Properties

Label 416.2.b.b.209.2
Level $416$
Weight $2$
Character 416.209
Analytic conductor $3.322$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,2,Mod(209,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(3.32177672409\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 209.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 416.209
Dual form 416.2.b.b.209.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{3} +3.46410i q^{5} +1.26795 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-2.00000i q^{3} +3.46410i q^{5} +1.26795 q^{7} -1.00000 q^{9} +4.73205i q^{11} +1.00000i q^{13} +6.92820 q^{15} +5.46410 q^{17} -0.732051i q^{19} -2.53590i q^{21} -4.00000 q^{23} -7.00000 q^{25} -4.00000i q^{27} +2.00000i q^{29} +6.73205 q^{31} +9.46410 q^{33} +4.39230i q^{35} -8.92820i q^{37} +2.00000 q^{39} +8.92820 q^{41} +0.535898i q^{43} -3.46410i q^{45} -6.73205 q^{47} -5.39230 q^{49} -10.9282i q^{51} -2.92820i q^{53} -16.3923 q^{55} -1.46410 q^{57} +10.1962i q^{59} +2.92820i q^{61} -1.26795 q^{63} -3.46410 q^{65} -0.732051i q^{67} +8.00000i q^{69} +8.19615 q^{71} -7.46410 q^{73} +14.0000i q^{75} +6.00000i q^{77} -5.46410 q^{79} -11.0000 q^{81} -3.26795i q^{83} +18.9282i q^{85} +4.00000 q^{87} -17.3205 q^{89} +1.26795i q^{91} -13.4641i q^{93} +2.53590 q^{95} +6.39230 q^{97} -4.73205i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{7} - 4 q^{9} + 8 q^{17} - 16 q^{23} - 28 q^{25} + 20 q^{31} + 24 q^{33} + 8 q^{39} + 8 q^{41} - 20 q^{47} + 20 q^{49} - 24 q^{55} + 8 q^{57} - 12 q^{63} + 12 q^{71} - 16 q^{73} - 8 q^{79} - 44 q^{81} + 16 q^{87} + 24 q^{95} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) − 2.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) 0 0
\(5\) 3.46410i 1.54919i 0.632456 + 0.774597i \(0.282047\pi\)
−0.632456 + 0.774597i \(0.717953\pi\)
\(6\) 0 0
\(7\) 1.26795 0.479240 0.239620 0.970867i \(-0.422977\pi\)
0.239620 + 0.970867i \(0.422977\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.73205i 1.42677i 0.700774 + 0.713384i \(0.252838\pi\)
−0.700774 + 0.713384i \(0.747162\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 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
\(19\) − 0.732051i − 0.167944i −0.996468 0.0839720i \(-0.973239\pi\)
0.996468 0.0839720i \(-0.0267606\pi\)
\(20\) 0 0
\(21\) − 2.53590i − 0.553378i
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) − 4.00000i − 0.769800i
\(28\) 0 0
\(29\) 2.00000i 0.371391i 0.982607 + 0.185695i \(0.0594537\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 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
\(35\) 4.39230i 0.742435i
\(36\) 0 0
\(37\) − 8.92820i − 1.46779i −0.679264 0.733894i \(-0.737701\pi\)
0.679264 0.733894i \(-0.262299\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 8.92820 1.39435 0.697176 0.716900i \(-0.254440\pi\)
0.697176 + 0.716900i \(0.254440\pi\)
\(42\) 0 0
\(43\) 0.535898i 0.0817237i 0.999165 + 0.0408619i \(0.0130104\pi\)
−0.999165 + 0.0408619i \(0.986990\pi\)
\(44\) 0 0
\(45\) − 3.46410i − 0.516398i
\(46\) 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
\(51\) − 10.9282i − 1.53025i
\(52\) 0 0
\(53\) − 2.92820i − 0.402220i −0.979569 0.201110i \(-0.935545\pi\)
0.979569 0.201110i \(-0.0644548\pi\)
\(54\) 0 0
\(55\) −16.3923 −2.21034
\(56\) 0 0
\(57\) −1.46410 −0.193925
\(58\) 0 0
\(59\) 10.1962i 1.32743i 0.747987 + 0.663713i \(0.231020\pi\)
−0.747987 + 0.663713i \(0.768980\pi\)
\(60\) 0 0
\(61\) 2.92820i 0.374918i 0.982272 + 0.187459i \(0.0600252\pi\)
−0.982272 + 0.187459i \(0.939975\pi\)
\(62\) 0 0
\(63\) −1.26795 −0.159747
\(64\) 0 0
\(65\) −3.46410 −0.429669
\(66\) 0 0
\(67\) − 0.732051i − 0.0894342i −0.999000 0.0447171i \(-0.985761\pi\)
0.999000 0.0447171i \(-0.0142386\pi\)
\(68\) 0 0
\(69\) 8.00000i 0.963087i
\(70\) 0 0
\(71\) 8.19615 0.972704 0.486352 0.873763i \(-0.338327\pi\)
0.486352 + 0.873763i \(0.338327\pi\)
\(72\) 0 0
\(73\) −7.46410 −0.873607 −0.436804 0.899557i \(-0.643889\pi\)
−0.436804 + 0.899557i \(0.643889\pi\)
\(74\) 0 0
\(75\) 14.0000i 1.61658i
\(76\) 0 0
\(77\) 6.00000i 0.683763i
\(78\) 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
\(83\) − 3.26795i − 0.358704i −0.983785 0.179352i \(-0.942600\pi\)
0.983785 0.179352i \(-0.0574001\pi\)
\(84\) 0 0
\(85\) 18.9282i 2.05305i
\(86\) 0 0
\(87\) 4.00000 0.428845
\(88\) 0 0
\(89\) −17.3205 −1.83597 −0.917985 0.396615i \(-0.870185\pi\)
−0.917985 + 0.396615i \(0.870185\pi\)
\(90\) 0 0
\(91\) 1.26795i 0.132917i
\(92\) 0 0
\(93\) − 13.4641i − 1.39616i
\(94\) 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
\(99\) − 4.73205i − 0.475589i
\(100\) 0 0
\(101\) − 12.0000i − 1.19404i −0.802225 0.597022i \(-0.796350\pi\)
0.802225 0.597022i \(-0.203650\pi\)
\(102\) 0 0
\(103\) −6.92820 −0.682656 −0.341328 0.939944i \(-0.610877\pi\)
−0.341328 + 0.939944i \(0.610877\pi\)
\(104\) 0 0
\(105\) 8.78461 0.857290
\(106\) 0 0
\(107\) − 4.92820i − 0.476427i −0.971213 0.238214i \(-0.923438\pi\)
0.971213 0.238214i \(-0.0765619\pi\)
\(108\) 0 0
\(109\) − 2.00000i − 0.191565i −0.995402 0.0957826i \(-0.969465\pi\)
0.995402 0.0957826i \(-0.0305354\pi\)
\(110\) 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
\(115\) − 13.8564i − 1.29212i
\(116\) 0 0
\(117\) − 1.00000i − 0.0924500i
\(118\) 0 0
\(119\) 6.92820 0.635107
\(120\) 0 0
\(121\) −11.3923 −1.03566
\(122\) 0 0
\(123\) − 17.8564i − 1.61006i
\(124\) 0 0
\(125\) − 6.92820i − 0.619677i
\(126\) 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
\(131\) − 19.8564i − 1.73486i −0.497557 0.867431i \(-0.665770\pi\)
0.497557 0.867431i \(-0.334230\pi\)
\(132\) 0 0
\(133\) − 0.928203i − 0.0804854i
\(134\) 0 0
\(135\) 13.8564 1.19257
\(136\) 0 0
\(137\) 12.9282 1.10453 0.552265 0.833668i \(-0.313763\pi\)
0.552265 + 0.833668i \(0.313763\pi\)
\(138\) 0 0
\(139\) 10.0000i 0.848189i 0.905618 + 0.424094i \(0.139408\pi\)
−0.905618 + 0.424094i \(0.860592\pi\)
\(140\) 0 0
\(141\) 13.4641i 1.13388i
\(142\) 0 0
\(143\) −4.73205 −0.395714
\(144\) 0 0
\(145\) −6.92820 −0.575356
\(146\) 0 0
\(147\) 10.7846i 0.889500i
\(148\) 0 0
\(149\) − 12.9282i − 1.05912i −0.848273 0.529560i \(-0.822357\pi\)
0.848273 0.529560i \(-0.177643\pi\)
\(150\) 0 0
\(151\) 7.12436 0.579772 0.289886 0.957061i \(-0.406383\pi\)
0.289886 + 0.957061i \(0.406383\pi\)
\(152\) 0 0
\(153\) −5.46410 −0.441746
\(154\) 0 0
\(155\) 23.3205i 1.87315i
\(156\) 0 0
\(157\) − 16.9282i − 1.35102i −0.737352 0.675509i \(-0.763924\pi\)
0.737352 0.675509i \(-0.236076\pi\)
\(158\) 0 0
\(159\) −5.85641 −0.464443
\(160\) 0 0
\(161\) −5.07180 −0.399714
\(162\) 0 0
\(163\) − 16.7321i − 1.31056i −0.755388 0.655278i \(-0.772552\pi\)
0.755388 0.655278i \(-0.227448\pi\)
\(164\) 0 0
\(165\) 32.7846i 2.55228i
\(166\) 0 0
\(167\) 5.66025 0.438004 0.219002 0.975724i \(-0.429720\pi\)
0.219002 + 0.975724i \(0.429720\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0.732051i 0.0559813i
\(172\) 0 0
\(173\) 6.92820i 0.526742i 0.964695 + 0.263371i \(0.0848343\pi\)
−0.964695 + 0.263371i \(0.915166\pi\)
\(174\) 0 0
\(175\) −8.87564 −0.670936
\(176\) 0 0
\(177\) 20.3923 1.53278
\(178\) 0 0
\(179\) 10.3923i 0.776757i 0.921500 + 0.388379i \(0.126965\pi\)
−0.921500 + 0.388379i \(0.873035\pi\)
\(180\) 0 0
\(181\) − 8.92820i − 0.663628i −0.943345 0.331814i \(-0.892339\pi\)
0.943345 0.331814i \(-0.107661\pi\)
\(182\) 0 0
\(183\) 5.85641 0.432918
\(184\) 0 0
\(185\) 30.9282 2.27389
\(186\) 0 0
\(187\) 25.8564i 1.89081i
\(188\) 0 0
\(189\) − 5.07180i − 0.368919i
\(190\) 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
\(195\) 6.92820i 0.496139i
\(196\) 0 0
\(197\) − 3.07180i − 0.218856i −0.993995 0.109428i \(-0.965098\pi\)
0.993995 0.109428i \(-0.0349020\pi\)
\(198\) 0 0
\(199\) −16.7846 −1.18983 −0.594915 0.803789i \(-0.702814\pi\)
−0.594915 + 0.803789i \(0.702814\pi\)
\(200\) 0 0
\(201\) −1.46410 −0.103270
\(202\) 0 0
\(203\) 2.53590i 0.177985i
\(204\) 0 0
\(205\) 30.9282i 2.16012i
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 3.46410 0.239617
\(210\) 0 0
\(211\) 7.85641i 0.540857i 0.962740 + 0.270429i \(0.0871654\pi\)
−0.962740 + 0.270429i \(0.912835\pi\)
\(212\) 0 0
\(213\) − 16.3923i − 1.12318i
\(214\) 0 0
\(215\) −1.85641 −0.126606
\(216\) 0 0
\(217\) 8.53590 0.579455
\(218\) 0 0
\(219\) 14.9282i 1.00875i
\(220\) 0 0
\(221\) 5.46410i 0.367555i
\(222\) 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
\(227\) − 2.87564i − 0.190863i −0.995436 0.0954316i \(-0.969577\pi\)
0.995436 0.0954316i \(-0.0304231\pi\)
\(228\) 0 0
\(229\) − 5.32051i − 0.351589i −0.984427 0.175795i \(-0.943751\pi\)
0.984427 0.175795i \(-0.0562494\pi\)
\(230\) 0 0
\(231\) 12.0000 0.789542
\(232\) 0 0
\(233\) 16.9282 1.10900 0.554502 0.832183i \(-0.312909\pi\)
0.554502 + 0.832183i \(0.312909\pi\)
\(234\) 0 0
\(235\) − 23.3205i − 1.52126i
\(236\) 0 0
\(237\) 10.9282i 0.709863i
\(238\) 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
\(243\) 10.0000i 0.641500i
\(244\) 0 0
\(245\) − 18.6795i − 1.19339i
\(246\) 0 0
\(247\) 0.732051 0.0465793
\(248\) 0 0
\(249\) −6.53590 −0.414196
\(250\) 0 0
\(251\) − 6.39230i − 0.403479i −0.979439 0.201739i \(-0.935341\pi\)
0.979439 0.201739i \(-0.0646594\pi\)
\(252\) 0 0
\(253\) − 18.9282i − 1.19001i
\(254\) 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
\(259\) − 11.3205i − 0.703422i
\(260\) 0 0
\(261\) − 2.00000i − 0.123797i
\(262\) 0 0
\(263\) 27.3205 1.68465 0.842327 0.538966i \(-0.181185\pi\)
0.842327 + 0.538966i \(0.181185\pi\)
\(264\) 0 0
\(265\) 10.1436 0.623116
\(266\) 0 0
\(267\) 34.6410i 2.12000i
\(268\) 0 0
\(269\) − 7.85641i − 0.479014i −0.970895 0.239507i \(-0.923014\pi\)
0.970895 0.239507i \(-0.0769857\pi\)
\(270\) 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
\(275\) − 33.1244i − 1.99747i
\(276\) 0 0
\(277\) 1.85641i 0.111541i 0.998444 + 0.0557703i \(0.0177615\pi\)
−0.998444 + 0.0557703i \(0.982239\pi\)
\(278\) 0 0
\(279\) −6.73205 −0.403037
\(280\) 0 0
\(281\) 9.32051 0.556015 0.278007 0.960579i \(-0.410326\pi\)
0.278007 + 0.960579i \(0.410326\pi\)
\(282\) 0 0
\(283\) − 19.4641i − 1.15702i −0.815675 0.578510i \(-0.803634\pi\)
0.815675 0.578510i \(-0.196366\pi\)
\(284\) 0 0
\(285\) − 5.07180i − 0.300427i
\(286\) 0 0
\(287\) 11.3205 0.668228
\(288\) 0 0
\(289\) 12.8564 0.756259
\(290\) 0 0
\(291\) − 12.7846i − 0.749447i
\(292\) 0 0
\(293\) 32.9282i 1.92369i 0.273602 + 0.961843i \(0.411785\pi\)
−0.273602 + 0.961843i \(0.588215\pi\)
\(294\) 0 0
\(295\) −35.3205 −2.05644
\(296\) 0 0
\(297\) 18.9282 1.09833
\(298\) 0 0
\(299\) − 4.00000i − 0.231326i
\(300\) 0 0
\(301\) 0.679492i 0.0391653i
\(302\) 0 0
\(303\) −24.0000 −1.37876
\(304\) 0 0
\(305\) −10.1436 −0.580820
\(306\) 0 0
\(307\) − 0.732051i − 0.0417803i −0.999782 0.0208902i \(-0.993350\pi\)
0.999782 0.0208902i \(-0.00665003\pi\)
\(308\) 0 0
\(309\) 13.8564i 0.788263i
\(310\) 0 0
\(311\) −1.07180 −0.0607760 −0.0303880 0.999538i \(-0.509674\pi\)
−0.0303880 + 0.999538i \(0.509674\pi\)
\(312\) 0 0
\(313\) 0.392305 0.0221744 0.0110872 0.999939i \(-0.496471\pi\)
0.0110872 + 0.999939i \(0.496471\pi\)
\(314\) 0 0
\(315\) − 4.39230i − 0.247478i
\(316\) 0 0
\(317\) 3.46410i 0.194563i 0.995257 + 0.0972817i \(0.0310148\pi\)
−0.995257 + 0.0972817i \(0.968985\pi\)
\(318\) 0 0
\(319\) −9.46410 −0.529888
\(320\) 0 0
\(321\) −9.85641 −0.550131
\(322\) 0 0
\(323\) − 4.00000i − 0.222566i
\(324\) 0 0
\(325\) − 7.00000i − 0.388290i
\(326\) 0 0
\(327\) −4.00000 −0.221201
\(328\) 0 0
\(329\) −8.53590 −0.470599
\(330\) 0 0
\(331\) − 17.5167i − 0.962803i −0.876500 0.481401i \(-0.840128\pi\)
0.876500 0.481401i \(-0.159872\pi\)
\(332\) 0 0
\(333\) 8.92820i 0.489263i
\(334\) 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
\(339\) − 5.07180i − 0.275462i
\(340\) 0 0
\(341\) 31.8564i 1.72512i
\(342\) 0 0
\(343\) −15.7128 −0.848412
\(344\) 0 0
\(345\) −27.7128 −1.49201
\(346\) 0 0
\(347\) 7.07180i 0.379634i 0.981819 + 0.189817i \(0.0607895\pi\)
−0.981819 + 0.189817i \(0.939211\pi\)
\(348\) 0 0
\(349\) 9.60770i 0.514288i 0.966373 + 0.257144i \(0.0827815\pi\)
−0.966373 + 0.257144i \(0.917219\pi\)
\(350\) 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
\(355\) 28.3923i 1.50691i
\(356\) 0 0
\(357\) − 13.8564i − 0.733359i
\(358\) 0 0
\(359\) −31.5167 −1.66339 −0.831693 0.555236i \(-0.812628\pi\)
−0.831693 + 0.555236i \(0.812628\pi\)
\(360\) 0 0
\(361\) 18.4641 0.971795
\(362\) 0 0
\(363\) 22.7846i 1.19588i
\(364\) 0 0
\(365\) − 25.8564i − 1.35339i
\(366\) 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
\(371\) − 3.71281i − 0.192760i
\(372\) 0 0
\(373\) − 14.7846i − 0.765518i −0.923848 0.382759i \(-0.874974\pi\)
0.923848 0.382759i \(-0.125026\pi\)
\(374\) 0 0
\(375\) −13.8564 −0.715542
\(376\) 0 0
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) − 14.5885i − 0.749359i −0.927154 0.374679i \(-0.877753\pi\)
0.927154 0.374679i \(-0.122247\pi\)
\(380\) 0 0
\(381\) − 8.00000i − 0.409852i
\(382\) 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
\(387\) − 0.535898i − 0.0272412i
\(388\) 0 0
\(389\) 2.00000i 0.101404i 0.998714 + 0.0507020i \(0.0161459\pi\)
−0.998714 + 0.0507020i \(0.983854\pi\)
\(390\) 0 0
\(391\) −21.8564 −1.10533
\(392\) 0 0
\(393\) −39.7128 −2.00325
\(394\) 0 0
\(395\) − 18.9282i − 0.952381i
\(396\) 0 0
\(397\) − 4.53590i − 0.227650i −0.993501 0.113825i \(-0.963690\pi\)
0.993501 0.113825i \(-0.0363104\pi\)
\(398\) 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
\(403\) 6.73205i 0.335347i
\(404\) 0 0
\(405\) − 38.1051i − 1.89346i
\(406\) 0 0
\(407\) 42.2487 2.09419
\(408\) 0 0
\(409\) 12.9282 0.639259 0.319629 0.947543i \(-0.396442\pi\)
0.319629 + 0.947543i \(0.396442\pi\)
\(410\) 0 0
\(411\) − 25.8564i − 1.27540i
\(412\) 0 0
\(413\) 12.9282i 0.636155i
\(414\) 0 0
\(415\) 11.3205 0.555702
\(416\) 0 0
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) 30.7846i 1.50393i 0.659205 + 0.751963i \(0.270893\pi\)
−0.659205 + 0.751963i \(0.729107\pi\)
\(420\) 0 0
\(421\) 24.2487i 1.18181i 0.806741 + 0.590905i \(0.201229\pi\)
−0.806741 + 0.590905i \(0.798771\pi\)
\(422\) 0 0
\(423\) 6.73205 0.327324
\(424\) 0 0
\(425\) −38.2487 −1.85534
\(426\) 0 0
\(427\) 3.71281i 0.179676i
\(428\) 0 0
\(429\) 9.46410i 0.456931i
\(430\) 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
\(435\) 13.8564i 0.664364i
\(436\) 0 0
\(437\) 2.92820i 0.140075i
\(438\) 0 0
\(439\) −2.92820 −0.139756 −0.0698778 0.997556i \(-0.522261\pi\)
−0.0698778 + 0.997556i \(0.522261\pi\)
\(440\) 0 0
\(441\) 5.39230 0.256776
\(442\) 0 0
\(443\) 30.0000i 1.42534i 0.701498 + 0.712672i \(0.252515\pi\)
−0.701498 + 0.712672i \(0.747485\pi\)
\(444\) 0 0
\(445\) − 60.0000i − 2.84427i
\(446\) 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
\(451\) 42.2487i 1.98941i
\(452\) 0 0
\(453\) − 14.2487i − 0.669463i
\(454\) 0 0
\(455\) −4.39230 −0.205914
\(456\) 0 0
\(457\) −14.0000 −0.654892 −0.327446 0.944870i \(-0.606188\pi\)
−0.327446 + 0.944870i \(0.606188\pi\)
\(458\) 0 0
\(459\) − 21.8564i − 1.02017i
\(460\) 0 0
\(461\) 20.9282i 0.974724i 0.873200 + 0.487362i \(0.162041\pi\)
−0.873200 + 0.487362i \(0.837959\pi\)
\(462\) 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
\(467\) − 36.2487i − 1.67739i −0.544601 0.838695i \(-0.683319\pi\)
0.544601 0.838695i \(-0.316681\pi\)
\(468\) 0 0
\(469\) − 0.928203i − 0.0428604i
\(470\) 0 0
\(471\) −33.8564 −1.56002
\(472\) 0 0
\(473\) −2.53590 −0.116601
\(474\) 0 0
\(475\) 5.12436i 0.235122i
\(476\) 0 0
\(477\) 2.92820i 0.134073i
\(478\) 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
\(483\) 10.1436i 0.461549i
\(484\) 0 0
\(485\) 22.1436i 1.00549i
\(486\) 0 0
\(487\) 42.4449 1.92336 0.961680 0.274174i \(-0.0884043\pi\)
0.961680 + 0.274174i \(0.0884043\pi\)
\(488\) 0 0
\(489\) −33.4641 −1.51330
\(490\) 0 0
\(491\) 24.2487i 1.09433i 0.837025 + 0.547165i \(0.184293\pi\)
−0.837025 + 0.547165i \(0.815707\pi\)
\(492\) 0 0
\(493\) 10.9282i 0.492182i
\(494\) 0 0
\(495\) 16.3923 0.736779
\(496\) 0 0
\(497\) 10.3923 0.466159
\(498\) 0 0
\(499\) 7.26795i 0.325358i 0.986679 + 0.162679i \(0.0520135\pi\)
−0.986679 + 0.162679i \(0.947986\pi\)
\(500\) 0 0
\(501\) − 11.3205i − 0.505763i
\(502\) 0 0
\(503\) −30.5359 −1.36153 −0.680764 0.732503i \(-0.738352\pi\)
−0.680764 + 0.732503i \(0.738352\pi\)
\(504\) 0 0
\(505\) 41.5692 1.84981
\(506\) 0 0
\(507\) 2.00000i 0.0888231i
\(508\) 0 0
\(509\) 14.3923i 0.637928i 0.947767 + 0.318964i \(0.103335\pi\)
−0.947767 + 0.318964i \(0.896665\pi\)
\(510\) 0 0
\(511\) −9.46410 −0.418667
\(512\) 0 0
\(513\) −2.92820 −0.129283
\(514\) 0 0
\(515\) − 24.0000i − 1.05757i
\(516\) 0 0
\(517\) − 31.8564i − 1.40104i
\(518\) 0 0
\(519\) 13.8564 0.608229
\(520\) 0 0
\(521\) −25.1769 −1.10302 −0.551510 0.834168i \(-0.685948\pi\)
−0.551510 + 0.834168i \(0.685948\pi\)
\(522\) 0 0
\(523\) − 14.0000i − 0.612177i −0.952003 0.306089i \(-0.900980\pi\)
0.952003 0.306089i \(-0.0990204\pi\)
\(524\) 0 0
\(525\) 17.7513i 0.774730i
\(526\) 0 0
\(527\) 36.7846 1.60236
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) − 10.1962i − 0.442475i
\(532\) 0 0
\(533\) 8.92820i 0.386723i
\(534\) 0 0
\(535\) 17.0718 0.738078
\(536\) 0 0
\(537\) 20.7846 0.896922
\(538\) 0 0
\(539\) − 25.5167i − 1.09908i
\(540\) 0 0
\(541\) − 3.07180i − 0.132067i −0.997817 0.0660334i \(-0.978966\pi\)
0.997817 0.0660334i \(-0.0210344\pi\)
\(542\) 0 0
\(543\) −17.8564 −0.766292
\(544\) 0 0
\(545\) 6.92820 0.296772
\(546\) 0 0
\(547\) 11.8564i 0.506943i 0.967343 + 0.253472i \(0.0815725\pi\)
−0.967343 + 0.253472i \(0.918428\pi\)
\(548\) 0 0
\(549\) − 2.92820i − 0.124973i
\(550\) 0 0
\(551\) 1.46410 0.0623728
\(552\) 0 0
\(553\) −6.92820 −0.294617
\(554\) 0 0
\(555\) − 61.8564i − 2.62566i
\(556\) 0 0
\(557\) 34.7846i 1.47387i 0.675963 + 0.736936i \(0.263728\pi\)
−0.675963 + 0.736936i \(0.736272\pi\)
\(558\) 0 0
\(559\) −0.535898 −0.0226661
\(560\) 0 0
\(561\) 51.7128 2.18332
\(562\) 0 0
\(563\) 7.46410i 0.314574i 0.987553 + 0.157287i \(0.0502748\pi\)
−0.987553 + 0.157287i \(0.949725\pi\)
\(564\) 0 0
\(565\) 8.78461i 0.369571i
\(566\) 0 0
\(567\) −13.9474 −0.585737
\(568\) 0 0
\(569\) −14.0000 −0.586911 −0.293455 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(570\) 0 0
\(571\) − 2.67949i − 0.112133i −0.998427 0.0560666i \(-0.982144\pi\)
0.998427 0.0560666i \(-0.0178559\pi\)
\(572\) 0 0
\(573\) − 29.0718i − 1.21449i
\(574\) 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
\(579\) 3.21539i 0.133627i
\(580\) 0 0
\(581\) − 4.14359i − 0.171905i
\(582\) 0 0
\(583\) 13.8564 0.573874
\(584\) 0 0
\(585\) 3.46410 0.143223
\(586\) 0 0
\(587\) 4.33975i 0.179120i 0.995981 + 0.0895602i \(0.0285462\pi\)
−0.995981 + 0.0895602i \(0.971454\pi\)
\(588\) 0 0
\(589\) − 4.92820i − 0.203063i
\(590\) 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
\(595\) 24.0000i 0.983904i
\(596\) 0 0
\(597\) 33.5692i 1.37390i
\(598\) 0 0
\(599\) 45.1769 1.84588 0.922939 0.384945i \(-0.125780\pi\)
0.922939 + 0.384945i \(0.125780\pi\)
\(600\) 0 0
\(601\) −4.39230 −0.179166 −0.0895829 0.995979i \(-0.528553\pi\)
−0.0895829 + 0.995979i \(0.528553\pi\)
\(602\) 0 0
\(603\) 0.732051i 0.0298114i
\(604\) 0 0
\(605\) − 39.4641i − 1.60444i
\(606\) 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
\(611\) − 6.73205i − 0.272350i
\(612\) 0 0
\(613\) 24.6410i 0.995241i 0.867395 + 0.497621i \(0.165793\pi\)
−0.867395 + 0.497621i \(0.834207\pi\)
\(614\) 0 0
\(615\) 61.8564 2.49429
\(616\) 0 0
\(617\) 41.3205 1.66350 0.831751 0.555150i \(-0.187339\pi\)
0.831751 + 0.555150i \(0.187339\pi\)
\(618\) 0 0
\(619\) 18.5885i 0.747133i 0.927603 + 0.373567i \(0.121865\pi\)
−0.927603 + 0.373567i \(0.878135\pi\)
\(620\) 0 0
\(621\) 16.0000i 0.642058i
\(622\) 0 0
\(623\) −21.9615 −0.879870
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) − 6.92820i − 0.276686i
\(628\) 0 0
\(629\) − 48.7846i − 1.94517i
\(630\) 0 0
\(631\) 42.0526 1.67409 0.837043 0.547137i \(-0.184282\pi\)
0.837043 + 0.547137i \(0.184282\pi\)
\(632\) 0 0
\(633\) 15.7128 0.624528
\(634\) 0 0
\(635\) 13.8564i 0.549875i
\(636\) 0 0
\(637\) − 5.39230i − 0.213651i
\(638\) 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
\(643\) 12.7321i 0.502103i 0.967974 + 0.251052i \(0.0807764\pi\)
−0.967974 + 0.251052i \(0.919224\pi\)
\(644\) 0 0
\(645\) 3.71281i 0.146192i
\(646\) 0 0
\(647\) −37.8564 −1.48829 −0.744144 0.668019i \(-0.767143\pi\)
−0.744144 + 0.668019i \(0.767143\pi\)
\(648\) 0 0
\(649\) −48.2487 −1.89393
\(650\) 0 0
\(651\) − 17.0718i − 0.669096i
\(652\) 0 0
\(653\) − 3.07180i − 0.120209i −0.998192 0.0601043i \(-0.980857\pi\)
0.998192 0.0601043i \(-0.0191433\pi\)
\(654\) 0 0
\(655\) 68.7846 2.68764
\(656\) 0 0
\(657\) 7.46410 0.291202
\(658\) 0 0
\(659\) − 14.0000i − 0.545363i −0.962104 0.272681i \(-0.912090\pi\)
0.962104 0.272681i \(-0.0879105\pi\)
\(660\) 0 0
\(661\) 17.3205i 0.673690i 0.941560 + 0.336845i \(0.109360\pi\)
−0.941560 + 0.336845i \(0.890640\pi\)
\(662\) 0 0
\(663\) 10.9282 0.424416
\(664\) 0 0
\(665\) 3.21539 0.124687
\(666\) 0 0
\(667\) − 8.00000i − 0.309761i
\(668\) 0 0
\(669\) − 0.392305i − 0.0151674i
\(670\) 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
\(675\) 28.0000i 1.07772i
\(676\) 0 0
\(677\) 21.0718i 0.809855i 0.914349 + 0.404927i \(0.132703\pi\)
−0.914349 + 0.404927i \(0.867297\pi\)
\(678\) 0 0
\(679\) 8.10512 0.311046
\(680\) 0 0
\(681\) −5.75129 −0.220390
\(682\) 0 0
\(683\) 17.8038i 0.681245i 0.940200 + 0.340623i \(0.110638\pi\)
−0.940200 + 0.340623i \(0.889362\pi\)
\(684\) 0 0
\(685\) 44.7846i 1.71113i
\(686\) 0 0
\(687\) −10.6410 −0.405980
\(688\) 0 0
\(689\) 2.92820 0.111556
\(690\) 0 0
\(691\) 32.8372i 1.24918i 0.780951 + 0.624592i \(0.214735\pi\)
−0.780951 + 0.624592i \(0.785265\pi\)
\(692\) 0 0
\(693\) − 6.00000i − 0.227921i
\(694\) 0 0
\(695\) −34.6410 −1.31401
\(696\) 0 0
\(697\) 48.7846 1.84785
\(698\) 0 0
\(699\) − 33.8564i − 1.28057i
\(700\) 0 0
\(701\) − 40.6410i − 1.53499i −0.641055 0.767495i \(-0.721503\pi\)
0.641055 0.767495i \(-0.278497\pi\)
\(702\) 0 0
\(703\) −6.53590 −0.246506
\(704\) 0 0
\(705\) −46.6410 −1.75660
\(706\) 0 0
\(707\) − 15.2154i − 0.572234i
\(708\) 0 0
\(709\) 29.3205i 1.10115i 0.834784 + 0.550577i \(0.185592\pi\)
−0.834784 + 0.550577i \(0.814408\pi\)
\(710\) 0 0
\(711\) 5.46410 0.204920
\(712\) 0 0
\(713\) −26.9282 −1.00847
\(714\) 0 0
\(715\) − 16.3923i − 0.613037i
\(716\) 0 0
\(717\) 37.4641i 1.39912i
\(718\) 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
\(723\) 60.7846i 2.26060i
\(724\) 0 0
\(725\) − 14.0000i − 0.519947i
\(726\) 0 0
\(727\) 22.9282 0.850360 0.425180 0.905109i \(-0.360211\pi\)
0.425180 + 0.905109i \(0.360211\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 2.92820i 0.108304i
\(732\) 0 0
\(733\) 37.3205i 1.37846i 0.724541 + 0.689232i \(0.242052\pi\)
−0.724541 + 0.689232i \(0.757948\pi\)
\(734\) 0 0
\(735\) −37.3590 −1.37801
\(736\) 0 0
\(737\) 3.46410 0.127602
\(738\) 0 0
\(739\) − 39.7654i − 1.46279i −0.681953 0.731396i \(-0.738869\pi\)
0.681953 0.731396i \(-0.261131\pi\)
\(740\) 0 0
\(741\) − 1.46410i − 0.0537851i
\(742\) 0 0
\(743\) 32.1962 1.18116 0.590581 0.806978i \(-0.298899\pi\)
0.590581 + 0.806978i \(0.298899\pi\)
\(744\) 0 0
\(745\) 44.7846 1.64078
\(746\) 0 0
\(747\) 3.26795i 0.119568i
\(748\) 0 0
\(749\) − 6.24871i − 0.228323i
\(750\) 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
\(755\) 24.6795i 0.898179i
\(756\) 0 0
\(757\) 40.7846i 1.48234i 0.671316 + 0.741171i \(0.265729\pi\)
−0.671316 + 0.741171i \(0.734271\pi\)
\(758\) 0 0
\(759\) −37.8564 −1.37410
\(760\) 0 0
\(761\) 18.7846 0.680942 0.340471 0.940255i \(-0.389414\pi\)
0.340471 + 0.940255i \(0.389414\pi\)
\(762\) 0 0
\(763\) − 2.53590i − 0.0918057i
\(764\) 0 0
\(765\) − 18.9282i − 0.684351i
\(766\) 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
\(771\) 47.7128i 1.71833i
\(772\) 0 0
\(773\) 27.4641i 0.987815i 0.869514 + 0.493908i \(0.164432\pi\)
−0.869514 + 0.493908i \(0.835568\pi\)
\(774\) 0 0
\(775\) −47.1244 −1.69276
\(776\) 0 0
\(777\) −22.6410 −0.812242
\(778\) 0 0
\(779\) − 6.53590i − 0.234173i
\(780\) 0 0
\(781\) 38.7846i 1.38782i
\(782\) 0 0
\(783\) 8.00000 0.285897
\(784\) 0 0
\(785\) 58.6410 2.09299
\(786\) 0 0
\(787\) 29.1244i 1.03817i 0.854722 + 0.519086i \(0.173727\pi\)
−0.854722 + 0.519086i \(0.826273\pi\)
\(788\) 0 0
\(789\) − 54.6410i − 1.94527i
\(790\) 0 0
\(791\) 3.21539 0.114326
\(792\) 0 0
\(793\) −2.92820 −0.103984
\(794\) 0 0
\(795\) − 20.2872i − 0.719512i
\(796\) 0 0
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) 0 0
\(799\) −36.7846 −1.30135
\(800\) 0 0
\(801\) 17.3205 0.611990
\(802\) 0 0
\(803\) − 35.3205i − 1.24643i
\(804\) 0 0
\(805\) − 17.5692i − 0.619234i
\(806\) 0 0
\(807\) −15.7128 −0.553117
\(808\) 0 0
\(809\) −5.46410 −0.192108 −0.0960538 0.995376i \(-0.530622\pi\)
−0.0960538 + 0.995376i \(0.530622\pi\)
\(810\) 0 0
\(811\) − 21.4115i − 0.751861i −0.926648 0.375930i \(-0.877323\pi\)
0.926648 0.375930i \(-0.122677\pi\)
\(812\) 0 0
\(813\) − 40.3923i − 1.41662i
\(814\) 0 0
\(815\) 57.9615 2.03030
\(816\) 0 0
\(817\) 0.392305 0.0137250
\(818\) 0 0
\(819\) − 1.26795i − 0.0443057i
\(820\) 0 0
\(821\) − 48.2487i − 1.68389i −0.539562 0.841946i \(-0.681410\pi\)
0.539562 0.841946i \(-0.318590\pi\)
\(822\) 0 0
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) 0 0
\(825\) −66.2487 −2.30648
\(826\) 0 0
\(827\) − 8.73205i − 0.303643i −0.988408 0.151822i \(-0.951486\pi\)
0.988408 0.151822i \(-0.0485139\pi\)
\(828\) 0 0
\(829\) − 28.7846i − 0.999731i −0.866103 0.499865i \(-0.833383\pi\)
0.866103 0.499865i \(-0.166617\pi\)
\(830\) 0 0
\(831\) 3.71281 0.128796
\(832\) 0 0
\(833\) −29.4641 −1.02087
\(834\) 0 0
\(835\) 19.6077i 0.678552i
\(836\) 0 0
\(837\) − 26.9282i − 0.930775i
\(838\) 0 0
\(839\) 20.9808 0.724336 0.362168 0.932113i \(-0.382037\pi\)
0.362168 + 0.932113i \(0.382037\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) − 18.6410i − 0.642031i
\(844\) 0 0
\(845\) − 3.46410i − 0.119169i
\(846\) 0 0
\(847\) −14.4449 −0.496331
\(848\) 0 0
\(849\) −38.9282 −1.33601
\(850\) 0 0
\(851\) 35.7128i 1.22422i
\(852\) 0 0
\(853\) − 55.1769i − 1.88922i −0.328193 0.944611i \(-0.606440\pi\)
0.328193 0.944611i \(-0.393560\pi\)
\(854\) 0 0
\(855\) −2.53590 −0.0867259
\(856\) 0 0
\(857\) −7.85641 −0.268370 −0.134185 0.990956i \(-0.542842\pi\)
−0.134185 + 0.990956i \(0.542842\pi\)
\(858\) 0 0
\(859\) − 18.0000i − 0.614152i −0.951685 0.307076i \(-0.900649\pi\)
0.951685 0.307076i \(-0.0993506\pi\)
\(860\) 0 0
\(861\) − 22.6410i − 0.771604i
\(862\) 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
\(867\) − 25.7128i − 0.873253i
\(868\) 0 0
\(869\) − 25.8564i − 0.877119i
\(870\) 0 0
\(871\) 0.732051 0.0248046
\(872\) 0 0
\(873\) −6.39230 −0.216347
\(874\) 0 0
\(875\) − 8.78461i − 0.296974i
\(876\) 0 0
\(877\) − 23.4641i − 0.792326i −0.918180 0.396163i \(-0.870341\pi\)
0.918180 0.396163i \(-0.129659\pi\)
\(878\) 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
\(883\) − 31.4641i − 1.05885i −0.848356 0.529426i \(-0.822407\pi\)
0.848356 0.529426i \(-0.177593\pi\)
\(884\) 0 0
\(885\) 70.6410i 2.37457i
\(886\) 0 0
\(887\) −30.2487 −1.01565 −0.507826 0.861460i \(-0.669551\pi\)
−0.507826 + 0.861460i \(0.669551\pi\)
\(888\) 0 0
\(889\) 5.07180 0.170103
\(890\) 0 0
\(891\) − 52.0526i − 1.74383i
\(892\) 0 0
\(893\) 4.92820i 0.164916i
\(894\) 0 0
\(895\) −36.0000 −1.20335
\(896\) 0 0
\(897\) −8.00000 −0.267112
\(898\) 0 0
\(899\) 13.4641i 0.449053i
\(900\) 0 0
\(901\) − 16.0000i − 0.533037i
\(902\) 0 0
\(903\) 1.35898 0.0452242
\(904\) 0 0
\(905\) 30.9282 1.02809
\(906\) 0 0
\(907\) − 22.1051i − 0.733988i −0.930223 0.366994i \(-0.880387\pi\)
0.930223 0.366994i \(-0.119613\pi\)
\(908\) 0 0
\(909\) 12.0000i 0.398015i
\(910\) 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
\(915\) 20.2872i 0.670674i
\(916\) 0 0
\(917\) − 25.1769i − 0.831415i
\(918\) 0 0
\(919\) 35.3205 1.16512 0.582558 0.812789i \(-0.302052\pi\)
0.582558 + 0.812789i \(0.302052\pi\)
\(920\) 0 0
\(921\) −1.46410 −0.0482438
\(922\) 0 0
\(923\) 8.19615i 0.269780i
\(924\) 0 0
\(925\) 62.4974i 2.05490i
\(926\) 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
\(931\) 3.94744i 0.129372i
\(932\) 0 0
\(933\) 2.14359i 0.0701781i
\(934\) 0 0
\(935\) −89.5692 −2.92923
\(936\) 0 0
\(937\) −13.7128 −0.447978 −0.223989 0.974592i \(-0.571908\pi\)
−0.223989 + 0.974592i \(0.571908\pi\)
\(938\) 0 0
\(939\) − 0.784610i − 0.0256048i
\(940\) 0 0
\(941\) 32.2487i 1.05128i 0.850708 + 0.525639i \(0.176174\pi\)
−0.850708 + 0.525639i \(0.823826\pi\)
\(942\) 0 0
\(943\) −35.7128 −1.16297
\(944\) 0 0
\(945\) 17.5692 0.571527
\(946\) 0 0
\(947\) 48.4449i 1.57425i 0.616796 + 0.787123i \(0.288430\pi\)
−0.616796 + 0.787123i \(0.711570\pi\)
\(948\) 0 0
\(949\) − 7.46410i − 0.242295i
\(950\) 0 0
\(951\) 6.92820 0.224662
\(952\) 0 0
\(953\) −31.8564 −1.03193 −0.515965 0.856610i \(-0.672567\pi\)
−0.515965 + 0.856610i \(0.672567\pi\)
\(954\) 0 0
\(955\) 50.3538i 1.62941i
\(956\) 0 0
\(957\) 18.9282i 0.611862i
\(958\) 0 0
\(959\) 16.3923 0.529335
\(960\) 0 0
\(961\) 14.3205 0.461952
\(962\) 0 0
\(963\) 4.92820i 0.158809i
\(964\) 0 0
\(965\) − 5.56922i − 0.179280i
\(966\) 0 0
\(967\) 8.98076 0.288802 0.144401 0.989519i \(-0.453875\pi\)
0.144401 + 0.989519i \(0.453875\pi\)
\(968\) 0 0
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) 16.1436i 0.518073i 0.965868 + 0.259036i \(0.0834049\pi\)
−0.965868 + 0.259036i \(0.916595\pi\)
\(972\) 0 0
\(973\) 12.6795i 0.406486i
\(974\) 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
\(979\) − 81.9615i − 2.61950i
\(980\) 0 0
\(981\) 2.00000i 0.0638551i
\(982\) 0 0
\(983\) −16.1962 −0.516577 −0.258289 0.966068i \(-0.583159\pi\)
−0.258289 + 0.966068i \(0.583159\pi\)
\(984\) 0 0
\(985\) 10.6410 0.339051
\(986\) 0 0
\(987\) 17.0718i 0.543401i
\(988\) 0 0
\(989\) − 2.14359i − 0.0681623i
\(990\) 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
\(995\) − 58.1436i − 1.84328i
\(996\) 0 0
\(997\) 33.5692i 1.06315i 0.847012 + 0.531574i \(0.178399\pi\)
−0.847012 + 0.531574i \(0.821601\pi\)
\(998\) 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 416.2.b.b.209.2 4
3.2 odd 2 3744.2.g.b.1873.1 4
4.3 odd 2 104.2.b.b.53.1 4
8.3 odd 2 104.2.b.b.53.2 yes 4
8.5 even 2 inner 416.2.b.b.209.3 4
12.11 even 2 936.2.g.b.469.4 4
16.3 odd 4 3328.2.a.n.1.2 2
16.5 even 4 3328.2.a.m.1.1 2
16.11 odd 4 3328.2.a.bd.1.1 2
16.13 even 4 3328.2.a.bc.1.2 2
24.5 odd 2 3744.2.g.b.1873.3 4
24.11 even 2 936.2.g.b.469.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
104.2.b.b.53.1 4 4.3 odd 2
104.2.b.b.53.2 yes 4 8.3 odd 2
416.2.b.b.209.2 4 1.1 even 1 trivial
416.2.b.b.209.3 4 8.5 even 2 inner
936.2.g.b.469.3 4 24.11 even 2
936.2.g.b.469.4 4 12.11 even 2
3328.2.a.m.1.1 2 16.5 even 4
3328.2.a.n.1.2 2 16.3 odd 4
3328.2.a.bc.1.2 2 16.13 even 4
3328.2.a.bd.1.1 2 16.11 odd 4
3744.2.g.b.1873.1 4 3.2 odd 2
3744.2.g.b.1873.3 4 24.5 odd 2