Properties

Label 352.2.a.f.1.1
Level $352$
Weight $2$
Character 352.1
Self dual yes
Analytic conductor $2.811$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [352,2,Mod(1,352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(352, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("352.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 352 = 2^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 352.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(2.81073415115\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 352.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +1.00000 q^{5} +6.00000 q^{9} -1.00000 q^{11} -6.00000 q^{13} +3.00000 q^{15} -4.00000 q^{17} +6.00000 q^{19} +3.00000 q^{23} -4.00000 q^{25} +9.00000 q^{27} -4.00000 q^{29} -9.00000 q^{31} -3.00000 q^{33} +7.00000 q^{37} -18.0000 q^{39} -2.00000 q^{41} +6.00000 q^{43} +6.00000 q^{45} +12.0000 q^{47} -7.00000 q^{49} -12.0000 q^{51} +2.00000 q^{53} -1.00000 q^{55} +18.0000 q^{57} +9.00000 q^{59} +8.00000 q^{61} -6.00000 q^{65} -15.0000 q^{67} +9.00000 q^{69} -3.00000 q^{71} -6.00000 q^{73} -12.0000 q^{75} -6.00000 q^{79} +9.00000 q^{81} -6.00000 q^{83} -4.00000 q^{85} -12.0000 q^{87} -5.00000 q^{89} -27.0000 q^{93} +6.00000 q^{95} -3.00000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −9.00000 −1.61645 −0.808224 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) 0 0
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 0 0
\(39\) −18.0000 −2.88231
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −12.0000 −1.68034
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 18.0000 2.38416
\(58\) 0 0
\(59\) 9.00000 1.17170 0.585850 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) −15.0000 −1.83254 −0.916271 0.400559i \(-0.868816\pi\)
−0.916271 + 0.400559i \(0.868816\pi\)
\(68\) 0 0
\(69\) 9.00000 1.08347
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) −12.0000 −1.38564
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) −12.0000 −1.28654
\(88\) 0 0
\(89\) −5.00000 −0.529999 −0.264999 0.964249i \(-0.585372\pi\)
−0.264999 + 0.964249i \(0.585372\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −27.0000 −2.79977
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −3.00000 −0.304604 −0.152302 0.988334i \(-0.548669\pi\)
−0.152302 + 0.988334i \(0.548669\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 16.0000 1.59206 0.796030 0.605257i \(-0.206930\pi\)
0.796030 + 0.605257i \(0.206930\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 21.0000 1.99323
\(112\) 0 0
\(113\) 1.00000 0.0940721 0.0470360 0.998893i \(-0.485022\pi\)
0.0470360 + 0.998893i \(0.485022\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 0 0
\(117\) −36.0000 −3.32820
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 0 0
\(129\) 18.0000 1.58481
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 9.00000 0.774597
\(136\) 0 0
\(137\) 5.00000 0.427179 0.213589 0.976924i \(-0.431485\pi\)
0.213589 + 0.976924i \(0.431485\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 36.0000 3.03175
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 0 0
\(147\) −21.0000 −1.73205
\(148\) 0 0
\(149\) −16.0000 −1.31077 −0.655386 0.755295i \(-0.727494\pi\)
−0.655386 + 0.755295i \(0.727494\pi\)
\(150\) 0 0
\(151\) −6.00000 −0.488273 −0.244137 0.969741i \(-0.578505\pi\)
−0.244137 + 0.969741i \(0.578505\pi\)
\(152\) 0 0
\(153\) −24.0000 −1.94029
\(154\) 0 0
\(155\) −9.00000 −0.722897
\(156\) 0 0
\(157\) 13.0000 1.03751 0.518756 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 0 0
\(165\) −3.00000 −0.233550
\(166\) 0 0
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 36.0000 2.75299
\(172\) 0 0
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 27.0000 2.02944
\(178\) 0 0
\(179\) −15.0000 −1.12115 −0.560576 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) 0 0
\(181\) 3.00000 0.222988 0.111494 0.993765i \(-0.464436\pi\)
0.111494 + 0.993765i \(0.464436\pi\)
\(182\) 0 0
\(183\) 24.0000 1.77413
\(184\) 0 0
\(185\) 7.00000 0.514650
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 21.0000 1.51951 0.759753 0.650211i \(-0.225320\pi\)
0.759753 + 0.650211i \(0.225320\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) −18.0000 −1.28901
\(196\) 0 0
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) −45.0000 −3.17406
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) 0 0
\(207\) 18.0000 1.25109
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) −9.00000 −0.616670
\(214\) 0 0
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −18.0000 −1.21633
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) −21.0000 −1.40626 −0.703132 0.711059i \(-0.748216\pi\)
−0.703132 + 0.711059i \(0.748216\pi\)
\(224\) 0 0
\(225\) −24.0000 −1.60000
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 15.0000 0.991228 0.495614 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4.00000 −0.262049 −0.131024 0.991379i \(-0.541827\pi\)
−0.131024 + 0.991379i \(0.541827\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) 0 0
\(237\) −18.0000 −1.16923
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −7.00000 −0.447214
\(246\) 0 0
\(247\) −36.0000 −2.29063
\(248\) 0 0
\(249\) −18.0000 −1.14070
\(250\) 0 0
\(251\) 21.0000 1.32551 0.662754 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) 0 0
\(253\) −3.00000 −0.188608
\(254\) 0 0
\(255\) −12.0000 −0.751469
\(256\) 0 0
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −24.0000 −1.48556
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) −15.0000 −0.917985
\(268\) 0 0
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) −6.00000 −0.364474 −0.182237 0.983255i \(-0.558334\pi\)
−0.182237 + 0.983255i \(0.558334\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −6.00000 −0.360505 −0.180253 0.983620i \(-0.557691\pi\)
−0.180253 + 0.983620i \(0.557691\pi\)
\(278\) 0 0
\(279\) −54.0000 −3.23290
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 18.0000 1.06623
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −9.00000 −0.527589
\(292\) 0 0
\(293\) −22.0000 −1.28525 −0.642627 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) 0 0
\(295\) 9.00000 0.524000
\(296\) 0 0
\(297\) −9.00000 −0.522233
\(298\) 0 0
\(299\) −18.0000 −1.04097
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 48.0000 2.75753
\(304\) 0 0
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) 6.00000 0.342438 0.171219 0.985233i \(-0.445229\pi\)
0.171219 + 0.985233i \(0.445229\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −23.0000 −1.29181 −0.645904 0.763418i \(-0.723520\pi\)
−0.645904 + 0.763418i \(0.723520\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) 36.0000 2.00932
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 24.0000 1.33128
\(326\) 0 0
\(327\) −18.0000 −0.995402
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 15.0000 0.824475 0.412237 0.911077i \(-0.364747\pi\)
0.412237 + 0.911077i \(0.364747\pi\)
\(332\) 0 0
\(333\) 42.0000 2.30159
\(334\) 0 0
\(335\) −15.0000 −0.819538
\(336\) 0 0
\(337\) 24.0000 1.30736 0.653682 0.756770i \(-0.273224\pi\)
0.653682 + 0.756770i \(0.273224\pi\)
\(338\) 0 0
\(339\) 3.00000 0.162938
\(340\) 0 0
\(341\) 9.00000 0.487377
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 9.00000 0.484544
\(346\) 0 0
\(347\) 6.00000 0.322097 0.161048 0.986947i \(-0.448512\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 0 0
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) −54.0000 −2.88231
\(352\) 0 0
\(353\) −5.00000 −0.266123 −0.133062 0.991108i \(-0.542481\pi\)
−0.133062 + 0.991108i \(0.542481\pi\)
\(354\) 0 0
\(355\) −3.00000 −0.159223
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 3.00000 0.157459
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) 15.0000 0.782994 0.391497 0.920179i \(-0.371957\pi\)
0.391497 + 0.920179i \(0.371957\pi\)
\(368\) 0 0
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) −27.0000 −1.39427
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) −21.0000 −1.07870 −0.539349 0.842082i \(-0.681330\pi\)
−0.539349 + 0.842082i \(0.681330\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) −21.0000 −1.07305 −0.536525 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 36.0000 1.82998
\(388\) 0 0
\(389\) 5.00000 0.253510 0.126755 0.991934i \(-0.459544\pi\)
0.126755 + 0.991934i \(0.459544\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) 36.0000 1.81596
\(394\) 0 0
\(395\) −6.00000 −0.301893
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 0 0
\(403\) 54.0000 2.68993
\(404\) 0 0
\(405\) 9.00000 0.447214
\(406\) 0 0
\(407\) −7.00000 −0.346977
\(408\) 0 0
\(409\) 18.0000 0.890043 0.445021 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) 0 0
\(411\) 15.0000 0.739895
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) −36.0000 −1.76293
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 0 0
\(423\) 72.0000 3.50076
\(424\) 0 0
\(425\) 16.0000 0.776114
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 18.0000 0.869048
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −11.0000 −0.528626 −0.264313 0.964437i \(-0.585145\pi\)
−0.264313 + 0.964437i \(0.585145\pi\)
\(434\) 0 0
\(435\) −12.0000 −0.575356
\(436\) 0 0
\(437\) 18.0000 0.861057
\(438\) 0 0
\(439\) 36.0000 1.71819 0.859093 0.511819i \(-0.171028\pi\)
0.859093 + 0.511819i \(0.171028\pi\)
\(440\) 0 0
\(441\) −42.0000 −2.00000
\(442\) 0 0
\(443\) 9.00000 0.427603 0.213801 0.976877i \(-0.431415\pi\)
0.213801 + 0.976877i \(0.431415\pi\)
\(444\) 0 0
\(445\) −5.00000 −0.237023
\(446\) 0 0
\(447\) −48.0000 −2.27032
\(448\) 0 0
\(449\) −17.0000 −0.802280 −0.401140 0.916017i \(-0.631386\pi\)
−0.401140 + 0.916017i \(0.631386\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) 0 0
\(453\) −18.0000 −0.845714
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −24.0000 −1.12267 −0.561336 0.827588i \(-0.689713\pi\)
−0.561336 + 0.827588i \(0.689713\pi\)
\(458\) 0 0
\(459\) −36.0000 −1.68034
\(460\) 0 0
\(461\) −40.0000 −1.86299 −0.931493 0.363760i \(-0.881493\pi\)
−0.931493 + 0.363760i \(0.881493\pi\)
\(462\) 0 0
\(463\) −15.0000 −0.697109 −0.348555 0.937288i \(-0.613327\pi\)
−0.348555 + 0.937288i \(0.613327\pi\)
\(464\) 0 0
\(465\) −27.0000 −1.25210
\(466\) 0 0
\(467\) −15.0000 −0.694117 −0.347059 0.937843i \(-0.612820\pi\)
−0.347059 + 0.937843i \(0.612820\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 39.0000 1.79703
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) −24.0000 −1.10120
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) −42.0000 −1.91504
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.00000 −0.136223
\(486\) 0 0
\(487\) 3.00000 0.135943 0.0679715 0.997687i \(-0.478347\pi\)
0.0679715 + 0.997687i \(0.478347\pi\)
\(488\) 0 0
\(489\) 36.0000 1.62798
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 0 0
\(493\) 16.0000 0.720604
\(494\) 0 0
\(495\) −6.00000 −0.269680
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 0 0
\(501\) 54.0000 2.41254
\(502\) 0 0
\(503\) 42.0000 1.87269 0.936344 0.351085i \(-0.114187\pi\)
0.936344 + 0.351085i \(0.114187\pi\)
\(504\) 0 0
\(505\) 16.0000 0.711991
\(506\) 0 0
\(507\) 69.0000 3.06440
\(508\) 0 0
\(509\) −1.00000 −0.0443242 −0.0221621 0.999754i \(-0.507055\pi\)
−0.0221621 + 0.999754i \(0.507055\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 54.0000 2.38416
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) 0 0
\(519\) 30.0000 1.31685
\(520\) 0 0
\(521\) −31.0000 −1.35813 −0.679067 0.734076i \(-0.737616\pi\)
−0.679067 + 0.734076i \(0.737616\pi\)
\(522\) 0 0
\(523\) −30.0000 −1.31181 −0.655904 0.754844i \(-0.727712\pi\)
−0.655904 + 0.754844i \(0.727712\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 36.0000 1.56818
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 54.0000 2.34340
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 0 0
\(537\) −45.0000 −1.94189
\(538\) 0 0
\(539\) 7.00000 0.301511
\(540\) 0 0
\(541\) −6.00000 −0.257960 −0.128980 0.991647i \(-0.541170\pi\)
−0.128980 + 0.991647i \(0.541170\pi\)
\(542\) 0 0
\(543\) 9.00000 0.386227
\(544\) 0 0
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) 36.0000 1.53925 0.769624 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) 0 0
\(549\) 48.0000 2.04859
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 21.0000 0.891400
\(556\) 0 0
\(557\) −16.0000 −0.677942 −0.338971 0.940797i \(-0.610079\pi\)
−0.338971 + 0.940797i \(0.610079\pi\)
\(558\) 0 0
\(559\) −36.0000 −1.52264
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 1.00000 0.0420703
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 44.0000 1.84458 0.922288 0.386503i \(-0.126317\pi\)
0.922288 + 0.386503i \(0.126317\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 63.0000 2.63186
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) 29.0000 1.20729 0.603643 0.797255i \(-0.293715\pi\)
0.603643 + 0.797255i \(0.293715\pi\)
\(578\) 0 0
\(579\) 12.0000 0.498703
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −2.00000 −0.0828315
\(584\) 0 0
\(585\) −36.0000 −1.48842
\(586\) 0 0
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 0 0
\(589\) −54.0000 −2.22503
\(590\) 0 0
\(591\) 66.0000 2.71488
\(592\) 0 0
\(593\) 20.0000 0.821302 0.410651 0.911793i \(-0.365302\pi\)
0.410651 + 0.911793i \(0.365302\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 28.0000 1.14214 0.571072 0.820900i \(-0.306528\pi\)
0.571072 + 0.820900i \(0.306528\pi\)
\(602\) 0 0
\(603\) −90.0000 −3.66508
\(604\) 0 0
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −18.0000 −0.730597 −0.365299 0.930890i \(-0.619033\pi\)
−0.365299 + 0.930890i \(0.619033\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −72.0000 −2.91281
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 0 0
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) 15.0000 0.602901 0.301450 0.953482i \(-0.402529\pi\)
0.301450 + 0.953482i \(0.402529\pi\)
\(620\) 0 0
\(621\) 27.0000 1.08347
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) −18.0000 −0.718851
\(628\) 0 0
\(629\) −28.0000 −1.11643
\(630\) 0 0
\(631\) 15.0000 0.597141 0.298570 0.954388i \(-0.403490\pi\)
0.298570 + 0.954388i \(0.403490\pi\)
\(632\) 0 0
\(633\) 36.0000 1.43087
\(634\) 0 0
\(635\) −6.00000 −0.238103
\(636\) 0 0
\(637\) 42.0000 1.66410
\(638\) 0 0
\(639\) −18.0000 −0.712069
\(640\) 0 0
\(641\) −37.0000 −1.46141 −0.730706 0.682692i \(-0.760809\pi\)
−0.730706 + 0.682692i \(0.760809\pi\)
\(642\) 0 0
\(643\) 21.0000 0.828159 0.414080 0.910241i \(-0.364104\pi\)
0.414080 + 0.910241i \(0.364104\pi\)
\(644\) 0 0
\(645\) 18.0000 0.708749
\(646\) 0 0
\(647\) −3.00000 −0.117942 −0.0589711 0.998260i \(-0.518782\pi\)
−0.0589711 + 0.998260i \(0.518782\pi\)
\(648\) 0 0
\(649\) −9.00000 −0.353281
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −17.0000 −0.665261 −0.332631 0.943057i \(-0.607936\pi\)
−0.332631 + 0.943057i \(0.607936\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) −36.0000 −1.40449
\(658\) 0 0
\(659\) −18.0000 −0.701180 −0.350590 0.936529i \(-0.614019\pi\)
−0.350590 + 0.936529i \(0.614019\pi\)
\(660\) 0 0
\(661\) −23.0000 −0.894596 −0.447298 0.894385i \(-0.647614\pi\)
−0.447298 + 0.894385i \(0.647614\pi\)
\(662\) 0 0
\(663\) 72.0000 2.79625
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12.0000 −0.464642
\(668\) 0 0
\(669\) −63.0000 −2.43572
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) −36.0000 −1.38564
\(676\) 0 0
\(677\) −8.00000 −0.307465 −0.153732 0.988113i \(-0.549129\pi\)
−0.153732 + 0.988113i \(0.549129\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 36.0000 1.37952
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 5.00000 0.191040
\(686\) 0 0
\(687\) 45.0000 1.71686
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 33.0000 1.25538 0.627690 0.778464i \(-0.284001\pi\)
0.627690 + 0.778464i \(0.284001\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) 8.00000 0.303022
\(698\) 0 0
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) −40.0000 −1.51078 −0.755390 0.655276i \(-0.772552\pi\)
−0.755390 + 0.655276i \(0.772552\pi\)
\(702\) 0 0
\(703\) 42.0000 1.58406
\(704\) 0 0
\(705\) 36.0000 1.35584
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −33.0000 −1.23934 −0.619671 0.784862i \(-0.712734\pi\)
−0.619671 + 0.784862i \(0.712734\pi\)
\(710\) 0 0
\(711\) −36.0000 −1.35011
\(712\) 0 0
\(713\) −27.0000 −1.01116
\(714\) 0 0
\(715\) 6.00000 0.224387
\(716\) 0 0
\(717\) −18.0000 −0.672222
\(718\) 0 0
\(719\) 27.0000 1.00693 0.503465 0.864016i \(-0.332058\pi\)
0.503465 + 0.864016i \(0.332058\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −36.0000 −1.33885
\(724\) 0 0
\(725\) 16.0000 0.594225
\(726\) 0 0
\(727\) 3.00000 0.111264 0.0556319 0.998451i \(-0.482283\pi\)
0.0556319 + 0.998451i \(0.482283\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 48.0000 1.77292 0.886460 0.462805i \(-0.153157\pi\)
0.886460 + 0.462805i \(0.153157\pi\)
\(734\) 0 0
\(735\) −21.0000 −0.774597
\(736\) 0 0
\(737\) 15.0000 0.552532
\(738\) 0 0
\(739\) −6.00000 −0.220714 −0.110357 0.993892i \(-0.535199\pi\)
−0.110357 + 0.993892i \(0.535199\pi\)
\(740\) 0 0
\(741\) −108.000 −3.96748
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) −16.0000 −0.586195
\(746\) 0 0
\(747\) −36.0000 −1.31717
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −3.00000 −0.109472 −0.0547358 0.998501i \(-0.517432\pi\)
−0.0547358 + 0.998501i \(0.517432\pi\)
\(752\) 0 0
\(753\) 63.0000 2.29585
\(754\) 0 0
\(755\) −6.00000 −0.218362
\(756\) 0 0
\(757\) 6.00000 0.218074 0.109037 0.994038i \(-0.465223\pi\)
0.109037 + 0.994038i \(0.465223\pi\)
\(758\) 0 0
\(759\) −9.00000 −0.326679
\(760\) 0 0
\(761\) −26.0000 −0.942499 −0.471250 0.882000i \(-0.656197\pi\)
−0.471250 + 0.882000i \(0.656197\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −24.0000 −0.867722
\(766\) 0 0
\(767\) −54.0000 −1.94983
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 42.0000 1.51259
\(772\) 0 0
\(773\) 2.00000 0.0719350 0.0359675 0.999353i \(-0.488549\pi\)
0.0359675 + 0.999353i \(0.488549\pi\)
\(774\) 0 0
\(775\) 36.0000 1.29316
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 3.00000 0.107348
\(782\) 0 0
\(783\) −36.0000 −1.28654
\(784\) 0 0
\(785\) 13.0000 0.463990
\(786\) 0 0
\(787\) −36.0000 −1.28326 −0.641631 0.767014i \(-0.721742\pi\)
−0.641631 + 0.767014i \(0.721742\pi\)
\(788\) 0 0
\(789\) 72.0000 2.56327
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −48.0000 −1.70453
\(794\) 0 0
\(795\) 6.00000 0.212798
\(796\) 0 0
\(797\) 49.0000 1.73567 0.867835 0.496853i \(-0.165511\pi\)
0.867835 + 0.496853i \(0.165511\pi\)
\(798\) 0 0
\(799\) −48.0000 −1.69812
\(800\) 0 0
\(801\) −30.0000 −1.06000
\(802\) 0 0
\(803\) 6.00000 0.211735
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) 46.0000 1.61727 0.808637 0.588308i \(-0.200206\pi\)
0.808637 + 0.588308i \(0.200206\pi\)
\(810\) 0 0
\(811\) 30.0000 1.05344 0.526721 0.850038i \(-0.323421\pi\)
0.526721 + 0.850038i \(0.323421\pi\)
\(812\) 0 0
\(813\) −18.0000 −0.631288
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) 36.0000 1.25948
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 0 0
\(823\) −33.0000 −1.15031 −0.575154 0.818045i \(-0.695058\pi\)
−0.575154 + 0.818045i \(0.695058\pi\)
\(824\) 0 0
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) −18.0000 −0.625921 −0.312961 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) 0 0
\(829\) 21.0000 0.729360 0.364680 0.931133i \(-0.381178\pi\)
0.364680 + 0.931133i \(0.381178\pi\)
\(830\) 0 0
\(831\) −18.0000 −0.624413
\(832\) 0 0
\(833\) 28.0000 0.970143
\(834\) 0 0
\(835\) 18.0000 0.622916
\(836\) 0 0
\(837\) −81.0000 −2.79977
\(838\) 0 0
\(839\) −33.0000 −1.13929 −0.569643 0.821892i \(-0.692919\pi\)
−0.569643 + 0.821892i \(0.692919\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 30.0000 1.03325
\(844\) 0 0
\(845\) 23.0000 0.791224
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 21.0000 0.719871
\(852\) 0 0
\(853\) 46.0000 1.57501 0.787505 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 0 0
\(855\) 36.0000 1.23117
\(856\) 0 0
\(857\) −32.0000 −1.09310 −0.546550 0.837427i \(-0.684059\pi\)
−0.546550 + 0.837427i \(0.684059\pi\)
\(858\) 0 0
\(859\) −27.0000 −0.921228 −0.460614 0.887601i \(-0.652371\pi\)
−0.460614 + 0.887601i \(0.652371\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 10.0000 0.340010
\(866\) 0 0
\(867\) −3.00000 −0.101885
\(868\) 0 0
\(869\) 6.00000 0.203536
\(870\) 0 0
\(871\) 90.0000 3.04953
\(872\) 0 0
\(873\) −18.0000 −0.609208
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 0 0
\(879\) −66.0000 −2.22612
\(880\) 0 0
\(881\) 29.0000 0.977035 0.488517 0.872554i \(-0.337538\pi\)
0.488517 + 0.872554i \(0.337538\pi\)
\(882\) 0 0
\(883\) −12.0000 −0.403832 −0.201916 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(884\) 0 0
\(885\) 27.0000 0.907595
\(886\) 0 0
\(887\) 6.00000 0.201460 0.100730 0.994914i \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −9.00000 −0.301511
\(892\) 0 0
\(893\) 72.0000 2.40939
\(894\) 0 0
\(895\) −15.0000 −0.501395
\(896\) 0 0
\(897\) −54.0000 −1.80301
\(898\) 0 0
\(899\) 36.0000 1.20067
\(900\) 0 0
\(901\) −8.00000 −0.266519
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3.00000 0.0997234
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 0 0
\(909\) 96.0000 3.18412
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 6.00000 0.198571
\(914\) 0 0
\(915\) 24.0000 0.793416
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 30.0000 0.989609 0.494804 0.869004i \(-0.335240\pi\)
0.494804 + 0.869004i \(0.335240\pi\)
\(920\) 0 0
\(921\) 18.0000 0.593120
\(922\) 0 0
\(923\) 18.0000 0.592477
\(924\) 0 0
\(925\) −28.0000 −0.920634
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −22.0000 −0.721797 −0.360898 0.932605i \(-0.617530\pi\)
−0.360898 + 0.932605i \(0.617530\pi\)
\(930\) 0 0
\(931\) −42.0000 −1.37649
\(932\) 0 0
\(933\) −72.0000 −2.35717
\(934\) 0 0
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) −3.00000 −0.0979013
\(940\) 0 0
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 0 0
\(943\) −6.00000 −0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −15.0000 −0.487435 −0.243717 0.969846i \(-0.578367\pi\)
−0.243717 + 0.969846i \(0.578367\pi\)
\(948\) 0 0
\(949\) 36.0000 1.16861
\(950\) 0 0
\(951\) −69.0000 −2.23748
\(952\) 0 0
\(953\) −40.0000 −1.29573 −0.647864 0.761756i \(-0.724337\pi\)
−0.647864 + 0.761756i \(0.724337\pi\)
\(954\) 0 0
\(955\) 21.0000 0.679544
\(956\) 0 0
\(957\) 12.0000 0.387905
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 50.0000 1.61290
\(962\) 0 0
\(963\) 72.0000 2.32017
\(964\) 0 0
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) 12.0000 0.385894 0.192947 0.981209i \(-0.438195\pi\)
0.192947 + 0.981209i \(0.438195\pi\)
\(968\) 0 0
\(969\) −72.0000 −2.31297
\(970\) 0 0
\(971\) 15.0000 0.481373 0.240686 0.970603i \(-0.422627\pi\)
0.240686 + 0.970603i \(0.422627\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 72.0000 2.30585
\(976\) 0 0
\(977\) −23.0000 −0.735835 −0.367918 0.929858i \(-0.619929\pi\)
−0.367918 + 0.929858i \(0.619929\pi\)
\(978\) 0 0
\(979\) 5.00000 0.159801
\(980\) 0 0
\(981\) −36.0000 −1.14939
\(982\) 0 0
\(983\) −45.0000 −1.43528 −0.717639 0.696416i \(-0.754777\pi\)
−0.717639 + 0.696416i \(0.754777\pi\)
\(984\) 0 0
\(985\) 22.0000 0.700978
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 18.0000 0.572367
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 45.0000 1.42803
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 8.00000 0.253363 0.126681 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) 0 0
\(999\) 63.0000 1.99323
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 352.2.a.f.1.1 yes 1
3.2 odd 2 3168.2.a.i.1.1 1
4.3 odd 2 352.2.a.a.1.1 1
5.4 even 2 8800.2.a.a.1.1 1
8.3 odd 2 704.2.a.k.1.1 1
8.5 even 2 704.2.a.a.1.1 1
11.10 odd 2 3872.2.a.m.1.1 1
12.11 even 2 3168.2.a.h.1.1 1
16.3 odd 4 2816.2.c.g.1409.1 2
16.5 even 4 2816.2.c.h.1409.1 2
16.11 odd 4 2816.2.c.g.1409.2 2
16.13 even 4 2816.2.c.h.1409.2 2
20.19 odd 2 8800.2.a.bb.1.1 1
24.5 odd 2 6336.2.a.bs.1.1 1
24.11 even 2 6336.2.a.bt.1.1 1
44.43 even 2 3872.2.a.a.1.1 1
88.21 odd 2 7744.2.a.a.1.1 1
88.43 even 2 7744.2.a.bj.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
352.2.a.a.1.1 1 4.3 odd 2
352.2.a.f.1.1 yes 1 1.1 even 1 trivial
704.2.a.a.1.1 1 8.5 even 2
704.2.a.k.1.1 1 8.3 odd 2
2816.2.c.g.1409.1 2 16.3 odd 4
2816.2.c.g.1409.2 2 16.11 odd 4
2816.2.c.h.1409.1 2 16.5 even 4
2816.2.c.h.1409.2 2 16.13 even 4
3168.2.a.h.1.1 1 12.11 even 2
3168.2.a.i.1.1 1 3.2 odd 2
3872.2.a.a.1.1 1 44.43 even 2
3872.2.a.m.1.1 1 11.10 odd 2
6336.2.a.bs.1.1 1 24.5 odd 2
6336.2.a.bt.1.1 1 24.11 even 2
7744.2.a.a.1.1 1 88.21 odd 2
7744.2.a.bj.1.1 1 88.43 even 2
8800.2.a.a.1.1 1 5.4 even 2
8800.2.a.bb.1.1 1 20.19 odd 2