Properties

Label 4056.2.c.b.337.1
Level $4056$
Weight $2$
Character 4056.337
Analytic conductor $32.387$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4056,2,Mod(337,4056)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4056, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4056.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.c (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: \(32.3873230598\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4056.337
Dual form 4056.2.c.b.337.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -3.00000i q^{5} -4.00000i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -3.00000i q^{5} -4.00000i q^{7} +1.00000 q^{9} +4.00000i q^{11} +3.00000i q^{15} -3.00000 q^{17} +4.00000i q^{19} +4.00000i q^{21} +8.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} -5.00000 q^{29} +8.00000i q^{31} -4.00000i q^{33} -12.0000 q^{35} +7.00000i q^{37} +9.00000i q^{41} -8.00000 q^{43} -3.00000i q^{45} -4.00000i q^{47} -9.00000 q^{49} +3.00000 q^{51} -5.00000 q^{53} +12.0000 q^{55} -4.00000i q^{57} +4.00000i q^{59} -5.00000 q^{61} -4.00000i q^{63} -8.00000i q^{67} -8.00000 q^{69} +4.00000i q^{71} +11.0000i q^{73} +4.00000 q^{75} +16.0000 q^{77} -4.00000 q^{79} +1.00000 q^{81} +9.00000i q^{85} +5.00000 q^{87} -6.00000i q^{89} -8.00000i q^{93} +12.0000 q^{95} +14.0000i q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{9} - 6 q^{17} + 16 q^{23} - 8 q^{25} - 2 q^{27} - 10 q^{29} - 24 q^{35} - 16 q^{43} - 18 q^{49} + 6 q^{51} - 10 q^{53} + 24 q^{55} - 10 q^{61} - 16 q^{69} + 8 q^{75} + 32 q^{77} - 8 q^{79} + 2 q^{81} + 10 q^{87} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times\).

\(n\) \(1015\) \(2029\) \(2705\) \(3889\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) − 3.00000i − 1.34164i −0.741620 0.670820i \(-0.765942\pi\)
0.741620 0.670820i \(-0.234058\pi\)
\(6\) 0 0
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 3.00000i 0.774597i
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 4.00000i 0.872872i
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 8.00000i 1.43684i 0.695608 + 0.718421i \(0.255135\pi\)
−0.695608 + 0.718421i \(0.744865\pi\)
\(32\) 0 0
\(33\) − 4.00000i − 0.696311i
\(34\) 0 0
\(35\) −12.0000 −2.02837
\(36\) 0 0
\(37\) 7.00000i 1.15079i 0.817875 + 0.575396i \(0.195152\pi\)
−0.817875 + 0.575396i \(0.804848\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.00000i 1.40556i 0.711405 + 0.702782i \(0.248059\pi\)
−0.711405 + 0.702782i \(0.751941\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) − 3.00000i − 0.447214i
\(46\) 0 0
\(47\) − 4.00000i − 0.583460i −0.956501 0.291730i \(-0.905769\pi\)
0.956501 0.291730i \(-0.0942309\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) −5.00000 −0.686803 −0.343401 0.939189i \(-0.611579\pi\)
−0.343401 + 0.939189i \(0.611579\pi\)
\(54\) 0 0
\(55\) 12.0000 1.61808
\(56\) 0 0
\(57\) − 4.00000i − 0.529813i
\(58\) 0 0
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) 0 0
\(63\) − 4.00000i − 0.503953i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 4.00000i 0.474713i 0.971423 + 0.237356i \(0.0762809\pi\)
−0.971423 + 0.237356i \(0.923719\pi\)
\(72\) 0 0
\(73\) 11.0000i 1.28745i 0.765256 + 0.643726i \(0.222612\pi\)
−0.765256 + 0.643726i \(0.777388\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 16.0000 1.82337
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 9.00000i 0.976187i
\(86\) 0 0
\(87\) 5.00000 0.536056
\(88\) 0 0
\(89\) − 6.00000i − 0.635999i −0.948091 0.317999i \(-0.896989\pi\)
0.948091 0.317999i \(-0.103011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) 12.0000 1.23117
\(96\) 0 0
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) 0 0
\(99\) 4.00000i 0.402015i
\(100\) 0 0
\(101\) −11.0000 −1.09454 −0.547270 0.836956i \(-0.684333\pi\)
−0.547270 + 0.836956i \(0.684333\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 12.0000 1.17108
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 18.0000i 1.72409i 0.506834 + 0.862044i \(0.330816\pi\)
−0.506834 + 0.862044i \(0.669184\pi\)
\(110\) 0 0
\(111\) − 7.00000i − 0.664411i
\(112\) 0 0
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 0 0
\(115\) − 24.0000i − 2.23801i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 12.0000i 1.10004i
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) − 9.00000i − 0.811503i
\(124\) 0 0
\(125\) − 3.00000i − 0.268328i
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) 0 0
\(135\) 3.00000i 0.258199i
\(136\) 0 0
\(137\) − 9.00000i − 0.768922i −0.923141 0.384461i \(-0.874387\pi\)
0.923141 0.384461i \(-0.125613\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 4.00000i 0.336861i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 15.0000i 1.24568i
\(146\) 0 0
\(147\) 9.00000 0.742307
\(148\) 0 0
\(149\) − 3.00000i − 0.245770i −0.992421 0.122885i \(-0.960785\pi\)
0.992421 0.122885i \(-0.0392146\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 24.0000 1.92773
\(156\) 0 0
\(157\) 23.0000 1.83560 0.917800 0.397043i \(-0.129964\pi\)
0.917800 + 0.397043i \(0.129964\pi\)
\(158\) 0 0
\(159\) 5.00000 0.396526
\(160\) 0 0
\(161\) − 32.0000i − 2.52195i
\(162\) 0 0
\(163\) − 8.00000i − 0.626608i −0.949653 0.313304i \(-0.898564\pi\)
0.949653 0.313304i \(-0.101436\pi\)
\(164\) 0 0
\(165\) −12.0000 −0.934199
\(166\) 0 0
\(167\) − 12.0000i − 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 4.00000i 0.305888i
\(172\) 0 0
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 16.0000i 1.20949i
\(176\) 0 0
\(177\) − 4.00000i − 0.300658i
\(178\) 0 0
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) 1.00000 0.0743294 0.0371647 0.999309i \(-0.488167\pi\)
0.0371647 + 0.999309i \(0.488167\pi\)
\(182\) 0 0
\(183\) 5.00000 0.369611
\(184\) 0 0
\(185\) 21.0000 1.54395
\(186\) 0 0
\(187\) − 12.0000i − 0.877527i
\(188\) 0 0
\(189\) 4.00000i 0.290957i
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) − 13.0000i − 0.935760i −0.883792 0.467880i \(-0.845018\pi\)
0.883792 0.467880i \(-0.154982\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 26.0000i 1.85242i 0.377004 + 0.926212i \(0.376954\pi\)
−0.377004 + 0.926212i \(0.623046\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 8.00000i 0.564276i
\(202\) 0 0
\(203\) 20.0000i 1.40372i
\(204\) 0 0
\(205\) 27.0000 1.88576
\(206\) 0 0
\(207\) 8.00000 0.556038
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) − 4.00000i − 0.274075i
\(214\) 0 0
\(215\) 24.0000i 1.63679i
\(216\) 0 0
\(217\) 32.0000 2.17230
\(218\) 0 0
\(219\) − 11.0000i − 0.743311i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) 0 0
\(229\) − 10.0000i − 0.660819i −0.943838 0.330409i \(-0.892813\pi\)
0.943838 0.330409i \(-0.107187\pi\)
\(230\) 0 0
\(231\) −16.0000 −1.05272
\(232\) 0 0
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) 28.0000i 1.81117i 0.424165 + 0.905585i \(0.360568\pi\)
−0.424165 + 0.905585i \(0.639432\pi\)
\(240\) 0 0
\(241\) − 5.00000i − 0.322078i −0.986948 0.161039i \(-0.948515\pi\)
0.986948 0.161039i \(-0.0514845\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 27.0000i 1.72497i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 32.0000i 2.01182i
\(254\) 0 0
\(255\) − 9.00000i − 0.563602i
\(256\) 0 0
\(257\) −7.00000 −0.436648 −0.218324 0.975876i \(-0.570059\pi\)
−0.218324 + 0.975876i \(0.570059\pi\)
\(258\) 0 0
\(259\) 28.0000 1.73984
\(260\) 0 0
\(261\) −5.00000 −0.309492
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 15.0000i 0.921443i
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) 0 0
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) 20.0000i 1.21491i 0.794353 + 0.607457i \(0.207810\pi\)
−0.794353 + 0.607457i \(0.792190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 16.0000i − 0.964836i
\(276\) 0 0
\(277\) −19.0000 −1.14160 −0.570800 0.821089i \(-0.693367\pi\)
−0.570800 + 0.821089i \(0.693367\pi\)
\(278\) 0 0
\(279\) 8.00000i 0.478947i
\(280\) 0 0
\(281\) − 13.0000i − 0.775515i −0.921761 0.387757i \(-0.873250\pi\)
0.921761 0.387757i \(-0.126750\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) −12.0000 −0.710819
\(286\) 0 0
\(287\) 36.0000 2.12501
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) − 14.0000i − 0.820695i
\(292\) 0 0
\(293\) − 9.00000i − 0.525786i −0.964825 0.262893i \(-0.915323\pi\)
0.964825 0.262893i \(-0.0846766\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) − 4.00000i − 0.232104i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 32.0000i 1.84445i
\(302\) 0 0
\(303\) 11.0000 0.631933
\(304\) 0 0
\(305\) 15.0000i 0.858898i
\(306\) 0 0
\(307\) 8.00000i 0.456584i 0.973593 + 0.228292i \(0.0733141\pi\)
−0.973593 + 0.228292i \(0.926686\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) −12.0000 −0.676123
\(316\) 0 0
\(317\) 25.0000i 1.40414i 0.712108 + 0.702070i \(0.247741\pi\)
−0.712108 + 0.702070i \(0.752259\pi\)
\(318\) 0 0
\(319\) − 20.0000i − 1.11979i
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) − 12.0000i − 0.667698i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 18.0000i − 0.995402i
\(328\) 0 0
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) − 8.00000i − 0.439720i −0.975531 0.219860i \(-0.929440\pi\)
0.975531 0.219860i \(-0.0705600\pi\)
\(332\) 0 0
\(333\) 7.00000i 0.383598i
\(334\) 0 0
\(335\) −24.0000 −1.31126
\(336\) 0 0
\(337\) −23.0000 −1.25289 −0.626445 0.779466i \(-0.715491\pi\)
−0.626445 + 0.779466i \(0.715491\pi\)
\(338\) 0 0
\(339\) 9.00000 0.488813
\(340\) 0 0
\(341\) −32.0000 −1.73290
\(342\) 0 0
\(343\) 8.00000i 0.431959i
\(344\) 0 0
\(345\) 24.0000i 1.29212i
\(346\) 0 0
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) 0 0
\(349\) 30.0000i 1.60586i 0.596071 + 0.802932i \(0.296728\pi\)
−0.596071 + 0.802932i \(0.703272\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.00000i 0.0532246i 0.999646 + 0.0266123i \(0.00847196\pi\)
−0.999646 + 0.0266123i \(0.991528\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 0 0
\(357\) − 12.0000i − 0.635107i
\(358\) 0 0
\(359\) 8.00000i 0.422224i 0.977462 + 0.211112i \(0.0677085\pi\)
−0.977462 + 0.211112i \(0.932292\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 33.0000 1.72730
\(366\) 0 0
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 0 0
\(369\) 9.00000i 0.468521i
\(370\) 0 0
\(371\) 20.0000i 1.03835i
\(372\) 0 0
\(373\) −9.00000 −0.466002 −0.233001 0.972476i \(-0.574855\pi\)
−0.233001 + 0.972476i \(0.574855\pi\)
\(374\) 0 0
\(375\) 3.00000i 0.154919i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 12.0000i 0.616399i 0.951322 + 0.308199i \(0.0997264\pi\)
−0.951322 + 0.308199i \(0.900274\pi\)
\(380\) 0 0
\(381\) −20.0000 −1.02463
\(382\) 0 0
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 0 0
\(385\) − 48.0000i − 2.44631i
\(386\) 0 0
\(387\) −8.00000 −0.406663
\(388\) 0 0
\(389\) −27.0000 −1.36895 −0.684477 0.729034i \(-0.739969\pi\)
−0.684477 + 0.729034i \(0.739969\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 0 0
\(393\) −8.00000 −0.403547
\(394\) 0 0
\(395\) 12.0000i 0.603786i
\(396\) 0 0
\(397\) − 18.0000i − 0.903394i −0.892171 0.451697i \(-0.850819\pi\)
0.892171 0.451697i \(-0.149181\pi\)
\(398\) 0 0
\(399\) −16.0000 −0.801002
\(400\) 0 0
\(401\) − 29.0000i − 1.44819i −0.689700 0.724095i \(-0.742257\pi\)
0.689700 0.724095i \(-0.257743\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 3.00000i − 0.149071i
\(406\) 0 0
\(407\) −28.0000 −1.38791
\(408\) 0 0
\(409\) 25.0000i 1.23617i 0.786111 + 0.618085i \(0.212091\pi\)
−0.786111 + 0.618085i \(0.787909\pi\)
\(410\) 0 0
\(411\) 9.00000i 0.443937i
\(412\) 0 0
\(413\) 16.0000 0.787309
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 1.00000i 0.0487370i 0.999703 + 0.0243685i \(0.00775751\pi\)
−0.999703 + 0.0243685i \(0.992242\pi\)
\(422\) 0 0
\(423\) − 4.00000i − 0.194487i
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) 20.0000i 0.967868i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 8.00000i − 0.385346i −0.981263 0.192673i \(-0.938284\pi\)
0.981263 0.192673i \(-0.0617157\pi\)
\(432\) 0 0
\(433\) 29.0000 1.39365 0.696826 0.717241i \(-0.254595\pi\)
0.696826 + 0.717241i \(0.254595\pi\)
\(434\) 0 0
\(435\) − 15.0000i − 0.719195i
\(436\) 0 0
\(437\) 32.0000i 1.53077i
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) −9.00000 −0.428571
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 0 0
\(447\) 3.00000i 0.141895i
\(448\) 0 0
\(449\) − 30.0000i − 1.41579i −0.706319 0.707894i \(-0.749646\pi\)
0.706319 0.707894i \(-0.250354\pi\)
\(450\) 0 0
\(451\) −36.0000 −1.69517
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9.00000i 0.421002i 0.977594 + 0.210501i \(0.0675096\pi\)
−0.977594 + 0.210501i \(0.932490\pi\)
\(458\) 0 0
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) 5.00000i 0.232873i 0.993198 + 0.116437i \(0.0371472\pi\)
−0.993198 + 0.116437i \(0.962853\pi\)
\(462\) 0 0
\(463\) 32.0000i 1.48717i 0.668644 + 0.743583i \(0.266875\pi\)
−0.668644 + 0.743583i \(0.733125\pi\)
\(464\) 0 0
\(465\) −24.0000 −1.11297
\(466\) 0 0
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) −32.0000 −1.47762
\(470\) 0 0
\(471\) −23.0000 −1.05978
\(472\) 0 0
\(473\) − 32.0000i − 1.47136i
\(474\) 0 0
\(475\) − 16.0000i − 0.734130i
\(476\) 0 0
\(477\) −5.00000 −0.228934
\(478\) 0 0
\(479\) − 36.0000i − 1.64488i −0.568850 0.822441i \(-0.692612\pi\)
0.568850 0.822441i \(-0.307388\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 32.0000i 1.45605i
\(484\) 0 0
\(485\) 42.0000 1.90712
\(486\) 0 0
\(487\) − 40.0000i − 1.81257i −0.422664 0.906287i \(-0.638905\pi\)
0.422664 0.906287i \(-0.361095\pi\)
\(488\) 0 0
\(489\) 8.00000i 0.361773i
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 15.0000 0.675566
\(494\) 0 0
\(495\) 12.0000 0.539360
\(496\) 0 0
\(497\) 16.0000 0.717698
\(498\) 0 0
\(499\) 36.0000i 1.61158i 0.592200 + 0.805791i \(0.298259\pi\)
−0.592200 + 0.805791i \(0.701741\pi\)
\(500\) 0 0
\(501\) 12.0000i 0.536120i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 33.0000i 1.46848i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 31.0000i − 1.37405i −0.726633 0.687025i \(-0.758916\pi\)
0.726633 0.687025i \(-0.241084\pi\)
\(510\) 0 0
\(511\) 44.0000 1.94645
\(512\) 0 0
\(513\) − 4.00000i − 0.176604i
\(514\) 0 0
\(515\) − 12.0000i − 0.528783i
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −9.00000 −0.394297 −0.197149 0.980374i \(-0.563168\pi\)
−0.197149 + 0.980374i \(0.563168\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) − 16.0000i − 0.698297i
\(526\) 0 0
\(527\) − 24.0000i − 1.04546i
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 4.00000i 0.173585i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 24.0000i − 1.03761i
\(536\) 0 0
\(537\) 16.0000 0.690451
\(538\) 0 0
\(539\) − 36.0000i − 1.55063i
\(540\) 0 0
\(541\) 27.0000i 1.16082i 0.814324 + 0.580410i \(0.197108\pi\)
−0.814324 + 0.580410i \(0.802892\pi\)
\(542\) 0 0
\(543\) −1.00000 −0.0429141
\(544\) 0 0
\(545\) 54.0000 2.31311
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 0 0
\(549\) −5.00000 −0.213395
\(550\) 0 0
\(551\) − 20.0000i − 0.852029i
\(552\) 0 0
\(553\) 16.0000i 0.680389i
\(554\) 0 0
\(555\) −21.0000 −0.891400
\(556\) 0 0
\(557\) 35.0000i 1.48300i 0.670954 + 0.741499i \(0.265885\pi\)
−0.670954 + 0.741499i \(0.734115\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 12.0000i 0.506640i
\(562\) 0 0
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) 27.0000i 1.13590i
\(566\) 0 0
\(567\) − 4.00000i − 0.167984i
\(568\) 0 0
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 8.00000 0.334205
\(574\) 0 0
\(575\) −32.0000 −1.33449
\(576\) 0 0
\(577\) 9.00000i 0.374675i 0.982296 + 0.187337i \(0.0599858\pi\)
−0.982296 + 0.187337i \(0.940014\pi\)
\(578\) 0 0
\(579\) 13.0000i 0.540262i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 20.0000i − 0.828315i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 48.0000i − 1.98117i −0.136892 0.990586i \(-0.543711\pi\)
0.136892 0.990586i \(-0.456289\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) − 26.0000i − 1.06950i
\(592\) 0 0
\(593\) 7.00000i 0.287456i 0.989617 + 0.143728i \(0.0459090\pi\)
−0.989617 + 0.143728i \(0.954091\pi\)
\(594\) 0 0
\(595\) 36.0000 1.47586
\(596\) 0 0
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 35.0000 1.42768 0.713840 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) 0 0
\(603\) − 8.00000i − 0.325785i
\(604\) 0 0
\(605\) 15.0000i 0.609837i
\(606\) 0 0
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 0 0
\(609\) − 20.0000i − 0.810441i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 31.0000i − 1.25208i −0.779792 0.626039i \(-0.784675\pi\)
0.779792 0.626039i \(-0.215325\pi\)
\(614\) 0 0
\(615\) −27.0000 −1.08875
\(616\) 0 0
\(617\) − 27.0000i − 1.08698i −0.839416 0.543490i \(-0.817103\pi\)
0.839416 0.543490i \(-0.182897\pi\)
\(618\) 0 0
\(619\) − 20.0000i − 0.803868i −0.915669 0.401934i \(-0.868338\pi\)
0.915669 0.401934i \(-0.131662\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 0 0
\(623\) −24.0000 −0.961540
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 16.0000 0.638978
\(628\) 0 0
\(629\) − 21.0000i − 0.837325i
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 60.0000i − 2.38103i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 4.00000i 0.158238i
\(640\) 0 0
\(641\) −35.0000 −1.38242 −0.691208 0.722655i \(-0.742921\pi\)
−0.691208 + 0.722655i \(0.742921\pi\)
\(642\) 0 0
\(643\) 44.0000i 1.73519i 0.497271 + 0.867595i \(0.334335\pi\)
−0.497271 + 0.867595i \(0.665665\pi\)
\(644\) 0 0
\(645\) − 24.0000i − 0.944999i
\(646\) 0 0
\(647\) −8.00000 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 0 0
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) 0 0
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) − 24.0000i − 0.937758i
\(656\) 0 0
\(657\) 11.0000i 0.429151i
\(658\) 0 0
\(659\) 40.0000 1.55818 0.779089 0.626913i \(-0.215682\pi\)
0.779089 + 0.626913i \(0.215682\pi\)
\(660\) 0 0
\(661\) − 5.00000i − 0.194477i −0.995261 0.0972387i \(-0.968999\pi\)
0.995261 0.0972387i \(-0.0310010\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 48.0000i − 1.86136i
\(666\) 0 0
\(667\) −40.0000 −1.54881
\(668\) 0 0
\(669\) − 16.0000i − 0.618596i
\(670\) 0 0
\(671\) − 20.0000i − 0.772091i
\(672\) 0 0
\(673\) −43.0000 −1.65753 −0.828764 0.559598i \(-0.810955\pi\)
−0.828764 + 0.559598i \(0.810955\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 0 0
\(679\) 56.0000 2.14908
\(680\) 0 0
\(681\) − 4.00000i − 0.153280i
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) −27.0000 −1.03162
\(686\) 0 0
\(687\) 10.0000i 0.381524i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 20.0000i 0.760836i 0.924815 + 0.380418i \(0.124220\pi\)
−0.924815 + 0.380418i \(0.875780\pi\)
\(692\) 0 0
\(693\) 16.0000 0.607790
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 27.0000i − 1.02270i
\(698\) 0 0
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) −28.0000 −1.05604
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) 0 0
\(707\) 44.0000i 1.65479i
\(708\) 0 0
\(709\) − 9.00000i − 0.338002i −0.985616 0.169001i \(-0.945946\pi\)
0.985616 0.169001i \(-0.0540541\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 64.0000i 2.39682i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 28.0000i − 1.04568i
\(718\) 0 0
\(719\) −4.00000 −0.149175 −0.0745874 0.997214i \(-0.523764\pi\)
−0.0745874 + 0.997214i \(0.523764\pi\)
\(720\) 0 0
\(721\) − 16.0000i − 0.595871i
\(722\) 0 0
\(723\) 5.00000i 0.185952i
\(724\) 0 0
\(725\) 20.0000 0.742781
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 13.0000i 0.480166i 0.970752 + 0.240083i \(0.0771747\pi\)
−0.970752 + 0.240083i \(0.922825\pi\)
\(734\) 0 0
\(735\) − 27.0000i − 0.995910i
\(736\) 0 0
\(737\) 32.0000 1.17874
\(738\) 0 0
\(739\) − 16.0000i − 0.588570i −0.955718 0.294285i \(-0.904919\pi\)
0.955718 0.294285i \(-0.0950814\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 8.00000i − 0.293492i −0.989174 0.146746i \(-0.953120\pi\)
0.989174 0.146746i \(-0.0468799\pi\)
\(744\) 0 0
\(745\) −9.00000 −0.329734
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 32.0000i − 1.16925i
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) 0 0
\(759\) − 32.0000i − 1.16153i
\(760\) 0 0
\(761\) 42.0000i 1.52250i 0.648459 + 0.761249i \(0.275414\pi\)
−0.648459 + 0.761249i \(0.724586\pi\)
\(762\) 0 0
\(763\) 72.0000 2.60658
\(764\) 0 0
\(765\) 9.00000i 0.325396i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) − 34.0000i − 1.22607i −0.790055 0.613036i \(-0.789948\pi\)
0.790055 0.613036i \(-0.210052\pi\)
\(770\) 0 0
\(771\) 7.00000 0.252099
\(772\) 0 0
\(773\) − 38.0000i − 1.36677i −0.730061 0.683383i \(-0.760508\pi\)
0.730061 0.683383i \(-0.239492\pi\)
\(774\) 0 0
\(775\) − 32.0000i − 1.14947i
\(776\) 0 0
\(777\) −28.0000 −1.00449
\(778\) 0 0
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 5.00000 0.178685
\(784\) 0 0
\(785\) − 69.0000i − 2.46272i
\(786\) 0 0
\(787\) 16.0000i 0.570338i 0.958477 + 0.285169i \(0.0920498\pi\)
−0.958477 + 0.285169i \(0.907950\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 36.0000i 1.28001i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) − 15.0000i − 0.531995i
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 12.0000i 0.424529i
\(800\) 0 0
\(801\) − 6.00000i − 0.212000i
\(802\) 0 0
\(803\) −44.0000 −1.55273
\(804\) 0 0
\(805\) −96.0000 −3.38356
\(806\) 0 0
\(807\) 2.00000 0.0704033
\(808\) 0 0
\(809\) −9.00000 −0.316423 −0.158212 0.987405i \(-0.550573\pi\)
−0.158212 + 0.987405i \(0.550573\pi\)
\(810\) 0 0
\(811\) 24.0000i 0.842754i 0.906886 + 0.421377i \(0.138453\pi\)
−0.906886 + 0.421377i \(0.861547\pi\)
\(812\) 0 0
\(813\) − 20.0000i − 0.701431i
\(814\) 0 0
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) − 32.0000i − 1.11954i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 42.0000i 1.46581i 0.680331 + 0.732905i \(0.261836\pi\)
−0.680331 + 0.732905i \(0.738164\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) 16.0000i 0.557048i
\(826\) 0 0
\(827\) 36.0000i 1.25184i 0.779886 + 0.625921i \(0.215277\pi\)
−0.779886 + 0.625921i \(0.784723\pi\)
\(828\) 0 0
\(829\) −11.0000 −0.382046 −0.191023 0.981586i \(-0.561180\pi\)
−0.191023 + 0.981586i \(0.561180\pi\)
\(830\) 0 0
\(831\) 19.0000 0.659103
\(832\) 0 0
\(833\) 27.0000 0.935495
\(834\) 0 0
\(835\) −36.0000 −1.24583
\(836\) 0 0
\(837\) − 8.00000i − 0.276520i
\(838\) 0 0
\(839\) 20.0000i 0.690477i 0.938515 + 0.345238i \(0.112202\pi\)
−0.938515 + 0.345238i \(0.887798\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) 13.0000i 0.447744i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 20.0000i 0.687208i
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 56.0000i 1.91966i
\(852\) 0 0
\(853\) 19.0000i 0.650548i 0.945620 + 0.325274i \(0.105456\pi\)
−0.945620 + 0.325274i \(0.894544\pi\)
\(854\) 0 0
\(855\) 12.0000 0.410391
\(856\) 0 0
\(857\) 33.0000 1.12726 0.563629 0.826028i \(-0.309405\pi\)
0.563629 + 0.826028i \(0.309405\pi\)
\(858\) 0 0
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 0 0
\(861\) −36.0000 −1.22688
\(862\) 0 0
\(863\) − 44.0000i − 1.49778i −0.662696 0.748889i \(-0.730588\pi\)
0.662696 0.748889i \(-0.269412\pi\)
\(864\) 0 0
\(865\) 42.0000i 1.42804i
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) − 16.0000i − 0.542763i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 14.0000i 0.473828i
\(874\) 0 0
\(875\) −12.0000 −0.405674
\(876\) 0 0
\(877\) − 35.0000i − 1.18187i −0.806721 0.590933i \(-0.798760\pi\)
0.806721 0.590933i \(-0.201240\pi\)
\(878\) 0 0
\(879\) 9.00000i 0.303562i
\(880\) 0 0
\(881\) −15.0000 −0.505363 −0.252681 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(882\) 0 0
\(883\) −52.0000 −1.74994 −0.874970 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 0 0
\(885\) −12.0000 −0.403376
\(886\) 0 0
\(887\) 28.0000 0.940148 0.470074 0.882627i \(-0.344227\pi\)
0.470074 + 0.882627i \(0.344227\pi\)
\(888\) 0 0
\(889\) − 80.0000i − 2.68311i
\(890\) 0 0
\(891\) 4.00000i 0.134005i
\(892\) 0 0
\(893\) 16.0000 0.535420
\(894\) 0 0
\(895\) 48.0000i 1.60446i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 40.0000i − 1.33407i
\(900\) 0 0
\(901\) 15.0000 0.499722
\(902\) 0 0
\(903\) − 32.0000i − 1.06489i
\(904\) 0 0
\(905\) − 3.00000i − 0.0997234i
\(906\) 0 0
\(907\) 8.00000 0.265636 0.132818 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) 0 0
\(909\) −11.0000 −0.364847
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) − 15.0000i − 0.495885i
\(916\) 0 0
\(917\) − 32.0000i − 1.05673i
\(918\) 0 0
\(919\) −12.0000 −0.395843 −0.197922 0.980218i \(-0.563419\pi\)
−0.197922 + 0.980218i \(0.563419\pi\)
\(920\) 0 0
\(921\) − 8.00000i − 0.263609i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) − 28.0000i − 0.920634i
\(926\) 0 0
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) − 11.0000i − 0.360898i −0.983584 0.180449i \(-0.942245\pi\)
0.983584 0.180449i \(-0.0577551\pi\)
\(930\) 0 0
\(931\) − 36.0000i − 1.17985i
\(932\) 0 0
\(933\) −12.0000 −0.392862
\(934\) 0 0
\(935\) −36.0000 −1.17733
\(936\) 0 0
\(937\) 59.0000 1.92745 0.963723 0.266904i \(-0.0860008\pi\)
0.963723 + 0.266904i \(0.0860008\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 18.0000i 0.586783i 0.955992 + 0.293392i \(0.0947840\pi\)
−0.955992 + 0.293392i \(0.905216\pi\)
\(942\) 0 0
\(943\) 72.0000i 2.34464i
\(944\) 0 0
\(945\) 12.0000 0.390360
\(946\) 0 0
\(947\) 36.0000i 1.16984i 0.811090 + 0.584921i \(0.198875\pi\)
−0.811090 + 0.584921i \(0.801125\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) − 25.0000i − 0.810681i
\(952\) 0 0
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 0 0
\(955\) 24.0000i 0.776622i
\(956\) 0 0
\(957\) 20.0000i 0.646508i
\(958\) 0 0
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) −33.0000 −1.06452
\(962\) 0 0
\(963\) 8.00000 0.257796
\(964\) 0 0
\(965\) −39.0000 −1.25545
\(966\) 0 0
\(967\) 12.0000i 0.385894i 0.981209 + 0.192947i \(0.0618045\pi\)
−0.981209 + 0.192947i \(0.938195\pi\)
\(968\) 0 0
\(969\) 12.0000i 0.385496i
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 29.0000i 0.927792i 0.885890 + 0.463896i \(0.153549\pi\)
−0.885890 + 0.463896i \(0.846451\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) 18.0000i 0.574696i
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 78.0000 2.48529
\(986\) 0 0
\(987\) 16.0000 0.509286
\(988\) 0 0
\(989\) −64.0000 −2.03508
\(990\) 0 0
\(991\) −36.0000 −1.14358 −0.571789 0.820401i \(-0.693750\pi\)
−0.571789 + 0.820401i \(0.693750\pi\)
\(992\) 0 0
\(993\) 8.00000i 0.253872i
\(994\) 0 0
\(995\) − 24.0000i − 0.760851i
\(996\) 0 0
\(997\) 35.0000 1.10846 0.554231 0.832363i \(-0.313013\pi\)
0.554231 + 0.832363i \(0.313013\pi\)
\(998\) 0 0
\(999\) − 7.00000i − 0.221470i
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 4056.2.c.b.337.1 2
13.5 odd 4 4056.2.a.b.1.1 1
13.7 odd 12 312.2.q.c.289.1 yes 2
13.8 odd 4 4056.2.a.j.1.1 1
13.11 odd 12 312.2.q.c.217.1 2
13.12 even 2 inner 4056.2.c.b.337.2 2
39.11 even 12 936.2.t.b.217.1 2
39.20 even 12 936.2.t.b.289.1 2
52.7 even 12 624.2.q.e.289.1 2
52.11 even 12 624.2.q.e.529.1 2
52.31 even 4 8112.2.a.r.1.1 1
52.47 even 4 8112.2.a.bh.1.1 1
156.11 odd 12 1872.2.t.a.1153.1 2
156.59 odd 12 1872.2.t.a.289.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.2.q.c.217.1 2 13.11 odd 12
312.2.q.c.289.1 yes 2 13.7 odd 12
624.2.q.e.289.1 2 52.7 even 12
624.2.q.e.529.1 2 52.11 even 12
936.2.t.b.217.1 2 39.11 even 12
936.2.t.b.289.1 2 39.20 even 12
1872.2.t.a.289.1 2 156.59 odd 12
1872.2.t.a.1153.1 2 156.11 odd 12
4056.2.a.b.1.1 1 13.5 odd 4
4056.2.a.j.1.1 1 13.8 odd 4
4056.2.c.b.337.1 2 1.1 even 1 trivial
4056.2.c.b.337.2 2 13.12 even 2 inner
8112.2.a.r.1.1 1 52.31 even 4
8112.2.a.bh.1.1 1 52.47 even 4