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Properties

Label 7744.2.a.bi.1.1
Level $7744$
Weight $2$
Character 7744.1
Self dual yes
Analytic conductor $61.836$
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] = [7744,2,Mod(1,7744)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7744, 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("7744.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7744 = 2^{6} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7744.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: \(61.8361513253\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 242)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7744.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{3} +3.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{3} +3.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} -5.00000 q^{13} +6.00000 q^{15} +3.00000 q^{17} -2.00000 q^{19} +4.00000 q^{21} +6.00000 q^{23} +4.00000 q^{25} -4.00000 q^{27} +3.00000 q^{29} +2.00000 q^{31} +6.00000 q^{35} +7.00000 q^{37} -10.0000 q^{39} +3.00000 q^{41} -8.00000 q^{43} +3.00000 q^{45} +6.00000 q^{47} -3.00000 q^{49} +6.00000 q^{51} +3.00000 q^{53} -4.00000 q^{57} +10.0000 q^{61} +2.00000 q^{63} -15.0000 q^{65} +10.0000 q^{67} +12.0000 q^{69} +12.0000 q^{71} +14.0000 q^{73} +8.00000 q^{75} +2.00000 q^{79} -11.0000 q^{81} -18.0000 q^{83} +9.00000 q^{85} +6.00000 q^{87} -9.00000 q^{89} -10.0000 q^{91} +4.00000 q^{93} -6.00000 q^{95} +11.0000 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 0 0
\(5\) 3.00000 1.34164 0.670820 − 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 − 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 6.00000 1.54919
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 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
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.00000 1.01419
\(36\) 0 0
\(37\) 7.00000 1.15079 0.575396 − 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 0 0
\(39\) −10.0000 −1.60128
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 − 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) −15.0000 −1.86052
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 − 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 0 0
\(69\) 12.0000 1.44463
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 0 0
\(75\) 8.00000 0.923760
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 − 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −18.0000 −1.97576 −0.987878 − 0.155230i \(-0.950388\pi\)
−0.987878 + 0.155230i \(0.950388\pi\)
\(84\) 0 0
\(85\) 9.00000 0.976187
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 − 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) −10.0000 −1.04828
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) 11.0000 1.11688 0.558440 − 0.829545i \(-0.311400\pi\)
0.558440 + 0.829545i \(0.311400\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 12.0000 1.17108
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) −17.0000 −1.62830 −0.814152 − 0.580651i \(-0.802798\pi\)
−0.814152 + 0.580651i \(0.802798\pi\)
\(110\) 0 0
\(111\) 14.0000 1.32882
\(112\) 0 0
\(113\) −9.00000 −0.846649 −0.423324 − 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 0 0
\(115\) 18.0000 1.67851
\(116\) 0 0
\(117\) −5.00000 −0.462250
\(118\) 0 0
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) −16.0000 −1.40872
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 0 0
\(135\) −12.0000 −1.03280
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 22.0000 1.86602 0.933008 − 0.359856i \(-0.117174\pi\)
0.933008 + 0.359856i \(0.117174\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 9.00000 0.747409
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) 0 0
\(149\) 3.00000 0.245770 0.122885 − 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 3.00000 0.242536
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) 22.0000 1.72317 0.861586 − 0.507611i \(-0.169471\pi\)
0.861586 + 0.507611i \(0.169471\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 8.00000 0.604743
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 − 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 0 0
\(183\) 20.0000 1.47844
\(184\) 0 0
\(185\) 21.0000 1.54395
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −8.00000 −0.581914
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −13.0000 −0.935760 −0.467880 − 0.883792i \(-0.654982\pi\)
−0.467880 + 0.883792i \(0.654982\pi\)
\(194\) 0 0
\(195\) −30.0000 −2.14834
\(196\) 0 0
\(197\) 15.0000 1.06871 0.534353 − 0.845262i \(-0.320555\pi\)
0.534353 + 0.845262i \(0.320555\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 − 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 20.0000 1.41069
\(202\) 0 0
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) 9.00000 0.628587
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 24.0000 1.64445
\(214\) 0 0
\(215\) −24.0000 −1.63679
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) 28.0000 1.89206
\(220\) 0 0
\(221\) −15.0000 −1.00901
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 − 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −5.00000 −0.330409 −0.165205 − 0.986259i \(-0.552828\pi\)
−0.165205 + 0.986259i \(0.552828\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −21.0000 −1.37576 −0.687878 − 0.725826i \(-0.741458\pi\)
−0.687878 + 0.725826i \(0.741458\pi\)
\(234\) 0 0
\(235\) 18.0000 1.17419
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) −30.0000 −1.94054 −0.970269 − 0.242028i \(-0.922188\pi\)
−0.970269 + 0.242028i \(0.922188\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) 0 0
\(245\) −9.00000 −0.574989
\(246\) 0 0
\(247\) 10.0000 0.636285
\(248\) 0 0
\(249\) −36.0000 −2.28141
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 − 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 18.0000 1.12720
\(256\) 0 0
\(257\) 3.00000 0.187135 0.0935674 − 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) 0 0
\(259\) 14.0000 0.869918
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 0 0
\(263\) 6.00000 0.369976 0.184988 − 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) 9.00000 0.552866
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) 0 0
\(269\) 3.00000 0.182913 0.0914566 − 0.995809i \(-0.470848\pi\)
0.0914566 + 0.995809i \(0.470848\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 − 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) −20.0000 −1.21046
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −5.00000 −0.300421 −0.150210 − 0.988654i \(-0.547995\pi\)
−0.150210 + 0.988654i \(0.547995\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) −12.0000 −0.710819
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 22.0000 1.28966
\(292\) 0 0
\(293\) −21.0000 −1.22683 −0.613417 − 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −30.0000 −1.73494
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 0 0
\(303\) −12.0000 −0.689382
\(304\) 0 0
\(305\) 30.0000 1.71780
\(306\) 0 0
\(307\) 10.0000 0.570730 0.285365 − 0.958419i \(-0.407885\pi\)
0.285365 + 0.958419i \(0.407885\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 − 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) 0 0
\(315\) 6.00000 0.338062
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) −20.0000 −1.10940
\(326\) 0 0
\(327\) −34.0000 −1.88020
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 − 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) 7.00000 0.383598
\(334\) 0 0
\(335\) 30.0000 1.63908
\(336\) 0 0
\(337\) −13.0000 −0.708155 −0.354078 − 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 36.0000 1.93817
\(346\) 0 0
\(347\) −36.0000 −1.93258 −0.966291 − 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) 0 0
\(349\) −17.0000 −0.909989 −0.454995 − 0.890494i \(-0.650359\pi\)
−0.454995 + 0.890494i \(0.650359\pi\)
\(350\) 0 0
\(351\) 20.0000 1.06752
\(352\) 0 0
\(353\) −9.00000 −0.479022 −0.239511 − 0.970894i \(-0.576987\pi\)
−0.239511 + 0.970894i \(0.576987\pi\)
\(354\) 0 0
\(355\) 36.0000 1.91068
\(356\) 0 0
\(357\) 12.0000 0.635107
\(358\) 0 0
\(359\) 18.0000 0.950004 0.475002 − 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 42.0000 2.19838
\(366\) 0 0
\(367\) −10.0000 −0.521996 −0.260998 − 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 0 0
\(369\) 3.00000 0.156174
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 − 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) −6.00000 −0.309839
\(376\) 0 0
\(377\) −15.0000 −0.772539
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 − 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 − 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.00000 −0.406663
\(388\) 0 0
\(389\) −33.0000 −1.67317 −0.836583 − 0.547840i \(-0.815450\pi\)
−0.836583 + 0.547840i \(0.815450\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6.00000 0.301893
\(396\) 0 0
\(397\) −17.0000 −0.853206 −0.426603 − 0.904439i \(-0.640290\pi\)
−0.426603 + 0.904439i \(0.640290\pi\)
\(398\) 0 0
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) −33.0000 −1.64794 −0.823971 − 0.566632i \(-0.808246\pi\)
−0.823971 + 0.566632i \(0.808246\pi\)
\(402\) 0 0
\(403\) −10.0000 −0.498135
\(404\) 0 0
\(405\) −33.0000 −1.63978
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −13.0000 −0.642809 −0.321404 − 0.946942i \(-0.604155\pi\)
−0.321404 + 0.946942i \(0.604155\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −54.0000 −2.65076
\(416\) 0 0
\(417\) 44.0000 2.15469
\(418\) 0 0
\(419\) 18.0000 0.879358 0.439679 − 0.898155i \(-0.355092\pi\)
0.439679 + 0.898155i \(0.355092\pi\)
\(420\) 0 0
\(421\) −17.0000 −0.828529 −0.414265 − 0.910156i \(-0.635961\pi\)
−0.414265 + 0.910156i \(0.635961\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) 20.0000 0.967868
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 − 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) −37.0000 −1.77811 −0.889053 − 0.457804i \(-0.848636\pi\)
−0.889053 + 0.457804i \(0.848636\pi\)
\(434\) 0 0
\(435\) 18.0000 0.863034
\(436\) 0 0
\(437\) −12.0000 −0.574038
\(438\) 0 0
\(439\) −22.0000 −1.05000 −0.525001 − 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 − 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 0 0
\(445\) −27.0000 −1.27992
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) −21.0000 −0.991051 −0.495526 − 0.868593i \(-0.665025\pi\)
−0.495526 + 0.868593i \(0.665025\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 16.0000 0.751746
\(454\) 0 0
\(455\) −30.0000 −1.40642
\(456\) 0 0
\(457\) −1.00000 −0.0467780 −0.0233890 − 0.999726i \(-0.507446\pi\)
−0.0233890 + 0.999726i \(0.507446\pi\)
\(458\) 0 0
\(459\) −12.0000 −0.560112
\(460\) 0 0
\(461\) −21.0000 −0.978068 −0.489034 − 0.872265i \(-0.662651\pi\)
−0.489034 + 0.872265i \(0.662651\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 − 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 0 0
\(465\) 12.0000 0.556487
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 − 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 20.0000 0.923514
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −8.00000 −0.367065
\(476\) 0 0
\(477\) 3.00000 0.137361
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −35.0000 −1.59586
\(482\) 0 0
\(483\) 24.0000 1.09204
\(484\) 0 0
\(485\) 33.0000 1.49845
\(486\) 0 0
\(487\) −34.0000 −1.54069 −0.770344 − 0.637629i \(-0.779915\pi\)
−0.770344 + 0.637629i \(0.779915\pi\)
\(488\) 0 0
\(489\) 44.0000 1.98975
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 − 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 0 0
\(493\) 9.00000 0.405340
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) 16.0000 0.716258 0.358129 − 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) −24.0000 −1.07224
\(502\) 0 0
\(503\) 30.0000 1.33763 0.668817 − 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) 24.0000 1.06588
\(508\) 0 0
\(509\) −6.00000 −0.265945 −0.132973 − 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 28.0000 1.23865
\(512\) 0 0
\(513\) 8.00000 0.353209
\(514\) 0 0
\(515\) 24.0000 1.05757
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 − 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 16.0000 0.698297
\(526\) 0 0
\(527\) 6.00000 0.261364
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −15.0000 −0.649722
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 − 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) −10.0000 −0.429141
\(544\) 0 0
\(545\) −51.0000 −2.18460
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 − 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 0 0
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 0 0
\(555\) 42.0000 1.78280
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 − 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 40.0000 1.69182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 42.0000 1.77009 0.885044 − 0.465506i \(-0.154128\pi\)
0.885044 + 0.465506i \(0.154128\pi\)
\(564\) 0 0
\(565\) −27.0000 −1.13590
\(566\) 0 0
\(567\) −22.0000 −0.923913
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 − 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 10.0000 0.418487 0.209243 − 0.977864i \(-0.432900\pi\)
0.209243 + 0.977864i \(0.432900\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 24.0000 1.00087
\(576\) 0 0
\(577\) 11.0000 0.457936 0.228968 − 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 0 0
\(579\) −26.0000 −1.08052
\(580\) 0 0
\(581\) −36.0000 −1.49353
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −15.0000 −0.620174
\(586\) 0 0
\(587\) 18.0000 0.742940 0.371470 − 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 30.0000 1.23404
\(592\) 0 0
\(593\) 27.0000 1.10876 0.554379 − 0.832265i \(-0.312956\pi\)
0.554379 + 0.832265i \(0.312956\pi\)
\(594\) 0 0
\(595\) 18.0000 0.737928
\(596\) 0 0
\(597\) −32.0000 −1.30967
\(598\) 0 0
\(599\) −42.0000 −1.71607 −0.858037 − 0.513588i \(-0.828316\pi\)
−0.858037 + 0.513588i \(0.828316\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 − 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) 0 0
\(603\) 10.0000 0.407231
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −22.0000 −0.892952 −0.446476 − 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) −30.0000 −1.21367
\(612\) 0 0
\(613\) −5.00000 −0.201948 −0.100974 − 0.994889i \(-0.532196\pi\)
−0.100974 + 0.994889i \(0.532196\pi\)
\(614\) 0 0
\(615\) 18.0000 0.725830
\(616\) 0 0
\(617\) 39.0000 1.57008 0.785040 − 0.619445i \(-0.212642\pi\)
0.785040 + 0.619445i \(0.212642\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 − 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) −24.0000 −0.963087
\(622\) 0 0
\(623\) −18.0000 −0.721155
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 21.0000 0.837325
\(630\) 0 0
\(631\) 38.0000 1.51276 0.756378 − 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) 0 0
\(633\) 8.00000 0.317971
\(634\) 0 0
\(635\) 24.0000 0.952411
\(636\) 0 0
\(637\) 15.0000 0.594322
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −33.0000 −1.30342 −0.651711 − 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) 0 0
\(643\) −2.00000 −0.0788723 −0.0394362 − 0.999222i \(-0.512556\pi\)
−0.0394362 + 0.999222i \(0.512556\pi\)
\(644\) 0 0
\(645\) −48.0000 −1.89000
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 − 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) 0 0
\(653\) −18.0000 −0.704394 −0.352197 − 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 − 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) −17.0000 −0.661223 −0.330612 − 0.943767i \(-0.607255\pi\)
−0.330612 + 0.943767i \(0.607255\pi\)
\(662\) 0 0
\(663\) −30.0000 −1.16510
\(664\) 0 0
\(665\) −12.0000 −0.465340
\(666\) 0 0
\(667\) 18.0000 0.696963
\(668\) 0 0
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 38.0000 1.46479 0.732396 − 0.680879i \(-0.238402\pi\)
0.732396 + 0.680879i \(0.238402\pi\)
\(674\) 0 0
\(675\) −16.0000 −0.615840
\(676\) 0 0
\(677\) 15.0000 0.576497 0.288248 − 0.957556i \(-0.406927\pi\)
0.288248 + 0.957556i \(0.406927\pi\)
\(678\) 0 0
\(679\) 22.0000 0.844283
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −6.00000 −0.229584 −0.114792 − 0.993390i \(-0.536620\pi\)
−0.114792 + 0.993390i \(0.536620\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) 0 0
\(689\) −15.0000 −0.571454
\(690\) 0 0
\(691\) 4.00000 0.152167 0.0760836 − 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 66.0000 2.50352
\(696\) 0 0
\(697\) 9.00000 0.340899
\(698\) 0 0
\(699\) −42.0000 −1.58859
\(700\) 0 0
\(701\) 27.0000 1.01978 0.509888 − 0.860241i \(-0.329687\pi\)
0.509888 + 0.860241i \(0.329687\pi\)
\(702\) 0 0
\(703\) −14.0000 −0.528020
\(704\) 0 0
\(705\) 36.0000 1.35584
\(706\) 0 0
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 − 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 2.00000 0.0750059
\(712\) 0 0
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −60.0000 −2.24074
\(718\) 0 0
\(719\) 18.0000 0.671287 0.335643 − 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) −20.0000 −0.743808
\(724\) 0 0
\(725\) 12.0000 0.445669
\(726\) 0 0
\(727\) 26.0000 0.964287 0.482143 − 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) −5.00000 −0.184679 −0.0923396 − 0.995728i \(-0.529435\pi\)
−0.0923396 + 0.995728i \(0.529435\pi\)
\(734\) 0 0
\(735\) −18.0000 −0.663940
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −26.0000 −0.956425 −0.478213 − 0.878244i \(-0.658715\pi\)
−0.478213 + 0.878244i \(0.658715\pi\)
\(740\) 0 0
\(741\) 20.0000 0.734718
\(742\) 0 0
\(743\) −18.0000 −0.660356 −0.330178 − 0.943919i \(-0.607109\pi\)
−0.330178 + 0.943919i \(0.607109\pi\)
\(744\) 0 0
\(745\) 9.00000 0.329734
\(746\) 0 0
\(747\) −18.0000 −0.658586
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 44.0000 1.60558 0.802791 − 0.596260i \(-0.203347\pi\)
0.802791 + 0.596260i \(0.203347\pi\)
\(752\) 0 0
\(753\) −36.0000 −1.31191
\(754\) 0 0
\(755\) 24.0000 0.873449
\(756\) 0 0
\(757\) −41.0000 −1.49017 −0.745085 − 0.666969i \(-0.767591\pi\)
−0.745085 + 0.666969i \(0.767591\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.00000 0.108750 0.0543750 − 0.998521i \(-0.482683\pi\)
0.0543750 + 0.998521i \(0.482683\pi\)
\(762\) 0 0
\(763\) −34.0000 −1.23088
\(764\) 0 0
\(765\) 9.00000 0.325396
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 35.0000 1.26213 0.631066 − 0.775729i \(-0.282618\pi\)
0.631066 + 0.775729i \(0.282618\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 0 0
\(773\) 42.0000 1.51064 0.755318 − 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) 28.0000 1.00449
\(778\) 0 0
\(779\) −6.00000 −0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −12.0000 −0.428845
\(784\) 0 0
\(785\) −6.00000 −0.214149
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 − 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 0 0
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) −50.0000 −1.77555
\(794\) 0 0
\(795\) 18.0000 0.638394
\(796\) 0 0
\(797\) 42.0000 1.48772 0.743858 − 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) 0 0
\(801\) −9.00000 −0.317999
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 36.0000 1.26883
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 − 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 − 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 40.0000 1.40286
\(814\) 0 0
\(815\) 66.0000 2.31188
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) −10.0000 −0.349428
\(820\) 0 0
\(821\) −30.0000 −1.04701 −0.523504 − 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 − 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −18.0000 −0.625921 −0.312961 − 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) 0 0
\(829\) 43.0000 1.49345 0.746726 − 0.665132i \(-0.231625\pi\)
0.746726 + 0.665132i \(0.231625\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) −9.00000 −0.311832
\(834\) 0 0
\(835\) −36.0000 −1.24583
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 0 0
\(839\) 18.0000 0.621429 0.310715 − 0.950503i \(-0.399432\pi\)
0.310715 + 0.950503i \(0.399432\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 12.0000 0.413302
\(844\) 0 0
\(845\) 36.0000 1.23844
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −40.0000 −1.37280
\(850\) 0 0
\(851\) 42.0000 1.43974
\(852\) 0 0
\(853\) −29.0000 −0.992941 −0.496471 − 0.868054i \(-0.665371\pi\)
−0.496471 + 0.868054i \(0.665371\pi\)
\(854\) 0 0
\(855\) −6.00000 −0.205196
\(856\) 0 0
\(857\) −54.0000 −1.84460 −0.922302 − 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) 0 0
\(859\) −32.0000 −1.09183 −0.545913 − 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) 0 0
\(863\) 6.00000 0.204242 0.102121 − 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) −16.0000 −0.543388
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −50.0000 −1.69419
\(872\) 0 0
\(873\) 11.0000 0.372294
\(874\) 0 0
\(875\) −6.00000 −0.202837
\(876\) 0 0
\(877\) −41.0000 −1.38447 −0.692236 − 0.721671i \(-0.743374\pi\)
−0.692236 + 0.721671i \(0.743374\pi\)
\(878\) 0 0
\(879\) −42.0000 −1.41662
\(880\) 0 0
\(881\) −45.0000 −1.51609 −0.758044 − 0.652203i \(-0.773845\pi\)
−0.758044 + 0.652203i \(0.773845\pi\)
\(882\) 0 0
\(883\) 4.00000 0.134611 0.0673054 − 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −18.0000 −0.604381 −0.302190 − 0.953248i \(-0.597718\pi\)
−0.302190 + 0.953248i \(0.597718\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −60.0000 −2.00334
\(898\) 0 0
\(899\) 6.00000 0.200111
\(900\) 0 0
\(901\) 9.00000 0.299833
\(902\) 0 0
\(903\) −32.0000 −1.06489
\(904\) 0 0
\(905\) −15.0000 −0.498617
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 − 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 − 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 60.0000 1.98354
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −28.0000 −0.923635 −0.461817 − 0.886975i \(-0.652802\pi\)
−0.461817 + 0.886975i \(0.652802\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 0 0
\(923\) −60.0000 −1.97492
\(924\) 0 0
\(925\) 28.0000 0.920634
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 51.0000 1.67326 0.836628 − 0.547772i \(-0.184524\pi\)
0.836628 + 0.547772i \(0.184524\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 23.0000 0.751377 0.375689 − 0.926746i \(-0.377406\pi\)
0.375689 + 0.926746i \(0.377406\pi\)
\(938\) 0 0
\(939\) −2.00000 −0.0652675
\(940\) 0 0
\(941\) 39.0000 1.27136 0.635682 − 0.771951i \(-0.280719\pi\)
0.635682 + 0.771951i \(0.280719\pi\)
\(942\) 0 0
\(943\) 18.0000 0.586161
\(944\) 0 0
\(945\) −24.0000 −0.780720
\(946\) 0 0
\(947\) 30.0000 0.974869 0.487435 − 0.873160i \(-0.337933\pi\)
0.487435 + 0.873160i \(0.337933\pi\)
\(948\) 0 0
\(949\) −70.0000 −2.27230
\(950\) 0 0
\(951\) 36.0000 1.16738
\(952\) 0 0
\(953\) 39.0000 1.26333 0.631667 − 0.775240i \(-0.282371\pi\)
0.631667 + 0.775240i \(0.282371\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) 0 0
\(965\) −39.0000 −1.25545
\(966\) 0 0
\(967\) 50.0000 1.60789 0.803946 − 0.594703i \(-0.202730\pi\)
0.803946 + 0.594703i \(0.202730\pi\)
\(968\) 0 0
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 − 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 0 0
\(973\) 44.0000 1.41058
\(974\) 0 0
\(975\) −40.0000 −1.28103
\(976\) 0 0
\(977\) −33.0000 −1.05576 −0.527882 − 0.849318i \(-0.677014\pi\)
−0.527882 + 0.849318i \(0.677014\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −17.0000 −0.542768
\(982\) 0 0
\(983\) 60.0000 1.91370 0.956851 − 0.290578i \(-0.0938475\pi\)
0.956851 + 0.290578i \(0.0938475\pi\)
\(984\) 0 0
\(985\) 45.0000 1.43382
\(986\) 0 0
\(987\) 24.0000 0.763928
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) −52.0000 −1.65183 −0.825917 − 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) 0 0
\(993\) −40.0000 −1.26936
\(994\) 0 0
\(995\) −48.0000 −1.52170
\(996\) 0 0
\(997\) 31.0000 0.981780 0.490890 − 0.871222i \(-0.336672\pi\)
0.490890 + 0.871222i \(0.336672\pi\)
\(998\) 0 0
\(999\) −28.0000 −0.885881
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 7744.2.a.bi.1.1 1
4.3 odd 2 7744.2.a.g.1.1 1
8.3 odd 2 1936.2.a.j.1.1 1
8.5 even 2 242.2.a.a.1.1 ✓ 1
11.10 odd 2 7744.2.a.bh.1.1 1
24.5 odd 2 2178.2.a.l.1.1 1
40.29 even 2 6050.2.a.bl.1.1 1
44.43 even 2 7744.2.a.h.1.1 1
88.5 even 10 242.2.c.e.3.1 4
88.13 odd 10 242.2.c.b.81.1 4
88.21 odd 2 242.2.a.b.1.1 yes 1
88.29 odd 10 242.2.c.b.27.1 4
88.37 even 10 242.2.c.e.27.1 4
88.43 even 2 1936.2.a.k.1.1 1
88.53 even 10 242.2.c.e.81.1 4
88.61 odd 10 242.2.c.b.3.1 4
88.69 even 10 242.2.c.e.9.1 4
88.85 odd 10 242.2.c.b.9.1 4
264.197 even 2 2178.2.a.f.1.1 1
440.109 odd 2 6050.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
242.2.a.a.1.1 ✓ 1 8.5 even 2
242.2.a.b.1.1 yes 1 88.21 odd 2
242.2.c.b.3.1 4 88.61 odd 10
242.2.c.b.9.1 4 88.85 odd 10
242.2.c.b.27.1 4 88.29 odd 10
242.2.c.b.81.1 4 88.13 odd 10
242.2.c.e.3.1 4 88.5 even 10
242.2.c.e.9.1 4 88.69 even 10
242.2.c.e.27.1 4 88.37 even 10
242.2.c.e.81.1 4 88.53 even 10
1936.2.a.j.1.1 1 8.3 odd 2
1936.2.a.k.1.1 1 88.43 even 2
2178.2.a.f.1.1 1 264.197 even 2
2178.2.a.l.1.1 1 24.5 odd 2
6050.2.a.s.1.1 1 440.109 odd 2
6050.2.a.bl.1.1 1 40.29 even 2
7744.2.a.g.1.1 1 4.3 odd 2
7744.2.a.h.1.1 1 44.43 even 2
7744.2.a.bh.1.1 1 11.10 odd 2
7744.2.a.bi.1.1 1 1.1 even 1 trivial