Properties

Label 2704.2.f.b.337.2
Level $2704$
Weight $2$
Character 2704.337
Analytic conductor $21.592$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2704,2,Mod(337,2704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2704, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2704.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2704 = 2^{4} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2704.f (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: \(21.5915487066\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2704.337
Dual form 2704.2.f.b.337.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} +1.73205i q^{5} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} +1.73205i q^{5} +1.00000 q^{9} -3.46410i q^{15} -3.00000 q^{17} -3.46410i q^{19} +6.00000 q^{23} +2.00000 q^{25} +4.00000 q^{27} +3.00000 q^{29} +3.46410i q^{31} +8.66025i q^{37} -5.19615i q^{41} -8.00000 q^{43} +1.73205i q^{45} -3.46410i q^{47} +7.00000 q^{49} +6.00000 q^{51} -3.00000 q^{53} +6.92820i q^{57} +6.92820i q^{59} +1.00000 q^{61} -3.46410i q^{67} -12.0000 q^{69} +3.46410i q^{71} -1.73205i q^{73} -4.00000 q^{75} -4.00000 q^{79} -11.0000 q^{81} +13.8564i q^{83} -5.19615i q^{85} -6.00000 q^{87} -6.92820i q^{89} -6.92820i q^{93} +6.00000 q^{95} -6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} + 2 q^{9} - 6 q^{17} + 12 q^{23} + 4 q^{25} + 8 q^{27} + 6 q^{29} - 16 q^{43} + 14 q^{49} + 12 q^{51} - 6 q^{53} + 2 q^{61} - 24 q^{69} - 8 q^{75} - 8 q^{79} - 22 q^{81} - 12 q^{87} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1185\) \(2367\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 1.73205i 0.774597i 0.921954 + 0.387298i \(0.126592\pi\)
−0.921954 + 0.387298i \(0.873408\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) − 3.46410i − 0.894427i
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) − 3.46410i − 0.794719i −0.917663 0.397360i \(-0.869927\pi\)
0.917663 0.397360i \(-0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i 0.950382 + 0.311086i \(0.100693\pi\)
−0.950382 + 0.311086i \(0.899307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.66025i 1.42374i 0.702313 + 0.711868i \(0.252151\pi\)
−0.702313 + 0.711868i \(0.747849\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 5.19615i − 0.811503i −0.913984 0.405751i \(-0.867010\pi\)
0.913984 0.405751i \(-0.132990\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 1.73205i 0.258199i
\(46\) 0 0
\(47\) − 3.46410i − 0.505291i −0.967559 0.252646i \(-0.918699\pi\)
0.967559 0.252646i \(-0.0813007\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.92820i 0.917663i
\(58\) 0 0
\(59\) 6.92820i 0.901975i 0.892530 + 0.450988i \(0.148928\pi\)
−0.892530 + 0.450988i \(0.851072\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.46410i − 0.423207i −0.977356 0.211604i \(-0.932131\pi\)
0.977356 0.211604i \(-0.0678686\pi\)
\(68\) 0 0
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) 3.46410i 0.411113i 0.978645 + 0.205557i \(0.0659005\pi\)
−0.978645 + 0.205557i \(0.934100\pi\)
\(72\) 0 0
\(73\) − 1.73205i − 0.202721i −0.994850 0.101361i \(-0.967680\pi\)
0.994850 0.101361i \(-0.0323196\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 13.8564i 1.52094i 0.649374 + 0.760469i \(0.275031\pi\)
−0.649374 + 0.760469i \(0.724969\pi\)
\(84\) 0 0
\(85\) − 5.19615i − 0.563602i
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) − 6.92820i − 0.734388i −0.930144 0.367194i \(-0.880318\pi\)
0.930144 0.367194i \(-0.119682\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 6.92820i − 0.718421i
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) − 6.92820i − 0.703452i −0.936103 0.351726i \(-0.885595\pi\)
0.936103 0.351726i \(-0.114405\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) 10.0000 0.985329 0.492665 0.870219i \(-0.336023\pi\)
0.492665 + 0.870219i \(0.336023\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 0 0
\(109\) 13.8564i 1.32720i 0.748086 + 0.663602i \(0.230973\pi\)
−0.748086 + 0.663602i \(0.769027\pi\)
\(110\) 0 0
\(111\) − 17.3205i − 1.64399i
\(112\) 0 0
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) 10.3923i 0.969087i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 10.3923i 0.937043i
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 16.0000 1.40872
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 6.92820i 0.596285i
\(136\) 0 0
\(137\) 15.5885i 1.33181i 0.746036 + 0.665906i \(0.231955\pi\)
−0.746036 + 0.665906i \(0.768045\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 6.92820i 0.583460i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 5.19615i 0.431517i
\(146\) 0 0
\(147\) −14.0000 −1.15470
\(148\) 0 0
\(149\) 19.0526i 1.56085i 0.625252 + 0.780423i \(0.284996\pi\)
−0.625252 + 0.780423i \(0.715004\pi\)
\(150\) 0 0
\(151\) 17.3205i 1.40952i 0.709444 + 0.704761i \(0.248946\pi\)
−0.709444 + 0.704761i \(0.751054\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 20.7846i 1.62798i 0.580881 + 0.813988i \(0.302708\pi\)
−0.580881 + 0.813988i \(0.697292\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.8564i 1.07224i 0.844141 + 0.536120i \(0.180111\pi\)
−0.844141 + 0.536120i \(0.819889\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) − 3.46410i − 0.264906i
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 13.8564i − 1.04151i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 11.0000 0.817624 0.408812 0.912619i \(-0.365943\pi\)
0.408812 + 0.912619i \(0.365943\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) −15.0000 −1.10282
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 0 0
\(193\) − 5.19615i − 0.374027i −0.982357 0.187014i \(-0.940119\pi\)
0.982357 0.187014i \(-0.0598809\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.8564i 0.987228i 0.869681 + 0.493614i \(0.164324\pi\)
−0.869681 + 0.493614i \(0.835676\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) 6.92820i 0.488678i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 9.00000 0.628587
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 0 0
\(213\) − 6.92820i − 0.474713i
\(214\) 0 0
\(215\) − 13.8564i − 0.944999i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 3.46410i 0.234082i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 10.3923i 0.695920i 0.937509 + 0.347960i \(0.113126\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 0 0
\(225\) 2.00000 0.133333
\(226\) 0 0
\(227\) 24.2487i 1.60944i 0.593652 + 0.804722i \(0.297686\pi\)
−0.593652 + 0.804722i \(0.702314\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) − 20.7846i − 1.34444i −0.740349 0.672222i \(-0.765340\pi\)
0.740349 0.672222i \(-0.234660\pi\)
\(240\) 0 0
\(241\) 1.73205i 0.111571i 0.998443 + 0.0557856i \(0.0177663\pi\)
−0.998443 + 0.0557856i \(0.982234\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) 12.1244i 0.774597i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) − 27.7128i − 1.75623i
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 10.3923i 0.650791i
\(256\) 0 0
\(257\) 3.00000 0.187135 0.0935674 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) − 5.19615i − 0.319197i
\(266\) 0 0
\(267\) 13.8564i 0.847998i
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) − 20.7846i − 1.26258i −0.775549 0.631288i \(-0.782527\pi\)
0.775549 0.631288i \(-0.217473\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −7.00000 −0.420589 −0.210295 0.977638i \(-0.567442\pi\)
−0.210295 + 0.977638i \(0.567442\pi\)
\(278\) 0 0
\(279\) 3.46410i 0.207390i
\(280\) 0 0
\(281\) 22.5167i 1.34323i 0.740900 + 0.671616i \(0.234399\pi\)
−0.740900 + 0.671616i \(0.765601\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) −12.0000 −0.710819
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 13.8564i 0.812277i
\(292\) 0 0
\(293\) − 5.19615i − 0.303562i −0.988414 0.151781i \(-0.951499\pi\)
0.988414 0.151781i \(-0.0485009\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) 1.73205i 0.0991769i
\(306\) 0 0
\(307\) − 17.3205i − 0.988534i −0.869310 0.494267i \(-0.835437\pi\)
0.869310 0.494267i \(-0.164563\pi\)
\(308\) 0 0
\(309\) −20.0000 −1.13776
\(310\) 0 0
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.19615i 0.291845i 0.989296 + 0.145922i \(0.0466150\pi\)
−0.989296 + 0.145922i \(0.953385\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 10.3923i 0.578243i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 27.7128i − 1.53252i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 27.7128i − 1.52323i −0.648027 0.761617i \(-0.724406\pi\)
0.648027 0.761617i \(-0.275594\pi\)
\(332\) 0 0
\(333\) 8.66025i 0.474579i
\(334\) 0 0
\(335\) 6.00000 0.327815
\(336\) 0 0
\(337\) −23.0000 −1.25289 −0.626445 0.779466i \(-0.715491\pi\)
−0.626445 + 0.779466i \(0.715491\pi\)
\(338\) 0 0
\(339\) 30.0000 1.62938
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) − 20.7846i − 1.11901i
\(346\) 0 0
\(347\) 30.0000 1.61048 0.805242 0.592946i \(-0.202035\pi\)
0.805242 + 0.592946i \(0.202035\pi\)
\(348\) 0 0
\(349\) 13.8564i 0.741716i 0.928689 + 0.370858i \(0.120936\pi\)
−0.928689 + 0.370858i \(0.879064\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 32.9090i 1.75157i 0.482704 + 0.875784i \(0.339655\pi\)
−0.482704 + 0.875784i \(0.660345\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 6.92820i − 0.365657i −0.983145 0.182828i \(-0.941475\pi\)
0.983145 0.182828i \(-0.0585252\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) 0 0
\(363\) −22.0000 −1.15470
\(364\) 0 0
\(365\) 3.00000 0.157027
\(366\) 0 0
\(367\) 22.0000 1.14839 0.574195 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) 0 0
\(369\) − 5.19615i − 0.270501i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 19.0000 0.983783 0.491891 0.870657i \(-0.336306\pi\)
0.491891 + 0.870657i \(0.336306\pi\)
\(374\) 0 0
\(375\) − 24.2487i − 1.25220i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 24.2487i 1.24557i 0.782392 + 0.622786i \(0.213999\pi\)
−0.782392 + 0.622786i \(0.786001\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) − 20.7846i − 1.06204i −0.847358 0.531022i \(-0.821808\pi\)
0.847358 0.531022i \(-0.178192\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.00000 −0.406663
\(388\) 0 0
\(389\) −9.00000 −0.456318 −0.228159 0.973624i \(-0.573271\pi\)
−0.228159 + 0.973624i \(0.573271\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 36.0000 1.81596
\(394\) 0 0
\(395\) − 6.92820i − 0.348596i
\(396\) 0 0
\(397\) 13.8564i 0.695433i 0.937600 + 0.347717i \(0.113043\pi\)
−0.937600 + 0.347717i \(0.886957\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 1.73205i − 0.0864945i −0.999064 0.0432472i \(-0.986230\pi\)
0.999064 0.0432472i \(-0.0137703\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 19.0526i − 0.946729i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) − 15.5885i − 0.770800i −0.922750 0.385400i \(-0.874064\pi\)
0.922750 0.385400i \(-0.125936\pi\)
\(410\) 0 0
\(411\) − 31.1769i − 1.53784i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −24.0000 −1.17811
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 0 0
\(421\) − 15.5885i − 0.759735i −0.925041 0.379867i \(-0.875970\pi\)
0.925041 0.379867i \(-0.124030\pi\)
\(422\) 0 0
\(423\) − 3.46410i − 0.168430i
\(424\) 0 0
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.92820i 0.333720i 0.985981 + 0.166860i \(0.0533628\pi\)
−0.985981 + 0.166860i \(0.946637\pi\)
\(432\) 0 0
\(433\) −17.0000 −0.816968 −0.408484 0.912766i \(-0.633942\pi\)
−0.408484 + 0.912766i \(0.633942\pi\)
\(434\) 0 0
\(435\) − 10.3923i − 0.498273i
\(436\) 0 0
\(437\) − 20.7846i − 0.994263i
\(438\) 0 0
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 7.00000 0.333333
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) 0 0
\(447\) − 38.1051i − 1.80231i
\(448\) 0 0
\(449\) 6.92820i 0.326962i 0.986546 + 0.163481i \(0.0522723\pi\)
−0.986546 + 0.163481i \(0.947728\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) − 34.6410i − 1.62758i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.73205i 0.0810219i 0.999179 + 0.0405110i \(0.0128986\pi\)
−0.999179 + 0.0405110i \(0.987101\pi\)
\(458\) 0 0
\(459\) −12.0000 −0.560112
\(460\) 0 0
\(461\) − 22.5167i − 1.04871i −0.851501 0.524353i \(-0.824307\pi\)
0.851501 0.524353i \(-0.175693\pi\)
\(462\) 0 0
\(463\) 13.8564i 0.643962i 0.946746 + 0.321981i \(0.104349\pi\)
−0.946746 + 0.321981i \(0.895651\pi\)
\(464\) 0 0
\(465\) 12.0000 0.556487
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 26.0000 1.19802
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) − 6.92820i − 0.317888i
\(476\) 0 0
\(477\) −3.00000 −0.137361
\(478\) 0 0
\(479\) − 24.2487i − 1.10795i −0.832533 0.553976i \(-0.813110\pi\)
0.832533 0.553976i \(-0.186890\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.0000 0.544892
\(486\) 0 0
\(487\) − 6.92820i − 0.313947i −0.987603 0.156973i \(-0.949826\pi\)
0.987603 0.156973i \(-0.0501737\pi\)
\(488\) 0 0
\(489\) − 41.5692i − 1.87983i
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 31.1769i 1.39567i 0.716258 + 0.697835i \(0.245853\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) − 27.7128i − 1.23812i
\(502\) 0 0
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) − 5.19615i − 0.231226i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 19.0526i 0.844490i 0.906482 + 0.422245i \(0.138758\pi\)
−0.906482 + 0.422245i \(0.861242\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) − 13.8564i − 0.611775i
\(514\) 0 0
\(515\) 17.3205i 0.763233i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) 9.00000 0.394297 0.197149 0.980374i \(-0.436832\pi\)
0.197149 + 0.980374i \(0.436832\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 10.3923i − 0.452696i
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 6.92820i 0.300658i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 10.3923i − 0.449299i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) − 29.4449i − 1.26593i −0.774179 0.632967i \(-0.781837\pi\)
0.774179 0.632967i \(-0.218163\pi\)
\(542\) 0 0
\(543\) −22.0000 −0.944110
\(544\) 0 0
\(545\) −24.0000 −1.02805
\(546\) 0 0
\(547\) 22.0000 0.940652 0.470326 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) 0 0
\(549\) 1.00000 0.0426790
\(550\) 0 0
\(551\) − 10.3923i − 0.442727i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 30.0000 1.27343
\(556\) 0 0
\(557\) − 15.5885i − 0.660504i −0.943893 0.330252i \(-0.892866\pi\)
0.943893 0.330252i \(-0.107134\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) − 25.9808i − 1.09302i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −42.0000 −1.76073 −0.880366 0.474295i \(-0.842703\pi\)
−0.880366 + 0.474295i \(0.842703\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 36.0000 1.50392
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) 19.0526i 0.793168i 0.917998 + 0.396584i \(0.129805\pi\)
−0.917998 + 0.396584i \(0.870195\pi\)
\(578\) 0 0
\(579\) 10.3923i 0.431889i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 20.7846i 0.857873i 0.903335 + 0.428936i \(0.141112\pi\)
−0.903335 + 0.428936i \(0.858888\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) − 27.7128i − 1.13995i
\(592\) 0 0
\(593\) 25.9808i 1.06690i 0.845831 + 0.533451i \(0.179105\pi\)
−0.845831 + 0.533451i \(0.820895\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) 25.0000 1.01977 0.509886 0.860242i \(-0.329688\pi\)
0.509886 + 0.860242i \(0.329688\pi\)
\(602\) 0 0
\(603\) − 3.46410i − 0.141069i
\(604\) 0 0
\(605\) 19.0526i 0.774597i
\(606\) 0 0
\(607\) 34.0000 1.38002 0.690009 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 12.1244i − 0.489698i −0.969561 0.244849i \(-0.921262\pi\)
0.969561 0.244849i \(-0.0787384\pi\)
\(614\) 0 0
\(615\) −18.0000 −0.725830
\(616\) 0 0
\(617\) 22.5167i 0.906487i 0.891387 + 0.453243i \(0.149733\pi\)
−0.891387 + 0.453243i \(0.850267\pi\)
\(618\) 0 0
\(619\) − 20.7846i − 0.835404i −0.908584 0.417702i \(-0.862836\pi\)
0.908584 0.417702i \(-0.137164\pi\)
\(620\) 0 0
\(621\) 24.0000 0.963087
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 25.9808i − 1.03592i
\(630\) 0 0
\(631\) − 48.4974i − 1.93065i −0.261048 0.965326i \(-0.584068\pi\)
0.261048 0.965326i \(-0.415932\pi\)
\(632\) 0 0
\(633\) 20.0000 0.794929
\(634\) 0 0
\(635\) 3.46410i 0.137469i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 3.46410i 0.137038i
\(640\) 0 0
\(641\) 33.0000 1.30342 0.651711 0.758468i \(-0.274052\pi\)
0.651711 + 0.758468i \(0.274052\pi\)
\(642\) 0 0
\(643\) 13.8564i 0.546443i 0.961951 + 0.273222i \(0.0880892\pi\)
−0.961951 + 0.273222i \(0.911911\pi\)
\(644\) 0 0
\(645\) 27.7128i 1.09119i
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) − 31.1769i − 1.21818i
\(656\) 0 0
\(657\) − 1.73205i − 0.0675737i
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) − 46.7654i − 1.81896i −0.415745 0.909481i \(-0.636479\pi\)
0.415745 0.909481i \(-0.363521\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 18.0000 0.696963
\(668\) 0 0
\(669\) − 20.7846i − 0.803579i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) 0 0
\(675\) 8.00000 0.307920
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) − 48.4974i − 1.85843i
\(682\) 0 0
\(683\) 24.2487i 0.927851i 0.885874 + 0.463926i \(0.153559\pi\)
−0.885874 + 0.463926i \(0.846441\pi\)
\(684\) 0 0
\(685\) −27.0000 −1.03162
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 13.8564i − 0.527123i −0.964643 0.263561i \(-0.915103\pi\)
0.964643 0.263561i \(-0.0848971\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.92820i 0.262802i
\(696\) 0 0
\(697\) 15.5885i 0.590455i
\(698\) 0 0
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) 30.0000 1.13147
\(704\) 0 0
\(705\) −12.0000 −0.451946
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 5.19615i 0.195146i 0.995228 + 0.0975728i \(0.0311079\pi\)
−0.995228 + 0.0975728i \(0.968892\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 20.7846i 0.778390i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 41.5692i 1.55243i
\(718\) 0 0
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 3.46410i − 0.128831i
\(724\) 0 0
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) − 12.1244i − 0.447823i −0.974609 0.223912i \(-0.928117\pi\)
0.974609 0.223912i \(-0.0718827\pi\)
\(734\) 0 0
\(735\) − 24.2487i − 0.894427i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 20.7846i 0.764574i 0.924044 + 0.382287i \(0.124863\pi\)
−0.924044 + 0.382287i \(0.875137\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 34.6410i 1.27086i 0.772160 + 0.635428i \(0.219176\pi\)
−0.772160 + 0.635428i \(0.780824\pi\)
\(744\) 0 0
\(745\) −33.0000 −1.20903
\(746\) 0 0
\(747\) 13.8564i 0.506979i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 0 0
\(753\) −36.0000 −1.31191
\(754\) 0 0
\(755\) −30.0000 −1.09181
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 34.6410i − 1.25574i −0.778320 0.627868i \(-0.783928\pi\)
0.778320 0.627868i \(-0.216072\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) − 5.19615i − 0.187867i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 6.92820i 0.249837i 0.992167 + 0.124919i \(0.0398670\pi\)
−0.992167 + 0.124919i \(0.960133\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 0 0
\(773\) 34.6410i 1.24595i 0.782241 + 0.622975i \(0.214076\pi\)
−0.782241 + 0.622975i \(0.785924\pi\)
\(774\) 0 0
\(775\) 6.92820i 0.248868i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −18.0000 −0.644917
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 12.0000 0.428845
\(784\) 0 0
\(785\) − 22.5167i − 0.803654i
\(786\) 0 0
\(787\) 38.1051i 1.35830i 0.733999 + 0.679150i \(0.237652\pi\)
−0.733999 + 0.679150i \(0.762348\pi\)
\(788\) 0 0
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 10.3923i 0.368577i
\(796\) 0 0
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) 10.3923i 0.367653i
\(800\) 0 0
\(801\) − 6.92820i − 0.244796i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 12.0000 0.422420
\(808\) 0 0
\(809\) 33.0000 1.16022 0.580109 0.814539i \(-0.303010\pi\)
0.580109 + 0.814539i \(0.303010\pi\)
\(810\) 0 0
\(811\) − 38.1051i − 1.33805i −0.743239 0.669026i \(-0.766712\pi\)
0.743239 0.669026i \(-0.233288\pi\)
\(812\) 0 0
\(813\) 41.5692i 1.45790i
\(814\) 0 0
\(815\) −36.0000 −1.26102
\(816\) 0 0
\(817\) 27.7128i 0.969549i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 41.5692i − 1.45078i −0.688340 0.725388i \(-0.741660\pi\)
0.688340 0.725388i \(-0.258340\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 20.7846i 0.722752i 0.932420 + 0.361376i \(0.117693\pi\)
−0.932420 + 0.361376i \(0.882307\pi\)
\(828\) 0 0
\(829\) 25.0000 0.868286 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) 0 0
\(831\) 14.0000 0.485655
\(832\) 0 0
\(833\) −21.0000 −0.727607
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) 0 0
\(837\) 13.8564i 0.478947i
\(838\) 0 0
\(839\) 45.0333i 1.55472i 0.629054 + 0.777361i \(0.283442\pi\)
−0.629054 + 0.777361i \(0.716558\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) − 45.0333i − 1.55103i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) 51.9615i 1.78122i
\(852\) 0 0
\(853\) − 25.9808i − 0.889564i −0.895639 0.444782i \(-0.853281\pi\)
0.895639 0.444782i \(-0.146719\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) 0 0
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 27.7128i − 0.943355i −0.881771 0.471678i \(-0.843649\pi\)
0.881771 0.471678i \(-0.156351\pi\)
\(864\) 0 0
\(865\) 10.3923i 0.353349i
\(866\) 0 0
\(867\) 16.0000 0.543388
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) − 6.92820i − 0.234484i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 12.1244i − 0.409410i −0.978824 0.204705i \(-0.934376\pi\)
0.978824 0.204705i \(-0.0656236\pi\)
\(878\) 0 0
\(879\) 10.3923i 0.350524i
\(880\) 0 0
\(881\) 27.0000 0.909653 0.454827 0.890580i \(-0.349701\pi\)
0.454827 + 0.890580i \(0.349701\pi\)
\(882\) 0 0
\(883\) −10.0000 −0.336527 −0.168263 0.985742i \(-0.553816\pi\)
−0.168263 + 0.985742i \(0.553816\pi\)
\(884\) 0 0
\(885\) 24.0000 0.806751
\(886\) 0 0
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 10.3923i 0.346603i
\(900\) 0 0
\(901\) 9.00000 0.299833
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19.0526i 0.633328i
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 0 0
\(909\) −3.00000 −0.0995037
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) − 3.46410i − 0.114520i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −22.0000 −0.725713 −0.362857 0.931845i \(-0.618198\pi\)
−0.362857 + 0.931845i \(0.618198\pi\)
\(920\) 0 0
\(921\) 34.6410i 1.14146i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 17.3205i 0.569495i
\(926\) 0 0
\(927\) 10.0000 0.328443
\(928\) 0 0
\(929\) 46.7654i 1.53432i 0.641455 + 0.767161i \(0.278331\pi\)
−0.641455 + 0.767161i \(0.721669\pi\)
\(930\) 0 0
\(931\) − 24.2487i − 0.794719i
\(932\) 0 0
\(933\) −60.0000 −1.96431
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 7.00000 0.228680 0.114340 0.993442i \(-0.463525\pi\)
0.114340 + 0.993442i \(0.463525\pi\)
\(938\) 0 0
\(939\) −20.0000 −0.652675
\(940\) 0 0
\(941\) − 20.7846i − 0.677559i −0.940866 0.338779i \(-0.889986\pi\)
0.940866 0.338779i \(-0.110014\pi\)
\(942\) 0 0
\(943\) − 31.1769i − 1.01526i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 17.3205i 0.562841i 0.959585 + 0.281420i \(0.0908056\pi\)
−0.959585 + 0.281420i \(0.909194\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) − 10.3923i − 0.336994i
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) − 31.1769i − 1.00886i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 0 0
\(965\) 9.00000 0.289720
\(966\) 0 0
\(967\) 58.8897i 1.89377i 0.321578 + 0.946883i \(0.395787\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(968\) 0 0
\(969\) − 20.7846i − 0.667698i
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 43.3013i 1.38533i 0.721259 + 0.692665i \(0.243564\pi\)
−0.721259 + 0.692665i \(0.756436\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 13.8564i 0.442401i
\(982\) 0 0
\(983\) 51.9615i 1.65732i 0.559756 + 0.828658i \(0.310895\pi\)
−0.559756 + 0.828658i \(0.689105\pi\)
\(984\) 0 0
\(985\) −24.0000 −0.764704
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) −2.00000 −0.0635321 −0.0317660 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) 0 0
\(993\) 55.4256i 1.75888i
\(994\) 0 0
\(995\) 3.46410i 0.109819i
\(996\) 0 0
\(997\) 17.0000 0.538395 0.269198 0.963085i \(-0.413241\pi\)
0.269198 + 0.963085i \(0.413241\pi\)
\(998\) 0 0
\(999\) 34.6410i 1.09599i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2704.2.f.b.337.2 2
4.3 odd 2 169.2.b.a.168.1 2
12.11 even 2 1521.2.b.a.1351.2 2
13.5 odd 4 2704.2.a.o.1.2 2
13.8 odd 4 2704.2.a.o.1.1 2
13.9 even 3 208.2.w.b.49.1 2
13.10 even 6 208.2.w.b.17.1 2
13.12 even 2 inner 2704.2.f.b.337.1 2
39.23 odd 6 1872.2.by.d.433.1 2
39.35 odd 6 1872.2.by.d.1297.1 2
52.3 odd 6 169.2.e.a.147.1 2
52.7 even 12 169.2.c.a.146.1 4
52.11 even 12 169.2.c.a.22.1 4
52.15 even 12 169.2.c.a.22.2 4
52.19 even 12 169.2.c.a.146.2 4
52.23 odd 6 13.2.e.a.4.1 2
52.31 even 4 169.2.a.a.1.1 2
52.35 odd 6 13.2.e.a.10.1 yes 2
52.43 odd 6 169.2.e.a.23.1 2
52.47 even 4 169.2.a.a.1.2 2
52.51 odd 2 169.2.b.a.168.2 2
104.35 odd 6 832.2.w.d.257.1 2
104.61 even 6 832.2.w.a.257.1 2
104.75 odd 6 832.2.w.d.641.1 2
104.101 even 6 832.2.w.a.641.1 2
156.23 even 6 117.2.q.c.82.1 2
156.35 even 6 117.2.q.c.10.1 2
156.47 odd 4 1521.2.a.k.1.1 2
156.83 odd 4 1521.2.a.k.1.2 2
156.155 even 2 1521.2.b.a.1351.1 2
260.23 even 12 325.2.m.a.199.1 4
260.87 even 12 325.2.m.a.49.1 4
260.99 even 4 4225.2.a.v.1.1 2
260.127 even 12 325.2.m.a.199.2 4
260.139 odd 6 325.2.n.a.101.1 2
260.179 odd 6 325.2.n.a.251.1 2
260.239 even 4 4225.2.a.v.1.2 2
260.243 even 12 325.2.m.a.49.2 4
364.23 odd 6 637.2.k.a.459.1 2
364.75 even 6 637.2.k.c.459.1 2
364.83 odd 4 8281.2.a.q.1.1 2
364.87 even 6 637.2.k.c.569.1 2
364.139 even 6 637.2.q.a.491.1 2
364.179 odd 6 637.2.u.c.30.1 2
364.191 odd 6 637.2.u.c.361.1 2
364.243 even 6 637.2.u.b.361.1 2
364.283 even 6 637.2.u.b.30.1 2
364.307 odd 4 8281.2.a.q.1.2 2
364.335 even 6 637.2.q.a.589.1 2
364.347 odd 6 637.2.k.a.569.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.2.e.a.4.1 2 52.23 odd 6
13.2.e.a.10.1 yes 2 52.35 odd 6
117.2.q.c.10.1 2 156.35 even 6
117.2.q.c.82.1 2 156.23 even 6
169.2.a.a.1.1 2 52.31 even 4
169.2.a.a.1.2 2 52.47 even 4
169.2.b.a.168.1 2 4.3 odd 2
169.2.b.a.168.2 2 52.51 odd 2
169.2.c.a.22.1 4 52.11 even 12
169.2.c.a.22.2 4 52.15 even 12
169.2.c.a.146.1 4 52.7 even 12
169.2.c.a.146.2 4 52.19 even 12
169.2.e.a.23.1 2 52.43 odd 6
169.2.e.a.147.1 2 52.3 odd 6
208.2.w.b.17.1 2 13.10 even 6
208.2.w.b.49.1 2 13.9 even 3
325.2.m.a.49.1 4 260.87 even 12
325.2.m.a.49.2 4 260.243 even 12
325.2.m.a.199.1 4 260.23 even 12
325.2.m.a.199.2 4 260.127 even 12
325.2.n.a.101.1 2 260.139 odd 6
325.2.n.a.251.1 2 260.179 odd 6
637.2.k.a.459.1 2 364.23 odd 6
637.2.k.a.569.1 2 364.347 odd 6
637.2.k.c.459.1 2 364.75 even 6
637.2.k.c.569.1 2 364.87 even 6
637.2.q.a.491.1 2 364.139 even 6
637.2.q.a.589.1 2 364.335 even 6
637.2.u.b.30.1 2 364.283 even 6
637.2.u.b.361.1 2 364.243 even 6
637.2.u.c.30.1 2 364.179 odd 6
637.2.u.c.361.1 2 364.191 odd 6
832.2.w.a.257.1 2 104.61 even 6
832.2.w.a.641.1 2 104.101 even 6
832.2.w.d.257.1 2 104.35 odd 6
832.2.w.d.641.1 2 104.75 odd 6
1521.2.a.k.1.1 2 156.47 odd 4
1521.2.a.k.1.2 2 156.83 odd 4
1521.2.b.a.1351.1 2 156.155 even 2
1521.2.b.a.1351.2 2 12.11 even 2
1872.2.by.d.433.1 2 39.23 odd 6
1872.2.by.d.1297.1 2 39.35 odd 6
2704.2.a.o.1.1 2 13.8 odd 4
2704.2.a.o.1.2 2 13.5 odd 4
2704.2.f.b.337.1 2 13.12 even 2 inner
2704.2.f.b.337.2 2 1.1 even 1 trivial
4225.2.a.v.1.1 2 260.99 even 4
4225.2.a.v.1.2 2 260.239 even 4
8281.2.a.q.1.1 2 364.83 odd 4
8281.2.a.q.1.2 2 364.307 odd 4