 6 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} - 6 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} + 4 q^{17} + 2 q^{19} + 12 q^{21} + 8 q^{23} + 14 q^{25} + 8 q^{27} + 4 q^{29} + 10 q^{31} + 12 q^{33} - 12 q^{35} + 4 q^{37} - 4 q^{39} - 4 q^{41} - 8 q^{43} - 10 q^{47} + 10 q^{49} - 8 q^{51} - 8 q^{53} + 12 q^{55} - 4 q^{57} - 10 q^{59} - 8 q^{61} - 6 q^{63} + 2 q^{67} - 16 q^{69} - 6 q^{71} + 8 q^{73} - 28 q^{75} + 12 q^{77} - 4 q^{79} - 22 q^{81} - 10 q^{83} - 24 q^{85} - 8 q^{87} - 6 q^{91} - 20 q^{93} + 12 q^{95} - 8 q^{97} - 6 q^{99}+O(q^{100}) \) 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
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) −3.46410 −1.54919 −0.774597 0.632456i \(-0.782047\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 0 0
\(7\) −1.26795 −0.479240 −0.239620 0.970867i \(-0.577023\pi\)
−0.239620 + 0.970867i \(0.577023\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.73205 −1.42677 −0.713384 0.700774i \(-0.752838\pi\)
−0.713384 + 0.700774i \(0.752838\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 6.92820 1.78885
\(16\) 0 0
\(17\) 5.46410 1.32524 0.662620 0.748956i \(-0.269445\pi\)
0.662620 + 0.748956i \(0.269445\pi\)
\(18\) 0 0
\(19\) −0.732051 −0.167944 −0.0839720 0.996468i \(-0.526761\pi\)
−0.0839720 + 0.996468i \(0.526761\pi\)
\(20\) 0 0
\(21\) 2.53590 0.553378
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 6.73205 1.20911 0.604556 0.796563i \(-0.293351\pi\)
0.604556 + 0.796563i \(0.293351\pi\)
\(32\) 0 0
\(33\) 9.46410 1.64749
\(34\) 0 0
\(35\) 4.39230 0.742435
\(36\) 0 0
\(37\) 8.92820 1.46779 0.733894 0.679264i \(-0.237701\pi\)
0.733894 + 0.679264i \(0.237701\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −8.92820 −1.39435 −0.697176 0.716900i \(-0.745560\pi\)
−0.697176 + 0.716900i \(0.745560\pi\)
\(42\) 0 0
\(43\) −0.535898 −0.0817237 −0.0408619 0.999165i \(-0.513010\pi\)
−0.0408619 + 0.999165i \(0.513010\pi\)
\(44\) 0 0
\(45\) −3.46410 −0.516398
\(46\) 0 0
\(47\) −6.73205 −0.981971 −0.490985 0.871168i \(-0.663363\pi\)
−0.490985 + 0.871168i \(0.663363\pi\)
\(48\) 0 0
\(49\) −5.39230 −0.770329
\(50\) 0 0
\(51\) −10.9282 −1.53025
\(52\) 0 0
\(53\) 2.92820 0.402220 0.201110 0.979569i \(-0.435545\pi\)
0.201110 + 0.979569i \(0.435545\pi\)
\(54\) 0 0
\(55\) 16.3923 2.21034
\(56\) 0 0
\(57\) 1.46410 0.193925
\(58\) 0 0
\(59\) −10.1962 −1.32743 −0.663713 0.747987i \(-0.731020\pi\)
−0.663713 + 0.747987i \(0.731020\pi\)
\(60\) 0 0
\(61\) 2.92820 0.374918 0.187459 0.982272i \(-0.439975\pi\)
0.187459 + 0.982272i \(0.439975\pi\)
\(62\) 0 0
\(63\) −1.26795 −0.159747
\(64\) 0 0
\(65\) −3.46410 −0.429669
\(66\) 0 0
\(67\) −0.732051 −0.0894342 −0.0447171 0.999000i \(-0.514239\pi\)
−0.0447171 + 0.999000i \(0.514239\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) −8.19615 −0.972704 −0.486352 0.873763i \(-0.661673\pi\)
−0.486352 + 0.873763i \(0.661673\pi\)
\(72\) 0 0
\(73\) 7.46410 0.873607 0.436804 0.899557i \(-0.356111\pi\)
0.436804 + 0.899557i \(0.356111\pi\)
\(74\) 0 0
\(75\) −14.0000 −1.61658
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) −5.46410 −0.614759 −0.307380 0.951587i \(-0.599452\pi\)
−0.307380 + 0.951587i \(0.599452\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −3.26795 −0.358704 −0.179352 0.983785i \(-0.557400\pi\)
−0.179352 + 0.983785i \(0.557400\pi\)
\(84\) 0 0
\(85\) −18.9282 −2.05305
\(86\) 0 0
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) 17.3205 1.83597 0.917985 0.396615i \(-0.129815\pi\)
0.917985 + 0.396615i \(0.129815\pi\)
\(90\) 0 0
\(91\) −1.26795 −0.132917
\(92\) 0 0
\(93\) −13.4641 −1.39616
\(94\) 0 0
\(95\) 2.53590 0.260178
\(96\) 0 0
\(97\) 6.39230 0.649040 0.324520 0.945879i \(-0.394797\pi\)
0.324520 + 0.945879i \(0.394797\pi\)
\(98\) 0 0
\(99\) −4.73205 −0.475589
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) 6.92820 0.682656 0.341328 0.939944i \(-0.389123\pi\)
0.341328 + 0.939944i \(0.389123\pi\)
\(104\) 0 0
\(105\) −8.78461 −0.857290
\(106\) 0 0
\(107\) 4.92820 0.476427 0.238214 0.971213i \(-0.423438\pi\)
0.238214 + 0.971213i \(0.423438\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −17.8564 −1.69486
\(112\) 0 0
\(113\) 2.53590 0.238557 0.119279 0.992861i \(-0.461942\pi\)
0.119279 + 0.992861i \(0.461942\pi\)
\(114\) 0 0
\(115\) −13.8564 −1.29212
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −6.92820 −0.635107
\(120\) 0 0
\(121\) 11.3923 1.03566
\(122\) 0 0
\(123\) 17.8564 1.61006
\(124\) 0 0
\(125\) −6.92820 −0.619677
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) 1.07180 0.0943664
\(130\) 0 0
\(131\) −19.8564 −1.73486 −0.867431 0.497557i \(-0.834230\pi\)
−0.867431 + 0.497557i \(0.834230\pi\)
\(132\) 0 0
\(133\) 0.928203 0.0804854
\(134\) 0 0
\(135\) −13.8564 −1.19257
\(136\) 0 0
\(137\) −12.9282 −1.10453 −0.552265 0.833668i \(-0.686237\pi\)
−0.552265 + 0.833668i \(0.686237\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) 13.4641 1.13388
\(142\) 0 0
\(143\) −4.73205 −0.395714
\(144\) 0 0
\(145\) −6.92820 −0.575356
\(146\) 0 0
\(147\) 10.7846 0.889500
\(148\) 0 0
\(149\) 12.9282 1.05912 0.529560 0.848273i \(-0.322357\pi\)
0.529560 + 0.848273i \(0.322357\pi\)
\(150\) 0 0
\(151\) −7.12436 −0.579772 −0.289886 0.957061i \(-0.593617\pi\)
−0.289886 + 0.957061i \(0.593617\pi\)
\(152\) 0 0
\(153\) 5.46410 0.441746
\(154\) 0 0
\(155\) −23.3205 −1.87315
\(156\) 0 0
\(157\) −16.9282 −1.35102 −0.675509 0.737352i \(-0.736076\pi\)
−0.675509 + 0.737352i \(0.736076\pi\)
\(158\) 0 0
\(159\) −5.85641 −0.464443
\(160\) 0 0
\(161\) −5.07180 −0.399714
\(162\) 0 0
\(163\) −16.7321 −1.31056 −0.655278 0.755388i \(-0.727448\pi\)
−0.655278 + 0.755388i \(0.727448\pi\)
\(164\) 0 0
\(165\) −32.7846 −2.55228
\(166\) 0 0
\(167\) −5.66025 −0.438004 −0.219002 0.975724i \(-0.570280\pi\)
−0.219002 + 0.975724i \(0.570280\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −0.732051 −0.0559813
\(172\) 0 0
\(173\) 6.92820 0.526742 0.263371 0.964695i \(-0.415166\pi\)
0.263371 + 0.964695i \(0.415166\pi\)
\(174\) 0 0
\(175\) −8.87564 −0.670936
\(176\) 0 0
\(177\) 20.3923 1.53278
\(178\) 0 0
\(179\) 10.3923 0.776757 0.388379 0.921500i \(-0.373035\pi\)
0.388379 + 0.921500i \(0.373035\pi\)
\(180\) 0 0
\(181\) 8.92820 0.663628 0.331814 0.943345i \(-0.392339\pi\)
0.331814 + 0.943345i \(0.392339\pi\)
\(182\) 0 0
\(183\) −5.85641 −0.432918
\(184\) 0 0
\(185\) −30.9282 −2.27389
\(186\) 0 0
\(187\) −25.8564 −1.89081
\(188\) 0 0
\(189\) −5.07180 −0.368919
\(190\) 0 0
\(191\) 14.5359 1.05178 0.525890 0.850552i \(-0.323732\pi\)
0.525890 + 0.850552i \(0.323732\pi\)
\(192\) 0 0
\(193\) −1.60770 −0.115724 −0.0578622 0.998325i \(-0.518428\pi\)
−0.0578622 + 0.998325i \(0.518428\pi\)
\(194\) 0 0
\(195\) 6.92820 0.496139
\(196\) 0 0
\(197\) 3.07180 0.218856 0.109428 0.993995i \(-0.465098\pi\)
0.109428 + 0.993995i \(0.465098\pi\)
\(198\) 0 0
\(199\) 16.7846 1.18983 0.594915 0.803789i \(-0.297186\pi\)
0.594915 + 0.803789i \(0.297186\pi\)
\(200\) 0 0
\(201\) 1.46410 0.103270
\(202\) 0 0
\(203\) −2.53590 −0.177985
\(204\) 0 0
\(205\) 30.9282 2.16012
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 3.46410 0.239617
\(210\) 0 0
\(211\) 7.85641 0.540857 0.270429 0.962740i \(-0.412835\pi\)
0.270429 + 0.962740i \(0.412835\pi\)
\(212\) 0 0
\(213\) 16.3923 1.12318
\(214\) 0 0
\(215\) 1.85641 0.126606
\(216\) 0 0
\(217\) −8.53590 −0.579455
\(218\) 0 0
\(219\) −14.9282 −1.00875
\(220\) 0 0
\(221\) 5.46410 0.367555
\(222\) 0 0
\(223\) 0.196152 0.0131353 0.00656767 0.999978i \(-0.497909\pi\)
0.00656767 + 0.999978i \(0.497909\pi\)
\(224\) 0 0
\(225\) 7.00000 0.466667
\(226\) 0 0
\(227\) −2.87564 −0.190863 −0.0954316 0.995436i \(-0.530423\pi\)
−0.0954316 + 0.995436i \(0.530423\pi\)
\(228\) 0 0
\(229\) 5.32051 0.351589 0.175795 0.984427i \(-0.443751\pi\)
0.175795 + 0.984427i \(0.443751\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 0 0
\(233\) −16.9282 −1.10900 −0.554502 0.832183i \(-0.687091\pi\)
−0.554502 + 0.832183i \(0.687091\pi\)
\(234\) 0 0
\(235\) 23.3205 1.52126
\(236\) 0 0
\(237\) 10.9282 0.709863
\(238\) 0 0
\(239\) −18.7321 −1.21168 −0.605838 0.795588i \(-0.707162\pi\)
−0.605838 + 0.795588i \(0.707162\pi\)
\(240\) 0 0
\(241\) −30.3923 −1.95774 −0.978870 0.204482i \(-0.934449\pi\)
−0.978870 + 0.204482i \(0.934449\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) 18.6795 1.19339
\(246\) 0 0
\(247\) −0.732051 −0.0465793
\(248\) 0 0
\(249\) 6.53590 0.414196
\(250\) 0 0
\(251\) 6.39230 0.403479 0.201739 0.979439i \(-0.435341\pi\)
0.201739 + 0.979439i \(0.435341\pi\)
\(252\) 0 0
\(253\) −18.9282 −1.19001
\(254\) 0 0
\(255\) 37.8564 2.37066
\(256\) 0 0
\(257\) −23.8564 −1.48812 −0.744061 0.668112i \(-0.767103\pi\)
−0.744061 + 0.668112i \(0.767103\pi\)
\(258\) 0 0
\(259\) −11.3205 −0.703422
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) −27.3205 −1.68465 −0.842327 0.538966i \(-0.818815\pi\)
−0.842327 + 0.538966i \(0.818815\pi\)
\(264\) 0 0
\(265\) −10.1436 −0.623116
\(266\) 0 0
\(267\) −34.6410 −2.12000
\(268\) 0 0
\(269\) −7.85641 −0.479014 −0.239507 0.970895i \(-0.576986\pi\)
−0.239507 + 0.970895i \(0.576986\pi\)
\(270\) 0 0
\(271\) 20.1962 1.22683 0.613414 0.789761i \(-0.289796\pi\)
0.613414 + 0.789761i \(0.289796\pi\)
\(272\) 0 0
\(273\) 2.53590 0.153480
\(274\) 0 0
\(275\) −33.1244 −1.99747
\(276\) 0 0
\(277\) −1.85641 −0.111541 −0.0557703 0.998444i \(-0.517761\pi\)
−0.0557703 + 0.998444i \(0.517761\pi\)
\(278\) 0 0
\(279\) 6.73205 0.403037
\(280\) 0 0
\(281\) −9.32051 −0.556015 −0.278007 0.960579i \(-0.589674\pi\)
−0.278007 + 0.960579i \(0.589674\pi\)
\(282\) 0 0
\(283\) 19.4641 1.15702 0.578510 0.815675i \(-0.303634\pi\)
0.578510 + 0.815675i \(0.303634\pi\)
\(284\) 0 0
\(285\) −5.07180 −0.300427
\(286\) 0 0
\(287\) 11.3205 0.668228
\(288\) 0 0
\(289\) 12.8564 0.756259
\(290\) 0 0
\(291\) −12.7846 −0.749447
\(292\) 0 0
\(293\) −32.9282 −1.92369 −0.961843 0.273602i \(-0.911785\pi\)
−0.961843 + 0.273602i \(0.911785\pi\)
\(294\) 0 0
\(295\) 35.3205 2.05644
\(296\) 0 0
\(297\) −18.9282 −1.09833
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 0.679492 0.0391653
\(302\) 0 0
\(303\) −24.0000 −1.37876
\(304\) 0 0
\(305\) −10.1436 −0.580820
\(306\) 0 0
\(307\) −0.732051 −0.0417803 −0.0208902 0.999782i \(-0.506650\pi\)
−0.0208902 + 0.999782i \(0.506650\pi\)
\(308\) 0 0
\(309\) −13.8564 −0.788263
\(310\) 0 0
\(311\) 1.07180 0.0607760 0.0303880 0.999538i \(-0.490326\pi\)
0.0303880 + 0.999538i \(0.490326\pi\)
\(312\) 0 0
\(313\) −0.392305 −0.0221744 −0.0110872 0.999939i \(-0.503529\pi\)
−0.0110872 + 0.999939i \(0.503529\pi\)
\(314\) 0 0
\(315\) 4.39230 0.247478
\(316\) 0 0
\(317\) 3.46410 0.194563 0.0972817 0.995257i \(-0.468985\pi\)
0.0972817 + 0.995257i \(0.468985\pi\)
\(318\) 0 0
\(319\) −9.46410 −0.529888
\(320\) 0 0
\(321\) −9.85641 −0.550131
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) 7.00000 0.388290
\(326\) 0 0
\(327\) 4.00000 0.221201
\(328\) 0 0
\(329\) 8.53590 0.470599
\(330\) 0 0
\(331\) 17.5167 0.962803 0.481401 0.876500i \(-0.340128\pi\)
0.481401 + 0.876500i \(0.340128\pi\)
\(332\) 0 0
\(333\) 8.92820 0.489263
\(334\) 0 0
\(335\) 2.53590 0.138551
\(336\) 0 0
\(337\) 1.46410 0.0797547 0.0398773 0.999205i \(-0.487303\pi\)
0.0398773 + 0.999205i \(0.487303\pi\)
\(338\) 0 0
\(339\) −5.07180 −0.275462
\(340\) 0 0
\(341\) −31.8564 −1.72512
\(342\) 0 0
\(343\) 15.7128 0.848412
\(344\) 0 0
\(345\) 27.7128 1.49201
\(346\) 0 0
\(347\) −7.07180 −0.379634 −0.189817 0.981819i \(-0.560789\pi\)
−0.189817 + 0.981819i \(0.560789\pi\)
\(348\) 0 0
\(349\) 9.60770 0.514288 0.257144 0.966373i \(-0.417219\pi\)
0.257144 + 0.966373i \(0.417219\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 0 0
\(353\) −11.0718 −0.589292 −0.294646 0.955606i \(-0.595202\pi\)
−0.294646 + 0.955606i \(0.595202\pi\)
\(354\) 0 0
\(355\) 28.3923 1.50691
\(356\) 0 0
\(357\) 13.8564 0.733359
\(358\) 0 0
\(359\) 31.5167 1.66339 0.831693 0.555236i \(-0.187372\pi\)
0.831693 + 0.555236i \(0.187372\pi\)
\(360\) 0 0
\(361\) −18.4641 −0.971795
\(362\) 0 0
\(363\) −22.7846 −1.19588
\(364\) 0 0
\(365\) −25.8564 −1.35339
\(366\) 0 0
\(367\) 22.2487 1.16137 0.580687 0.814127i \(-0.302784\pi\)
0.580687 + 0.814127i \(0.302784\pi\)
\(368\) 0 0
\(369\) −8.92820 −0.464784
\(370\) 0 0
\(371\) −3.71281 −0.192760
\(372\) 0 0
\(373\) 14.7846 0.765518 0.382759 0.923848i \(-0.374974\pi\)
0.382759 + 0.923848i \(0.374974\pi\)
\(374\) 0 0
\(375\) 13.8564 0.715542
\(376\) 0 0
\(377\) 2.00000 0.103005
\(378\) 0 0
\(379\) 14.5885 0.749359 0.374679 0.927154i \(-0.377753\pi\)
0.374679 + 0.927154i \(0.377753\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) 10.3397 0.528336 0.264168 0.964477i \(-0.414903\pi\)
0.264168 + 0.964477i \(0.414903\pi\)
\(384\) 0 0
\(385\) −20.7846 −1.05928
\(386\) 0 0
\(387\) −0.535898 −0.0272412
\(388\) 0 0
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 0 0
\(391\) 21.8564 1.10533
\(392\) 0 0
\(393\) 39.7128 2.00325
\(394\) 0 0
\(395\) 18.9282 0.952381
\(396\) 0 0
\(397\) −4.53590 −0.227650 −0.113825 0.993501i \(-0.536310\pi\)
−0.113825 + 0.993501i \(0.536310\pi\)
\(398\) 0 0
\(399\) −1.85641 −0.0929366
\(400\) 0 0
\(401\) 4.53590 0.226512 0.113256 0.993566i \(-0.463872\pi\)
0.113256 + 0.993566i \(0.463872\pi\)
\(402\) 0 0
\(403\) 6.73205 0.335347
\(404\) 0 0
\(405\) 38.1051 1.89346
\(406\) 0 0
\(407\) −42.2487 −2.09419
\(408\) 0 0
\(409\) −12.9282 −0.639259 −0.319629 0.947543i \(-0.603558\pi\)
−0.319629 + 0.947543i \(0.603558\pi\)
\(410\) 0 0
\(411\) 25.8564 1.27540
\(412\) 0 0
\(413\) 12.9282 0.636155
\(414\) 0 0
\(415\) 11.3205 0.555702
\(416\) 0 0
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) 30.7846 1.50393 0.751963 0.659205i \(-0.229107\pi\)
0.751963 + 0.659205i \(0.229107\pi\)
\(420\) 0 0
\(421\) −24.2487 −1.18181 −0.590905 0.806741i \(-0.701229\pi\)
−0.590905 + 0.806741i \(0.701229\pi\)
\(422\) 0 0
\(423\) −6.73205 −0.327324
\(424\) 0 0
\(425\) 38.2487 1.85534
\(426\) 0 0
\(427\) −3.71281 −0.179676
\(428\) 0 0
\(429\) 9.46410 0.456931
\(430\) 0 0
\(431\) −39.1244 −1.88455 −0.942277 0.334835i \(-0.891320\pi\)
−0.942277 + 0.334835i \(0.891320\pi\)
\(432\) 0 0
\(433\) −6.78461 −0.326048 −0.163024 0.986622i \(-0.552125\pi\)
−0.163024 + 0.986622i \(0.552125\pi\)
\(434\) 0 0
\(435\) 13.8564 0.664364
\(436\) 0 0
\(437\) −2.92820 −0.140075
\(438\) 0 0
\(439\) 2.92820 0.139756 0.0698778 0.997556i \(-0.477739\pi\)
0.0698778 + 0.997556i \(0.477739\pi\)
\(440\) 0 0
\(441\) −5.39230 −0.256776
\(442\) 0 0
\(443\) −30.0000 −1.42534 −0.712672 0.701498i \(-0.752515\pi\)
−0.712672 + 0.701498i \(0.752515\pi\)
\(444\) 0 0
\(445\) −60.0000 −2.84427
\(446\) 0 0
\(447\) −25.8564 −1.22297
\(448\) 0 0
\(449\) 17.6077 0.830959 0.415479 0.909603i \(-0.363614\pi\)
0.415479 + 0.909603i \(0.363614\pi\)
\(450\) 0 0
\(451\) 42.2487 1.98941
\(452\) 0 0
\(453\) 14.2487 0.669463
\(454\) 0 0
\(455\) 4.39230 0.205914
\(456\) 0 0
\(457\) 14.0000 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) 0 0
\(459\) 21.8564 1.02017
\(460\) 0 0
\(461\) 20.9282 0.974724 0.487362 0.873200i \(-0.337959\pi\)
0.487362 + 0.873200i \(0.337959\pi\)
\(462\) 0 0
\(463\) −1.66025 −0.0771585 −0.0385793 0.999256i \(-0.512283\pi\)
−0.0385793 + 0.999256i \(0.512283\pi\)
\(464\) 0 0
\(465\) 46.6410 2.16293
\(466\) 0 0
\(467\) −36.2487 −1.67739 −0.838695 0.544601i \(-0.816681\pi\)
−0.838695 + 0.544601i \(0.816681\pi\)
\(468\) 0 0
\(469\) 0.928203 0.0428604
\(470\) 0 0
\(471\) 33.8564 1.56002
\(472\) 0 0
\(473\) 2.53590 0.116601
\(474\) 0 0
\(475\) −5.12436 −0.235122
\(476\) 0 0
\(477\) 2.92820 0.134073
\(478\) 0 0
\(479\) −21.2679 −0.971757 −0.485879 0.874026i \(-0.661500\pi\)
−0.485879 + 0.874026i \(0.661500\pi\)
\(480\) 0 0
\(481\) 8.92820 0.407091
\(482\) 0 0
\(483\) 10.1436 0.461549
\(484\) 0 0
\(485\) −22.1436 −1.00549
\(486\) 0 0
\(487\) −42.4449 −1.92336 −0.961680 0.274174i \(-0.911596\pi\)
−0.961680 + 0.274174i \(0.911596\pi\)
\(488\) 0 0
\(489\) 33.4641 1.51330
\(490\) 0 0
\(491\) −24.2487 −1.09433 −0.547165 0.837025i \(-0.684293\pi\)
−0.547165 + 0.837025i \(0.684293\pi\)
\(492\) 0 0
\(493\) 10.9282 0.492182
\(494\) 0 0
\(495\) 16.3923 0.736779
\(496\) 0 0
\(497\) 10.3923 0.466159
\(498\) 0 0
\(499\) 7.26795 0.325358 0.162679 0.986679i \(-0.447986\pi\)
0.162679 + 0.986679i \(0.447986\pi\)
\(500\) 0 0
\(501\) 11.3205 0.505763
\(502\) 0 0
\(503\) 30.5359 1.36153 0.680764 0.732503i \(-0.261648\pi\)
0.680764 + 0.732503i \(0.261648\pi\)
\(504\) 0 0
\(505\) −41.5692 −1.84981
\(506\) 0 0
\(507\) −2.00000 −0.0888231
\(508\) 0 0
\(509\) 14.3923 0.637928 0.318964 0.947767i \(-0.396665\pi\)
0.318964 + 0.947767i \(0.396665\pi\)
\(510\) 0 0
\(511\) −9.46410 −0.418667
\(512\) 0 0
\(513\) −2.92820 −0.129283
\(514\) 0 0
\(515\) −24.0000 −1.05757
\(516\) 0 0
\(517\) 31.8564 1.40104
\(518\) 0 0
\(519\) −13.8564 −0.608229
\(520\) 0 0
\(521\) 25.1769 1.10302 0.551510 0.834168i \(-0.314052\pi\)
0.551510 + 0.834168i \(0.314052\pi\)
\(522\) 0 0
\(523\) 14.0000 0.612177 0.306089 0.952003i \(-0.400980\pi\)
0.306089 + 0.952003i \(0.400980\pi\)
\(524\) 0 0
\(525\) 17.7513 0.774730
\(526\) 0 0
\(527\) 36.7846 1.60236
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −10.1962 −0.442475
\(532\) 0 0
\(533\) −8.92820 −0.386723
\(534\) 0 0
\(535\) −17.0718 −0.738078
\(536\) 0 0
\(537\) −20.7846 −0.896922
\(538\) 0 0
\(539\) 25.5167 1.09908
\(540\) 0 0
\(541\) −3.07180 −0.132067 −0.0660334 0.997817i \(-0.521034\pi\)
−0.0660334 + 0.997817i \(0.521034\pi\)
\(542\) 0 0
\(543\) −17.8564 −0.766292
\(544\) 0 0
\(545\) 6.92820 0.296772
\(546\) 0 0
\(547\) 11.8564 0.506943 0.253472 0.967343i \(-0.418428\pi\)
0.253472 + 0.967343i \(0.418428\pi\)
\(548\) 0 0
\(549\) 2.92820 0.124973
\(550\) 0 0
\(551\) −1.46410 −0.0623728
\(552\) 0 0
\(553\) 6.92820 0.294617
\(554\) 0 0
\(555\) 61.8564 2.62566
\(556\) 0 0
\(557\) 34.7846 1.47387 0.736936 0.675963i \(-0.236272\pi\)
0.736936 + 0.675963i \(0.236272\pi\)
\(558\) 0 0
\(559\) −0.535898 −0.0226661
\(560\) 0 0
\(561\) 51.7128 2.18332
\(562\) 0 0
\(563\) 7.46410 0.314574 0.157287 0.987553i \(-0.449725\pi\)
0.157287 + 0.987553i \(0.449725\pi\)
\(564\) 0 0
\(565\) −8.78461 −0.369571
\(566\) 0 0
\(567\) 13.9474 0.585737
\(568\) 0 0
\(569\) 14.0000 0.586911 0.293455 0.955973i \(-0.405195\pi\)
0.293455 + 0.955973i \(0.405195\pi\)
\(570\) 0 0
\(571\) 2.67949 0.112133 0.0560666 0.998427i \(-0.482144\pi\)
0.0560666 + 0.998427i \(0.482144\pi\)
\(572\) 0 0
\(573\) −29.0718 −1.21449
\(574\) 0 0
\(575\) 28.0000 1.16768
\(576\) 0 0
\(577\) −7.07180 −0.294403 −0.147201 0.989107i \(-0.547027\pi\)
−0.147201 + 0.989107i \(0.547027\pi\)
\(578\) 0 0
\(579\) 3.21539 0.133627
\(580\) 0 0
\(581\) 4.14359 0.171905
\(582\) 0 0
\(583\) −13.8564 −0.573874
\(584\) 0 0
\(585\) −3.46410 −0.143223
\(586\) 0 0
\(587\) −4.33975 −0.179120 −0.0895602 0.995981i \(-0.528546\pi\)
−0.0895602 + 0.995981i \(0.528546\pi\)
\(588\) 0 0
\(589\) −4.92820 −0.203063
\(590\) 0 0
\(591\) −6.14359 −0.252714
\(592\) 0 0
\(593\) −7.85641 −0.322624 −0.161312 0.986903i \(-0.551573\pi\)
−0.161312 + 0.986903i \(0.551573\pi\)
\(594\) 0 0
\(595\) 24.0000 0.983904
\(596\) 0 0
\(597\) −33.5692 −1.37390
\(598\) 0 0
\(599\) −45.1769 −1.84588 −0.922939 0.384945i \(-0.874220\pi\)
−0.922939 + 0.384945i \(0.874220\pi\)
\(600\) 0 0
\(601\) 4.39230 0.179166 0.0895829 0.995979i \(-0.471447\pi\)
0.0895829 + 0.995979i \(0.471447\pi\)
\(602\) 0 0
\(603\) −0.732051 −0.0298114
\(604\) 0 0
\(605\) −39.4641 −1.60444
\(606\) 0 0
\(607\) −7.32051 −0.297130 −0.148565 0.988903i \(-0.547465\pi\)
−0.148565 + 0.988903i \(0.547465\pi\)
\(608\) 0 0
\(609\) 5.07180 0.205520
\(610\) 0 0
\(611\) −6.73205 −0.272350
\(612\) 0 0
\(613\) −24.6410 −0.995241 −0.497621 0.867395i \(-0.665793\pi\)
−0.497621 + 0.867395i \(0.665793\pi\)
\(614\) 0 0
\(615\) −61.8564 −2.49429
\(616\) 0 0
\(617\) −41.3205 −1.66350 −0.831751 0.555150i \(-0.812661\pi\)
−0.831751 + 0.555150i \(0.812661\pi\)
\(618\) 0 0
\(619\) −18.5885 −0.747133 −0.373567 0.927603i \(-0.621865\pi\)
−0.373567 + 0.927603i \(0.621865\pi\)
\(620\) 0 0
\(621\) 16.0000 0.642058
\(622\) 0 0
\(623\) −21.9615 −0.879870
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) −6.92820 −0.276686
\(628\) 0 0
\(629\) 48.7846 1.94517
\(630\) 0 0
\(631\) −42.0526 −1.67409 −0.837043 0.547137i \(-0.815718\pi\)
−0.837043 + 0.547137i \(0.815718\pi\)
\(632\) 0 0
\(633\) −15.7128 −0.624528
\(634\) 0 0
\(635\) −13.8564 −0.549875
\(636\) 0 0
\(637\) −5.39230 −0.213651
\(638\) 0 0
\(639\) −8.19615 −0.324235
\(640\) 0 0
\(641\) −22.2487 −0.878771 −0.439386 0.898299i \(-0.644804\pi\)
−0.439386 + 0.898299i \(0.644804\pi\)
\(642\) 0 0
\(643\) 12.7321 0.502103 0.251052 0.967974i \(-0.419224\pi\)
0.251052 + 0.967974i \(0.419224\pi\)
\(644\) 0 0
\(645\) −3.71281 −0.146192
\(646\) 0 0
\(647\) 37.8564 1.48829 0.744144 0.668019i \(-0.232857\pi\)
0.744144 + 0.668019i \(0.232857\pi\)
\(648\) 0 0
\(649\) 48.2487 1.89393
\(650\) 0 0
\(651\) 17.0718 0.669096
\(652\) 0 0
\(653\) −3.07180 −0.120209 −0.0601043 0.998192i \(-0.519143\pi\)
−0.0601043 + 0.998192i \(0.519143\pi\)
\(654\) 0 0
\(655\) 68.7846 2.68764
\(656\) 0 0
\(657\) 7.46410 0.291202
\(658\) 0 0
\(659\) −14.0000 −0.545363 −0.272681 0.962104i \(-0.587910\pi\)
−0.272681 + 0.962104i \(0.587910\pi\)
\(660\) 0 0
\(661\) −17.3205 −0.673690 −0.336845 0.941560i \(-0.609360\pi\)
−0.336845 + 0.941560i \(0.609360\pi\)
\(662\) 0 0
\(663\) −10.9282 −0.424416
\(664\) 0 0
\(665\) −3.21539 −0.124687
\(666\) 0 0
\(667\) 8.00000 0.309761
\(668\) 0 0
\(669\) −0.392305 −0.0151674
\(670\) 0 0
\(671\) −13.8564 −0.534921
\(672\) 0 0
\(673\) 33.1769 1.27888 0.639438 0.768843i \(-0.279167\pi\)
0.639438 + 0.768843i \(0.279167\pi\)
\(674\) 0 0
\(675\) 28.0000 1.07772
\(676\) 0 0
\(677\) −21.0718 −0.809855 −0.404927 0.914349i \(-0.632703\pi\)
−0.404927 + 0.914349i \(0.632703\pi\)
\(678\) 0 0
\(679\) −8.10512 −0.311046
\(680\) 0 0
\(681\) 5.75129 0.220390
\(682\) 0 0
\(683\) −17.8038 −0.681245 −0.340623 0.940200i \(-0.610638\pi\)
−0.340623 + 0.940200i \(0.610638\pi\)
\(684\) 0 0
\(685\) 44.7846 1.71113
\(686\) 0 0
\(687\) −10.6410 −0.405980
\(688\) 0 0
\(689\) 2.92820 0.111556
\(690\) 0 0
\(691\) 32.8372 1.24918 0.624592 0.780951i \(-0.285265\pi\)
0.624592 + 0.780951i \(0.285265\pi\)
\(692\) 0 0
\(693\) 6.00000 0.227921
\(694\) 0 0
\(695\) 34.6410 1.31401
\(696\) 0 0
\(697\) −48.7846 −1.84785
\(698\) 0 0
\(699\) 33.8564 1.28057
\(700\) 0 0
\(701\) −40.6410 −1.53499 −0.767495 0.641055i \(-0.778497\pi\)
−0.767495 + 0.641055i \(0.778497\pi\)
\(702\) 0 0
\(703\) −6.53590 −0.246506
\(704\) 0 0
\(705\) −46.6410 −1.75660
\(706\) 0 0
\(707\) −15.2154 −0.572234
\(708\) 0 0
\(709\) −29.3205 −1.10115 −0.550577 0.834784i \(-0.685592\pi\)
−0.550577 + 0.834784i \(0.685592\pi\)
\(710\) 0 0
\(711\) −5.46410 −0.204920
\(712\) 0 0
\(713\) 26.9282 1.00847
\(714\) 0 0
\(715\) 16.3923 0.613037
\(716\) 0 0
\(717\) 37.4641 1.39912
\(718\) 0 0
\(719\) −30.9282 −1.15343 −0.576714 0.816946i \(-0.695665\pi\)
−0.576714 + 0.816946i \(0.695665\pi\)
\(720\) 0 0
\(721\) −8.78461 −0.327156
\(722\) 0 0
\(723\) 60.7846 2.26060
\(724\) 0 0
\(725\) 14.0000 0.519947
\(726\) 0 0
\(727\) −22.9282 −0.850360 −0.425180 0.905109i \(-0.639789\pi\)
−0.425180 + 0.905109i \(0.639789\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −2.92820 −0.108304
\(732\) 0 0
\(733\) 37.3205 1.37846 0.689232 0.724541i \(-0.257948\pi\)
0.689232 + 0.724541i \(0.257948\pi\)
\(734\) 0 0
\(735\) −37.3590 −1.37801
\(736\) 0 0
\(737\) 3.46410 0.127602
\(738\) 0 0
\(739\) −39.7654 −1.46279 −0.731396 0.681953i \(-0.761131\pi\)
−0.731396 + 0.681953i \(0.761131\pi\)
\(740\) 0 0
\(741\) 1.46410 0.0537851
\(742\) 0 0
\(743\) −32.1962 −1.18116 −0.590581 0.806978i \(-0.701101\pi\)
−0.590581 + 0.806978i \(0.701101\pi\)
\(744\) 0 0
\(745\) −44.7846 −1.64078
\(746\) 0 0
\(747\) −3.26795 −0.119568
\(748\) 0 0
\(749\) −6.24871 −0.228323
\(750\) 0 0
\(751\) −1.07180 −0.0391104 −0.0195552 0.999809i \(-0.506225\pi\)
−0.0195552 + 0.999809i \(0.506225\pi\)
\(752\) 0 0
\(753\) −12.7846 −0.465897
\(754\) 0 0
\(755\) 24.6795 0.898179
\(756\) 0 0
\(757\) −40.7846 −1.48234 −0.741171 0.671316i \(-0.765729\pi\)
−0.741171 + 0.671316i \(0.765729\pi\)
\(758\) 0 0
\(759\) 37.8564 1.37410
\(760\) 0 0
\(761\) −18.7846 −0.680942 −0.340471 0.940255i \(-0.610586\pi\)
−0.340471 + 0.940255i \(0.610586\pi\)
\(762\) 0 0
\(763\) 2.53590 0.0918057
\(764\) 0 0
\(765\) −18.9282 −0.684351
\(766\) 0 0
\(767\) −10.1962 −0.368162
\(768\) 0 0
\(769\) 31.4641 1.13462 0.567312 0.823503i \(-0.307983\pi\)
0.567312 + 0.823503i \(0.307983\pi\)
\(770\) 0 0
\(771\) 47.7128 1.71833
\(772\) 0 0
\(773\) −27.4641 −0.987815 −0.493908 0.869514i \(-0.664432\pi\)
−0.493908 + 0.869514i \(0.664432\pi\)
\(774\) 0 0
\(775\) 47.1244 1.69276
\(776\) 0 0
\(777\) 22.6410 0.812242
\(778\) 0 0
\(779\) 6.53590 0.234173
\(780\) 0 0
\(781\) 38.7846 1.38782
\(782\) 0 0
\(783\) 8.00000 0.285897
\(784\) 0 0
\(785\) 58.6410 2.09299
\(786\) 0 0
\(787\) 29.1244 1.03817 0.519086 0.854722i \(-0.326273\pi\)
0.519086 + 0.854722i \(0.326273\pi\)
\(788\) 0 0
\(789\) 54.6410 1.94527
\(790\) 0 0
\(791\) −3.21539 −0.114326
\(792\) 0 0
\(793\) 2.92820 0.103984
\(794\) 0 0
\(795\) 20.2872 0.719512
\(796\) 0 0
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) −36.7846 −1.30135
\(800\) 0 0
\(801\) 17.3205 0.611990
\(802\) 0 0
\(803\) −35.3205 −1.24643
\(804\) 0 0
\(805\) 17.5692 0.619234
\(806\) 0 0
\(807\) 15.7128 0.553117
\(808\) 0 0
\(809\) 5.46410 0.192108 0.0960538 0.995376i \(-0.469378\pi\)
0.0960538 + 0.995376i \(0.469378\pi\)
\(810\) 0 0
\(811\) 21.4115 0.751861 0.375930 0.926648i \(-0.377323\pi\)
0.375930 + 0.926648i \(0.377323\pi\)
\(812\) 0 0
\(813\) −40.3923 −1.41662
\(814\) 0 0
\(815\) 57.9615 2.03030
\(816\) 0 0
\(817\) 0.392305 0.0137250
\(818\) 0 0
\(819\) −1.26795 −0.0443057
\(820\) 0 0
\(821\) 48.2487 1.68389 0.841946 0.539562i \(-0.181410\pi\)
0.841946 + 0.539562i \(0.181410\pi\)
\(822\) 0 0
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) 0 0
\(825\) 66.2487 2.30648
\(826\) 0 0
\(827\) 8.73205 0.303643 0.151822 0.988408i \(-0.451486\pi\)
0.151822 + 0.988408i \(0.451486\pi\)
\(828\) 0 0
\(829\) −28.7846 −0.999731 −0.499865 0.866103i \(-0.666617\pi\)
−0.499865 + 0.866103i \(0.666617\pi\)
\(830\) 0 0
\(831\) 3.71281 0.128796
\(832\) 0 0
\(833\) −29.4641 −1.02087
\(834\) 0 0
\(835\) 19.6077 0.678552
\(836\) 0 0
\(837\) 26.9282 0.930775
\(838\) 0 0
\(839\) −20.9808 −0.724336 −0.362168 0.932113i \(-0.617963\pi\)
−0.362168 + 0.932113i \(0.617963\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 18.6410 0.642031
\(844\) 0 0
\(845\) −3.46410 −0.119169
\(846\) 0 0
\(847\) −14.4449 −0.496331
\(848\) 0 0
\(849\) −38.9282 −1.33601
\(850\) 0 0
\(851\) 35.7128 1.22422
\(852\) 0 0
\(853\) 55.1769 1.88922 0.944611 0.328193i \(-0.106440\pi\)
0.944611 + 0.328193i \(0.106440\pi\)
\(854\) 0 0
\(855\) 2.53590 0.0867259
\(856\) 0 0
\(857\) 7.85641 0.268370 0.134185 0.990956i \(-0.457158\pi\)
0.134185 + 0.990956i \(0.457158\pi\)
\(858\) 0 0
\(859\) 18.0000 0.614152 0.307076 0.951685i \(-0.400649\pi\)
0.307076 + 0.951685i \(0.400649\pi\)
\(860\) 0 0
\(861\) −22.6410 −0.771604
\(862\) 0 0
\(863\) −1.26795 −0.0431615 −0.0215807 0.999767i \(-0.506870\pi\)
−0.0215807 + 0.999767i \(0.506870\pi\)
\(864\) 0 0
\(865\) −24.0000 −0.816024
\(866\) 0 0
\(867\) −25.7128 −0.873253
\(868\) 0 0
\(869\) 25.8564 0.877119
\(870\) 0 0
\(871\) −0.732051 −0.0248046
\(872\) 0 0
\(873\) 6.39230 0.216347
\(874\) 0 0
\(875\) 8.78461 0.296974
\(876\) 0 0
\(877\) −23.4641 −0.792326 −0.396163 0.918180i \(-0.629659\pi\)
−0.396163 + 0.918180i \(0.629659\pi\)
\(878\) 0 0
\(879\) 65.8564 2.22128
\(880\) 0 0
\(881\) −33.7128 −1.13581 −0.567907 0.823093i \(-0.692247\pi\)
−0.567907 + 0.823093i \(0.692247\pi\)
\(882\) 0 0
\(883\) −31.4641 −1.05885 −0.529426 0.848356i \(-0.677593\pi\)
−0.529426 + 0.848356i \(0.677593\pi\)
\(884\) 0 0
\(885\) −70.6410 −2.37457
\(886\) 0 0
\(887\) 30.2487 1.01565 0.507826 0.861460i \(-0.330449\pi\)
0.507826 + 0.861460i \(0.330449\pi\)
\(888\) 0 0
\(889\) −5.07180 −0.170103
\(890\) 0 0
\(891\) 52.0526 1.74383
\(892\) 0 0
\(893\) 4.92820 0.164916
\(894\) 0 0
\(895\) −36.0000 −1.20335
\(896\) 0 0
\(897\) −8.00000 −0.267112
\(898\) 0 0
\(899\) 13.4641 0.449053
\(900\) 0 0
\(901\) 16.0000 0.533037
\(902\) 0 0
\(903\) −1.35898 −0.0452242
\(904\) 0 0
\(905\) −30.9282 −1.02809
\(906\) 0 0
\(907\) 22.1051 0.733988 0.366994 0.930223i \(-0.380387\pi\)
0.366994 + 0.930223i \(0.380387\pi\)
\(908\) 0 0
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) −3.32051 −0.110013 −0.0550067 0.998486i \(-0.517518\pi\)
−0.0550067 + 0.998486i \(0.517518\pi\)
\(912\) 0 0
\(913\) 15.4641 0.511787
\(914\) 0 0
\(915\) 20.2872 0.670674
\(916\) 0 0
\(917\) 25.1769 0.831415
\(918\) 0 0
\(919\) −35.3205 −1.16512 −0.582558 0.812789i \(-0.697948\pi\)
−0.582558 + 0.812789i \(0.697948\pi\)
\(920\) 0 0
\(921\) 1.46410 0.0482438
\(922\) 0 0
\(923\) −8.19615 −0.269780
\(924\) 0 0
\(925\) 62.4974 2.05490
\(926\) 0 0
\(927\) 6.92820 0.227552
\(928\) 0 0
\(929\) 20.5359 0.673761 0.336880 0.941547i \(-0.390628\pi\)
0.336880 + 0.941547i \(0.390628\pi\)
\(930\) 0 0
\(931\) 3.94744 0.129372
\(932\) 0 0
\(933\) −2.14359 −0.0701781
\(934\) 0 0
\(935\) 89.5692 2.92923
\(936\) 0 0
\(937\) 13.7128 0.447978 0.223989 0.974592i \(-0.428092\pi\)
0.223989 + 0.974592i \(0.428092\pi\)
\(938\) 0 0
\(939\) 0.784610 0.0256048
\(940\) 0 0
\(941\) 32.2487 1.05128 0.525639 0.850708i \(-0.323826\pi\)
0.525639 + 0.850708i \(0.323826\pi\)
\(942\) 0 0
\(943\) −35.7128 −1.16297
\(944\) 0 0
\(945\) 17.5692 0.571527
\(946\) 0 0
\(947\) 48.4449 1.57425 0.787123 0.616796i \(-0.211570\pi\)
0.787123 + 0.616796i \(0.211570\pi\)
\(948\) 0 0
\(949\) 7.46410 0.242295
\(950\) 0 0
\(951\) −6.92820 −0.224662
\(952\) 0 0
\(953\) 31.8564 1.03193 0.515965 0.856610i \(-0.327433\pi\)
0.515965 + 0.856610i \(0.327433\pi\)
\(954\) 0 0
\(955\) −50.3538 −1.62941
\(956\) 0 0
\(957\) 18.9282 0.611862
\(958\) 0 0
\(959\) 16.3923 0.529335
\(960\) 0 0
\(961\) 14.3205 0.461952
\(962\) 0 0
\(963\) 4.92820 0.158809
\(964\) 0 0
\(965\) 5.56922 0.179280
\(966\) 0 0
\(967\) −8.98076 −0.288802 −0.144401 0.989519i \(-0.546125\pi\)
−0.144401 + 0.989519i \(0.546125\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) −16.1436 −0.518073 −0.259036 0.965868i \(-0.583405\pi\)
−0.259036 + 0.965868i \(0.583405\pi\)
\(972\) 0 0
\(973\) 12.6795 0.406486
\(974\) 0 0
\(975\) −14.0000 −0.448359
\(976\) 0 0
\(977\) −58.3923 −1.86814 −0.934068 0.357096i \(-0.883767\pi\)
−0.934068 + 0.357096i \(0.883767\pi\)
\(978\) 0 0
\(979\) −81.9615 −2.61950
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 0 0
\(983\) 16.1962 0.516577 0.258289 0.966068i \(-0.416841\pi\)
0.258289 + 0.966068i \(0.416841\pi\)
\(984\) 0 0
\(985\) −10.6410 −0.339051
\(986\) 0 0
\(987\) −17.0718 −0.543401
\(988\) 0 0
\(989\) −2.14359 −0.0681623
\(990\) 0 0
\(991\) 8.67949 0.275713 0.137857 0.990452i \(-0.455979\pi\)
0.137857 + 0.990452i \(0.455979\pi\)
\(992\) 0 0
\(993\) −35.0333 −1.11175
\(994\) 0 0
\(995\) −58.1436 −1.84328
\(996\) 0 0
\(997\) −33.5692 −1.06315 −0.531574 0.847012i \(-0.678399\pi\)
−0.531574 + 0.847012i \(0.678399\pi\)
\(998\) 0 0
\(999\) 35.7128 1.12990
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 3328.2.a.m.1.1 2
4.3 odd 2 3328.2.a.bd.1.1 2
8.3 odd 2 3328.2.a.n.1.2 2
8.5 even 2 3328.2.a.bc.1.2 2
16.3 odd 4 104.2.b.b.53.1 4
16.5 even 4 416.2.b.b.209.3 4
16.11 odd 4 104.2.b.b.53.2 yes 4
16.13 even 4 416.2.b.b.209.2 4
48.5 odd 4 3744.2.g.b.1873.3 4
48.11 even 4 936.2.g.b.469.3 4
48.29 odd 4 3744.2.g.b.1873.1 4
48.35 even 4 936.2.g.b.469.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
104.2.b.b.53.1 4 16.3 odd 4
104.2.b.b.53.2 yes 4 16.11 odd 4
416.2.b.b.209.2 4 16.13 even 4
416.2.b.b.209.3 4 16.5 even 4
936.2.g.b.469.3 4 48.11 even 4
936.2.g.b.469.4 4 48.35 even 4
3328.2.a.m.1.1 2 1.1 even 1 trivial
3328.2.a.n.1.2 2 8.3 odd 2
3328.2.a.bc.1.2 2 8.5 even 2
3328.2.a.bd.1.1 2 4.3 odd 2
3744.2.g.b.1873.1 4 48.29 odd 4
3744.2.g.b.1873.3 4 48.5 odd 4