Properties

Label 4056.2.c.i.337.2
Level $4056$
Weight $2$
Character 4056.337
Analytic conductor $32.387$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 312)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.00000i q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.00000i q^{5} +1.00000 q^{9} +3.00000i q^{15} +1.00000 q^{17} -4.00000 q^{23} -4.00000 q^{25} +1.00000 q^{27} +3.00000 q^{29} +8.00000i q^{31} +5.00000i q^{37} +3.00000i q^{41} -4.00000 q^{43} +3.00000i q^{45} +8.00000i q^{47} +7.00000 q^{49} +1.00000 q^{51} -13.0000 q^{53} -12.0000i q^{59} +15.0000 q^{61} +12.0000i q^{67} -4.00000 q^{69} +8.00000i q^{71} -3.00000i q^{73} -4.00000 q^{75} -4.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +3.00000i q^{85} +3.00000 q^{87} -10.0000i q^{89} +8.00000i q^{93} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{9} + 2 q^{17} - 8 q^{23} - 8 q^{25} + 2 q^{27} + 6 q^{29} - 8 q^{43} + 14 q^{49} + 2 q^{51} - 26 q^{53} + 30 q^{61} - 8 q^{69} - 8 q^{75} - 8 q^{79} + 2 q^{81} + 6 q^{87}+O(q^{100}) \) 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 \(0.234058\pi\)
−0.741620 + 0.670820i \(0.765942\pi\)
\(6\) 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
\(15\) 3.00000i 0.774597i
\(16\) 0 0
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 8.00000i 1.43684i 0.695608 + 0.718421i \(0.255135\pi\)
−0.695608 + 0.718421i \(0.744865\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00000i 0.821995i 0.911636 + 0.410997i \(0.134819\pi\)
−0.911636 + 0.410997i \(0.865181\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000i 0.468521i 0.972174 + 0.234261i \(0.0752669\pi\)
−0.972174 + 0.234261i \(0.924733\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 3.00000i 0.447214i
\(46\) 0 0
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 0 0
\(53\) −13.0000 −1.78569 −0.892844 0.450367i \(-0.851293\pi\)
−0.892844 + 0.450367i \(0.851293\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 12.0000i − 1.56227i −0.624364 0.781133i \(-0.714642\pi\)
0.624364 0.781133i \(-0.285358\pi\)
\(60\) 0 0
\(61\) 15.0000 1.92055 0.960277 0.279050i \(-0.0900195\pi\)
0.960277 + 0.279050i \(0.0900195\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 8.00000i 0.949425i 0.880141 + 0.474713i \(0.157448\pi\)
−0.880141 + 0.474713i \(0.842552\pi\)
\(72\) 0 0
\(73\) − 3.00000i − 0.351123i −0.984468 0.175562i \(-0.943826\pi\)
0.984468 0.175562i \(-0.0561742\pi\)
\(74\) 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
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 3.00000i 0.325396i
\(86\) 0 0
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) − 10.0000i − 1.06000i −0.847998 0.529999i \(-0.822192\pi\)
0.847998 0.529999i \(-0.177808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.00000i 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 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
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −20.0000 −1.93347 −0.966736 0.255774i \(-0.917670\pi\)
−0.966736 + 0.255774i \(0.917670\pi\)
\(108\) 0 0
\(109\) − 18.0000i − 1.72409i −0.506834 0.862044i \(-0.669184\pi\)
0.506834 0.862044i \(-0.330816\pi\)
\(110\) 0 0
\(111\) 5.00000i 0.474579i
\(112\) 0 0
\(113\) −13.0000 −1.22294 −0.611469 0.791269i \(-0.709421\pi\)
−0.611469 + 0.791269i \(0.709421\pi\)
\(114\) 0 0
\(115\) − 12.0000i − 1.11901i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 3.00000i 0.270501i
\(124\) 0 0
\(125\) 3.00000i 0.268328i
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 3.00000i 0.258199i
\(136\) 0 0
\(137\) 5.00000i 0.427179i 0.976924 + 0.213589i \(0.0685155\pi\)
−0.976924 + 0.213589i \(0.931485\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 8.00000i 0.673722i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 9.00000i 0.747409i
\(146\) 0 0
\(147\) 7.00000 0.577350
\(148\) 0 0
\(149\) 19.0000i 1.55654i 0.627929 + 0.778270i \(0.283903\pi\)
−0.627929 + 0.778270i \(0.716097\pi\)
\(150\) 0 0
\(151\) 4.00000i 0.325515i 0.986666 + 0.162758i \(0.0520389\pi\)
−0.986666 + 0.162758i \(0.947961\pi\)
\(152\) 0 0
\(153\) 1.00000 0.0808452
\(154\) 0 0
\(155\) −24.0000 −1.92773
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 0 0
\(159\) −13.0000 −1.03097
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 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
\(171\) 0 0
\(172\) 0 0
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 12.0000i − 0.901975i
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −11.0000 −0.817624 −0.408812 0.912619i \(-0.634057\pi\)
−0.408812 + 0.912619i \(0.634057\pi\)
\(182\) 0 0
\(183\) 15.0000 1.10883
\(184\) 0 0
\(185\) −15.0000 −1.10282
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) − 11.0000i − 0.791797i −0.918294 0.395899i \(-0.870433\pi\)
0.918294 0.395899i \(-0.129567\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 12.0000i 0.846415i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −9.00000 −0.628587
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) 8.00000i 0.548151i
\(214\) 0 0
\(215\) − 12.0000i − 0.818393i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) − 3.00000i − 0.202721i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 8.00000i 0.535720i 0.963458 + 0.267860i \(0.0863164\pi\)
−0.963458 + 0.267860i \(0.913684\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) − 8.00000i − 0.530979i −0.964114 0.265489i \(-0.914466\pi\)
0.964114 0.265489i \(-0.0855335\pi\)
\(228\) 0 0
\(229\) 10.0000i 0.660819i 0.943838 + 0.330409i \(0.107187\pi\)
−0.943838 + 0.330409i \(0.892813\pi\)
\(230\) 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
\(235\) −24.0000 −1.56559
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) − 8.00000i − 0.517477i −0.965947 0.258738i \(-0.916693\pi\)
0.965947 0.258738i \(-0.0833068\pi\)
\(240\) 0 0
\(241\) − 11.0000i − 0.708572i −0.935137 0.354286i \(-0.884724\pi\)
0.935137 0.354286i \(-0.115276\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 21.0000i 1.34164i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 12.0000i 0.760469i
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 3.00000i 0.187867i
\(256\) 0 0
\(257\) 5.00000 0.311891 0.155946 0.987766i \(-0.450158\pi\)
0.155946 + 0.987766i \(0.450158\pi\)
\(258\) 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
\(265\) − 39.0000i − 2.39575i
\(266\) 0 0
\(267\) − 10.0000i − 0.611990i
\(268\) 0 0
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) − 28.0000i − 1.70088i −0.526073 0.850439i \(-0.676336\pi\)
0.526073 0.850439i \(-0.323664\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 25.0000 1.50210 0.751052 0.660243i \(-0.229547\pi\)
0.751052 + 0.660243i \(0.229547\pi\)
\(278\) 0 0
\(279\) 8.00000i 0.478947i
\(280\) 0 0
\(281\) − 15.0000i − 0.894825i −0.894328 0.447412i \(-0.852346\pi\)
0.894328 0.447412i \(-0.147654\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 2.00000i 0.117242i
\(292\) 0 0
\(293\) 1.00000i 0.0584206i 0.999573 + 0.0292103i \(0.00929925\pi\)
−0.999573 + 0.0292103i \(0.990701\pi\)
\(294\) 0 0
\(295\) 36.0000 2.09600
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −3.00000 −0.172345
\(304\) 0 0
\(305\) 45.0000i 2.57669i
\(306\) 0 0
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 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
\(317\) 31.0000i 1.74113i 0.492050 + 0.870567i \(0.336248\pi\)
−0.492050 + 0.870567i \(0.663752\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −20.0000 −1.11629
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 18.0000i − 0.995402i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 32.0000i − 1.75888i −0.476011 0.879440i \(-0.657918\pi\)
0.476011 0.879440i \(-0.342082\pi\)
\(332\) 0 0
\(333\) 5.00000i 0.273998i
\(334\) 0 0
\(335\) −36.0000 −1.96689
\(336\) 0 0
\(337\) 1.00000 0.0544735 0.0272367 0.999629i \(-0.491329\pi\)
0.0272367 + 0.999629i \(0.491329\pi\)
\(338\) 0 0
\(339\) −13.0000 −0.706063
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) − 12.0000i − 0.646058i
\(346\) 0 0
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 0 0
\(349\) 34.0000i 1.81998i 0.414632 + 0.909989i \(0.363910\pi\)
−0.414632 + 0.909989i \(0.636090\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 11.0000i 0.585471i 0.956193 + 0.292735i \(0.0945655\pi\)
−0.956193 + 0.292735i \(0.905434\pi\)
\(354\) 0 0
\(355\) −24.0000 −1.27379
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 28.0000i 1.47778i 0.673824 + 0.738892i \(0.264651\pi\)
−0.673824 + 0.738892i \(0.735349\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) 9.00000 0.471082
\(366\) 0 0
\(367\) 36.0000 1.87918 0.939592 0.342296i \(-0.111204\pi\)
0.939592 + 0.342296i \(0.111204\pi\)
\(368\) 0 0
\(369\) 3.00000i 0.156174i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 11.0000 0.569558 0.284779 0.958593i \(-0.408080\pi\)
0.284779 + 0.958593i \(0.408080\pi\)
\(374\) 0 0
\(375\) 3.00000i 0.154919i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) − 20.0000i − 1.02733i −0.857991 0.513665i \(-0.828287\pi\)
0.857991 0.513665i \(-0.171713\pi\)
\(380\) 0 0
\(381\) 12.0000 0.614779
\(382\) 0 0
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 21.0000 1.06474 0.532371 0.846511i \(-0.321301\pi\)
0.532371 + 0.846511i \(0.321301\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 12.0000i − 0.603786i
\(396\) 0 0
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9.00000i 0.449439i 0.974424 + 0.224719i \(0.0721465\pi\)
−0.974424 + 0.224719i \(0.927853\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 3.00000i 0.149071i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) − 9.00000i − 0.445021i −0.974930 0.222511i \(-0.928575\pi\)
0.974930 0.222511i \(-0.0714252\pi\)
\(410\) 0 0
\(411\) 5.00000i 0.246632i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −36.0000 −1.76717
\(416\) 0 0
\(417\) 8.00000 0.391762
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) 27.0000i 1.31590i 0.753062 + 0.657950i \(0.228576\pi\)
−0.753062 + 0.657950i \(0.771424\pi\)
\(422\) 0 0
\(423\) 8.00000i 0.388973i
\(424\) 0 0
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 28.0000i 1.34871i 0.738406 + 0.674356i \(0.235579\pi\)
−0.738406 + 0.674356i \(0.764421\pi\)
\(432\) 0 0
\(433\) 21.0000 1.00920 0.504598 0.863355i \(-0.331641\pi\)
0.504598 + 0.863355i \(0.331641\pi\)
\(434\) 0 0
\(435\) 9.00000i 0.431517i
\(436\) 0 0
\(437\) 0 0
\(438\) 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
\(443\) 40.0000 1.90046 0.950229 0.311553i \(-0.100849\pi\)
0.950229 + 0.311553i \(0.100849\pi\)
\(444\) 0 0
\(445\) 30.0000 1.42214
\(446\) 0 0
\(447\) 19.0000i 0.898669i
\(448\) 0 0
\(449\) − 2.00000i − 0.0943858i −0.998886 0.0471929i \(-0.984972\pi\)
0.998886 0.0471929i \(-0.0150276\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 4.00000i 0.187936i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 9.00000i − 0.421002i −0.977594 0.210501i \(-0.932490\pi\)
0.977594 0.210501i \(-0.0675096\pi\)
\(458\) 0 0
\(459\) 1.00000 0.0466760
\(460\) 0 0
\(461\) − 5.00000i − 0.232873i −0.993198 0.116437i \(-0.962853\pi\)
0.993198 0.116437i \(-0.0371472\pi\)
\(462\) 0 0
\(463\) − 20.0000i − 0.929479i −0.885448 0.464739i \(-0.846148\pi\)
0.885448 0.464739i \(-0.153852\pi\)
\(464\) 0 0
\(465\) −24.0000 −1.11297
\(466\) 0 0
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −13.0000 −0.599008
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −13.0000 −0.595229
\(478\) 0 0
\(479\) − 12.0000i − 0.548294i −0.961688 0.274147i \(-0.911605\pi\)
0.961688 0.274147i \(-0.0883955\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) − 12.0000i − 0.543772i −0.962329 0.271886i \(-0.912353\pi\)
0.962329 0.271886i \(-0.0876473\pi\)
\(488\) 0 0
\(489\) 16.0000i 0.723545i
\(490\) 0 0
\(491\) −4.00000 −0.180517 −0.0902587 0.995918i \(-0.528769\pi\)
−0.0902587 + 0.995918i \(0.528769\pi\)
\(492\) 0 0
\(493\) 3.00000 0.135113
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 12.0000i 0.537194i 0.963253 + 0.268597i \(0.0865599\pi\)
−0.963253 + 0.268597i \(0.913440\pi\)
\(500\) 0 0
\(501\) − 12.0000i − 0.536120i
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) − 9.00000i − 0.400495i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.00000i 0.310270i 0.987893 + 0.155135i \(0.0495812\pi\)
−0.987893 + 0.155135i \(0.950419\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 48.0000i − 2.11513i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 0 0
\(523\) −32.0000 −1.39926 −0.699631 0.714504i \(-0.746652\pi\)
−0.699631 + 0.714504i \(0.746652\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.00000i 0.348485i
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) − 12.0000i − 0.520756i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 60.0000i − 2.59403i
\(536\) 0 0
\(537\) −4.00000 −0.172613
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) − 7.00000i − 0.300954i −0.988614 0.150477i \(-0.951919\pi\)
0.988614 0.150477i \(-0.0480809\pi\)
\(542\) 0 0
\(543\) −11.0000 −0.472055
\(544\) 0 0
\(545\) 54.0000 2.31311
\(546\) 0 0
\(547\) −24.0000 −1.02617 −0.513083 0.858339i \(-0.671497\pi\)
−0.513083 + 0.858339i \(0.671497\pi\)
\(548\) 0 0
\(549\) 15.0000 0.640184
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −15.0000 −0.636715
\(556\) 0 0
\(557\) 29.0000i 1.22877i 0.789007 + 0.614385i \(0.210596\pi\)
−0.789007 + 0.614385i \(0.789404\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) − 39.0000i − 1.64074i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 38.0000 1.59304 0.796521 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) 0 0
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 16.0000 0.667246
\(576\) 0 0
\(577\) − 41.0000i − 1.70685i −0.521214 0.853426i \(-0.674521\pi\)
0.521214 0.853426i \(-0.325479\pi\)
\(578\) 0 0
\(579\) − 11.0000i − 0.457144i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.0000i 0.660391i 0.943913 + 0.330195i \(0.107115\pi\)
−0.943913 + 0.330195i \(0.892885\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000i 0.246807i
\(592\) 0 0
\(593\) − 3.00000i − 0.123195i −0.998101 0.0615976i \(-0.980380\pi\)
0.998101 0.0615976i \(-0.0196196\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) 19.0000 0.775026 0.387513 0.921864i \(-0.373334\pi\)
0.387513 + 0.921864i \(0.373334\pi\)
\(602\) 0 0
\(603\) 12.0000i 0.488678i
\(604\) 0 0
\(605\) 33.0000i 1.34164i
\(606\) 0 0
\(607\) −4.00000 −0.162355 −0.0811775 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 3.00000i 0.121169i 0.998163 + 0.0605844i \(0.0192964\pi\)
−0.998163 + 0.0605844i \(0.980704\pi\)
\(614\) 0 0
\(615\) −9.00000 −0.362915
\(616\) 0 0
\(617\) 15.0000i 0.603877i 0.953327 + 0.301939i \(0.0976338\pi\)
−0.953327 + 0.301939i \(0.902366\pi\)
\(618\) 0 0
\(619\) 28.0000i 1.12542i 0.826656 + 0.562708i \(0.190240\pi\)
−0.826656 + 0.562708i \(0.809760\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.00000i 0.199363i
\(630\) 0 0
\(631\) 8.00000i 0.318475i 0.987240 + 0.159237i \(0.0509036\pi\)
−0.987240 + 0.159237i \(0.949096\pi\)
\(632\) 0 0
\(633\) −8.00000 −0.317971
\(634\) 0 0
\(635\) 36.0000i 1.42862i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 8.00000i 0.316475i
\(640\) 0 0
\(641\) 9.00000 0.355479 0.177739 0.984078i \(-0.443122\pi\)
0.177739 + 0.984078i \(0.443122\pi\)
\(642\) 0 0
\(643\) − 20.0000i − 0.788723i −0.918955 0.394362i \(-0.870966\pi\)
0.918955 0.394362i \(-0.129034\pi\)
\(644\) 0 0
\(645\) − 12.0000i − 0.472500i
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 3.00000i − 0.117041i
\(658\) 0 0
\(659\) −40.0000 −1.55818 −0.779089 0.626913i \(-0.784318\pi\)
−0.779089 + 0.626913i \(0.784318\pi\)
\(660\) 0 0
\(661\) − 7.00000i − 0.272268i −0.990690 0.136134i \(-0.956532\pi\)
0.990690 0.136134i \(-0.0434678\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12.0000 −0.464642
\(668\) 0 0
\(669\) 8.00000i 0.309298i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 29.0000 1.11787 0.558934 0.829212i \(-0.311211\pi\)
0.558934 + 0.829212i \(0.311211\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) − 8.00000i − 0.306561i
\(682\) 0 0
\(683\) − 24.0000i − 0.918334i −0.888350 0.459167i \(-0.848148\pi\)
0.888350 0.459167i \(-0.151852\pi\)
\(684\) 0 0
\(685\) −15.0000 −0.573121
\(686\) 0 0
\(687\) 10.0000i 0.381524i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 8.00000i − 0.304334i −0.988355 0.152167i \(-0.951375\pi\)
0.988355 0.152167i \(-0.0486252\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 24.0000i 0.910372i
\(696\) 0 0
\(697\) 3.00000i 0.113633i
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −24.0000 −0.903892
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) − 19.0000i − 0.713560i −0.934188 0.356780i \(-0.883875\pi\)
0.934188 0.356780i \(-0.116125\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) − 32.0000i − 1.19841i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 8.00000i − 0.298765i
\(718\) 0 0
\(719\) −44.0000 −1.64092 −0.820462 0.571702i \(-0.806283\pi\)
−0.820462 + 0.571702i \(0.806283\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 11.0000i − 0.409094i
\(724\) 0 0
\(725\) −12.0000 −0.445669
\(726\) 0 0
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.00000 −0.147945
\(732\) 0 0
\(733\) − 17.0000i − 0.627909i −0.949438 0.313955i \(-0.898346\pi\)
0.949438 0.313955i \(-0.101654\pi\)
\(734\) 0 0
\(735\) 21.0000i 0.774597i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 32.0000i − 1.17714i −0.808447 0.588570i \(-0.799691\pi\)
0.808447 0.588570i \(-0.200309\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 32.0000i − 1.17397i −0.809599 0.586983i \(-0.800316\pi\)
0.809599 0.586983i \(-0.199684\pi\)
\(744\) 0 0
\(745\) −57.0000 −2.08832
\(746\) 0 0
\(747\) 12.0000i 0.439057i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 38.0000i 1.37750i 0.724999 + 0.688749i \(0.241840\pi\)
−0.724999 + 0.688749i \(0.758160\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 3.00000i 0.108465i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 18.0000i 0.649097i 0.945869 + 0.324548i \(0.105212\pi\)
−0.945869 + 0.324548i \(0.894788\pi\)
\(770\) 0 0
\(771\) 5.00000 0.180071
\(772\) 0 0
\(773\) − 10.0000i − 0.359675i −0.983696 0.179838i \(-0.942443\pi\)
0.983696 0.179838i \(-0.0575572\pi\)
\(774\) 0 0
\(775\) − 32.0000i − 1.14947i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 3.00000 0.107211
\(784\) 0 0
\(785\) − 39.0000i − 1.39197i
\(786\) 0 0
\(787\) 48.0000i 1.71102i 0.517790 + 0.855508i \(0.326755\pi\)
−0.517790 + 0.855508i \(0.673245\pi\)
\(788\) 0 0
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) − 39.0000i − 1.38319i
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 8.00000i 0.283020i
\(800\) 0 0
\(801\) − 10.0000i − 0.353333i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 14.0000 0.492823
\(808\) 0 0
\(809\) −21.0000 −0.738321 −0.369160 0.929366i \(-0.620355\pi\)
−0.369160 + 0.929366i \(0.620355\pi\)
\(810\) 0 0
\(811\) − 56.0000i − 1.96643i −0.182462 0.983213i \(-0.558407\pi\)
0.182462 0.983213i \(-0.441593\pi\)
\(812\) 0 0
\(813\) − 28.0000i − 0.982003i
\(814\) 0 0
\(815\) −48.0000 −1.68137
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 42.0000i − 1.46581i −0.680331 0.732905i \(-0.738164\pi\)
0.680331 0.732905i \(-0.261836\pi\)
\(822\) 0 0
\(823\) 36.0000 1.25488 0.627441 0.778664i \(-0.284103\pi\)
0.627441 + 0.778664i \(0.284103\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 28.0000i − 0.973655i −0.873498 0.486828i \(-0.838154\pi\)
0.873498 0.486828i \(-0.161846\pi\)
\(828\) 0 0
\(829\) 41.0000 1.42399 0.711994 0.702185i \(-0.247792\pi\)
0.711994 + 0.702185i \(0.247792\pi\)
\(830\) 0 0
\(831\) 25.0000 0.867240
\(832\) 0 0
\(833\) 7.00000 0.242536
\(834\) 0 0
\(835\) 36.0000 1.24583
\(836\) 0 0
\(837\) 8.00000i 0.276520i
\(838\) 0 0
\(839\) − 12.0000i − 0.414286i −0.978311 0.207143i \(-0.933583\pi\)
0.978311 0.207143i \(-0.0664165\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) − 15.0000i − 0.516627i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 16.0000 0.549119
\(850\) 0 0
\(851\) − 20.0000i − 0.685591i
\(852\) 0 0
\(853\) 17.0000i 0.582069i 0.956713 + 0.291034i \(0.0939994\pi\)
−0.956713 + 0.291034i \(0.906001\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −27.0000 −0.922302 −0.461151 0.887322i \(-0.652563\pi\)
−0.461151 + 0.887322i \(0.652563\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0000i 0.816970i 0.912765 + 0.408485i \(0.133943\pi\)
−0.912765 + 0.408485i \(0.866057\pi\)
\(864\) 0 0
\(865\) 54.0000i 1.83606i
\(866\) 0 0
\(867\) −16.0000 −0.543388
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 2.00000i 0.0676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 41.0000i − 1.38447i −0.721671 0.692236i \(-0.756626\pi\)
0.721671 0.692236i \(-0.243374\pi\)
\(878\) 0 0
\(879\) 1.00000i 0.0337292i
\(880\) 0 0
\(881\) 21.0000 0.707508 0.353754 0.935339i \(-0.384905\pi\)
0.353754 + 0.935339i \(0.384905\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) 36.0000 1.21013
\(886\) 0 0
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) − 12.0000i − 0.401116i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 24.0000i 0.800445i
\(900\) 0 0
\(901\) −13.0000 −0.433093
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 33.0000i − 1.09696i
\(906\) 0 0
\(907\) −48.0000 −1.59381 −0.796907 0.604102i \(-0.793532\pi\)
−0.796907 + 0.604102i \(0.793532\pi\)
\(908\) 0 0
\(909\) −3.00000 −0.0995037
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 45.0000i 1.48765i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −12.0000 −0.395843 −0.197922 0.980218i \(-0.563419\pi\)
−0.197922 + 0.980218i \(0.563419\pi\)
\(920\) 0 0
\(921\) 20.0000i 0.659022i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) − 20.0000i − 0.657596i
\(926\) 0 0
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) − 33.0000i − 1.08269i −0.840799 0.541347i \(-0.817914\pi\)
0.840799 0.541347i \(-0.182086\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −8.00000 −0.261908
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −37.0000 −1.20874 −0.604369 0.796705i \(-0.706575\pi\)
−0.604369 + 0.796705i \(0.706575\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 46.0000i 1.49956i 0.661689 + 0.749779i \(0.269840\pi\)
−0.661689 + 0.749779i \(0.730160\pi\)
\(942\) 0 0
\(943\) − 12.0000i − 0.390774i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28.0000i 0.909878i 0.890523 + 0.454939i \(0.150339\pi\)
−0.890523 + 0.454939i \(0.849661\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 31.0000i 1.00524i
\(952\) 0 0
\(953\) 38.0000 1.23094 0.615470 0.788160i \(-0.288966\pi\)
0.615470 + 0.788160i \(0.288966\pi\)
\(954\) 0 0
\(955\) − 24.0000i − 0.776622i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −33.0000 −1.06452
\(962\) 0 0
\(963\) −20.0000 −0.644491
\(964\) 0 0
\(965\) 33.0000 1.06231
\(966\) 0 0
\(967\) − 32.0000i − 1.02905i −0.857475 0.514525i \(-0.827968\pi\)
0.857475 0.514525i \(-0.172032\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 4.00000 0.128366 0.0641831 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 57.0000i − 1.82359i −0.410644 0.911796i \(-0.634696\pi\)
0.410644 0.911796i \(-0.365304\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) − 18.0000i − 0.574696i
\(982\) 0 0
\(983\) 16.0000i 0.510321i 0.966899 + 0.255160i \(0.0821283\pi\)
−0.966899 + 0.255160i \(0.917872\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 0 0
\(993\) − 32.0000i − 1.01549i
\(994\) 0 0
\(995\) − 12.0000i − 0.380426i
\(996\) 0 0
\(997\) 47.0000 1.48850 0.744252 0.667898i \(-0.232806\pi\)
0.744252 + 0.667898i \(0.232806\pi\)
\(998\) 0 0
\(999\) 5.00000i 0.158193i
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.i.337.2 2
13.2 odd 12 312.2.q.a.217.1 2
13.5 odd 4 4056.2.a.q.1.1 1
13.6 odd 12 312.2.q.a.289.1 yes 2
13.8 odd 4 4056.2.a.l.1.1 1
13.12 even 2 inner 4056.2.c.i.337.1 2
39.2 even 12 936.2.t.a.217.1 2
39.32 even 12 936.2.t.a.289.1 2
52.15 even 12 624.2.q.f.529.1 2
52.19 even 12 624.2.q.f.289.1 2
52.31 even 4 8112.2.a.n.1.1 1
52.47 even 4 8112.2.a.b.1.1 1
156.71 odd 12 1872.2.t.b.289.1 2
156.119 odd 12 1872.2.t.b.1153.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.2.q.a.217.1 2 13.2 odd 12
312.2.q.a.289.1 yes 2 13.6 odd 12
624.2.q.f.289.1 2 52.19 even 12
624.2.q.f.529.1 2 52.15 even 12
936.2.t.a.217.1 2 39.2 even 12
936.2.t.a.289.1 2 39.32 even 12
1872.2.t.b.289.1 2 156.71 odd 12
1872.2.t.b.1153.1 2 156.119 odd 12
4056.2.a.l.1.1 1 13.8 odd 4
4056.2.a.q.1.1 1 13.5 odd 4
4056.2.c.i.337.1 2 13.12 even 2 inner
4056.2.c.i.337.2 2 1.1 even 1 trivial
8112.2.a.b.1.1 1 52.47 even 4
8112.2.a.n.1.1 1 52.31 even 4