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Properties

Label 8820.2.a.p.1.1
Level $8820$
Weight $2$
Character 8820.1
Self dual yes
Analytic conductor $70.428$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8820,2,Mod(1,8820)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8820, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8820.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8820.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: \(70.4280545828\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8820.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +O(q^{10})\) \(q+1.00000 q^{5} -6.00000 q^{11} +2.00000 q^{13} +6.00000 q^{17} +8.00000 q^{19} -3.00000 q^{23} +1.00000 q^{25} -3.00000 q^{29} +2.00000 q^{31} +8.00000 q^{37} +3.00000 q^{41} +5.00000 q^{43} -12.0000 q^{53} -6.00000 q^{55} -1.00000 q^{61} +2.00000 q^{65} -7.00000 q^{67} -10.0000 q^{73} -4.00000 q^{79} -3.00000 q^{83} +6.00000 q^{85} +3.00000 q^{89} +8.00000 q^{95} -10.0000 q^{97} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 − 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 − 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 − 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 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
\(35\) 0 0
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 − 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 5.00000 0.762493 0.381246 − 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) −6.00000 −0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 − 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −7.00000 −0.855186 −0.427593 − 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 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
\(81\) 0 0
\(82\) 0 0
\(83\) −3.00000 −0.329293 −0.164646 − 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.00000 0.298511 0.149256 − 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) −7.00000 −0.689730 −0.344865 − 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.00000 0.290021 0.145010 − 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 0 0
\(109\) 17.0000 1.62830 0.814152 − 0.580651i \(-0.197202\pi\)
0.814152 + 0.580651i \(0.197202\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 − 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 − 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 − 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) −3.00000 −0.249136
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.0000 1.22885 0.614424 − 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 − 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 − 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 − 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.0000 1.62503 0.812514 − 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.00000 0.448461 0.224231 − 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) 17.0000 1.26360 0.631800 − 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) −36.0000 −2.63258
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 − 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 − 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 − 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 3.00000 0.209529
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −48.0000 −3.32023
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 − 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.00000 0.340997
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 − 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 − 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 − 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 16.0000 1.01806
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 − 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 24.0000 1.49708 0.748539 − 0.663090i \(-0.230755\pi\)
0.748539 + 0.663090i \(0.230755\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3.00000 0.184988 0.0924940 − 0.995713i \(-0.470516\pi\)
0.0924940 + 0.995713i \(0.470516\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.0000 0.914566 0.457283 − 0.889321i \(-0.348823\pi\)
0.457283 + 0.889321i \(0.348823\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.00000 −0.361814
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 − 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 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 \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 12.0000 0.701047 0.350524 − 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.00000 −0.0572598
\(306\) 0 0
\(307\) −19.0000 −1.08439 −0.542194 − 0.840254i \(-0.682406\pi\)
−0.542194 + 0.840254i \(0.682406\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −4.00000 −0.226093 −0.113047 − 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 18.0000 1.00781
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 48.0000 2.67079
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −22.0000 −1.20923 −0.604615 − 0.796518i \(-0.706673\pi\)
−0.604615 + 0.796518i \(0.706673\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −7.00000 −0.382451
\(336\) 0 0
\(337\) 20.0000 1.08947 0.544735 − 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 27.0000 1.44944 0.724718 − 0.689046i \(-0.241970\pi\)
0.724718 + 0.689046i \(0.241970\pi\)
\(348\) 0 0
\(349\) −1.00000 −0.0535288 −0.0267644 − 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.0000 0.638696 0.319348 − 0.947638i \(-0.396536\pi\)
0.319348 + 0.947638i \(0.396536\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 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
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) 5.00000 0.260998 0.130499 − 0.991448i \(-0.458342\pi\)
0.130499 + 0.991448i \(0.458342\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −16.0000 −0.828449 −0.414224 − 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 − 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.00000 −0.153293 −0.0766464 − 0.997058i \(-0.524421\pi\)
−0.0766464 + 0.997058i \(0.524421\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 − 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) 26.0000 1.30490 0.652451 − 0.757831i \(-0.273741\pi\)
0.652451 + 0.757831i \(0.273741\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.00000 −0.149813 −0.0749064 − 0.997191i \(-0.523866\pi\)
−0.0749064 + 0.997191i \(0.523866\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −48.0000 −2.37927
\(408\) 0 0
\(409\) 11.0000 0.543915 0.271957 − 0.962309i \(-0.412329\pi\)
0.271957 + 0.962309i \(0.412329\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −3.00000 −0.147264
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 − 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) 23.0000 1.12095 0.560476 − 0.828171i \(-0.310618\pi\)
0.560476 + 0.828171i \(0.310618\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 − 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) −28.0000 −1.34559 −0.672797 − 0.739827i \(-0.734907\pi\)
−0.672797 + 0.739827i \(0.734907\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −24.0000 −1.14808
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 − 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −9.00000 −0.427603 −0.213801 − 0.976877i \(-0.568585\pi\)
−0.213801 + 0.976877i \(0.568585\pi\)
\(444\) 0 0
\(445\) 3.00000 0.142214
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9.00000 0.424736 0.212368 − 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) 0 0
\(451\) −18.0000 −0.847587
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −4.00000 −0.187112 −0.0935561 − 0.995614i \(-0.529823\pi\)
−0.0935561 + 0.995614i \(0.529823\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 − 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 29.0000 1.34774 0.673872 − 0.738848i \(-0.264630\pi\)
0.673872 + 0.738848i \(0.264630\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −33.0000 −1.52706 −0.763529 − 0.645774i \(-0.776535\pi\)
−0.763529 + 0.645774i \(0.776535\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −30.0000 −1.37940
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 30.0000 1.37073 0.685367 − 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 − 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.00000 −0.270776 −0.135388 − 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 0 0
\(493\) −18.0000 −0.810679
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2.00000 0.0895323 0.0447661 − 0.998997i \(-0.485746\pi\)
0.0447661 + 0.998997i \(0.485746\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.00000 0.401290 0.200645 − 0.979664i \(-0.435696\pi\)
0.200645 + 0.979664i \(0.435696\pi\)
\(504\) 0 0
\(505\) 3.00000 0.133498
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −39.0000 −1.72864 −0.864322 − 0.502938i \(-0.832252\pi\)
−0.864322 + 0.502938i \(0.832252\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7.00000 −0.308457
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 − 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) 44.0000 1.92399 0.961993 − 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0000 0.522728
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.00000 0.259889
\(534\) 0 0
\(535\) 3.00000 0.129701
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 17.0000 0.730887 0.365444 − 0.930834i \(-0.380917\pi\)
0.365444 + 0.930834i \(0.380917\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 17.0000 0.728200
\(546\) 0 0
\(547\) −19.0000 −0.812381 −0.406191 − 0.913788i \(-0.633143\pi\)
−0.406191 + 0.913788i \(0.633143\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −30.0000 −1.27114 −0.635570 − 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) 10.0000 0.422955
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.00000 0.379305 0.189652 − 0.981851i \(-0.439264\pi\)
0.189652 + 0.981851i \(0.439264\pi\)
\(564\) 0 0
\(565\) 12.0000 0.504844
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 − 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −16.0000 −0.669579 −0.334790 − 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3.00000 −0.125109
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 − 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 72.0000 2.98194
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 − 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 12.0000 0.492781 0.246390 − 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 − 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 25.0000 1.01639
\(606\) 0 0
\(607\) −43.0000 −1.74532 −0.872658 − 0.488332i \(-0.837606\pi\)
−0.872658 + 0.488332i \(0.837606\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 − 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.0000 0.483102 0.241551 − 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 0 0
\(619\) 14.0000 0.562708 0.281354 − 0.959604i \(-0.409217\pi\)
0.281354 + 0.959604i \(0.409217\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 48.0000 1.91389
\(630\) 0 0
\(631\) 38.0000 1.51276 0.756378 − 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 39.0000 1.54041 0.770204 − 0.637798i \(-0.220155\pi\)
0.770204 + 0.637798i \(0.220155\pi\)
\(642\) 0 0
\(643\) 20.0000 0.788723 0.394362 − 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −21.0000 −0.825595 −0.412798 − 0.910823i \(-0.635448\pi\)
−0.412798 + 0.910823i \(0.635448\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 42.0000 1.64359 0.821794 − 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 − 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −25.0000 −0.972387 −0.486194 − 0.873851i \(-0.661615\pi\)
−0.486194 + 0.873851i \(0.661615\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9.00000 0.348481
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 6.00000 0.231627
\(672\) 0 0
\(673\) 32.0000 1.23351 0.616755 − 0.787155i \(-0.288447\pi\)
0.616755 + 0.787155i \(0.288447\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 − 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 15.0000 0.573959 0.286980 − 0.957937i \(-0.407349\pi\)
0.286980 + 0.957937i \(0.407349\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 14.0000 0.532585 0.266293 − 0.963892i \(-0.414201\pi\)
0.266293 + 0.963892i \(0.414201\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.00000 0.0758643
\(696\) 0 0
\(697\) 18.0000 0.681799
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −27.0000 −1.01978 −0.509888 − 0.860241i \(-0.670313\pi\)
−0.509888 + 0.860241i \(0.670313\pi\)
\(702\) 0 0
\(703\) 64.0000 2.41381
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.00000 −0.0375558 −0.0187779 − 0.999824i \(-0.505978\pi\)
−0.0187779 + 0.999824i \(0.505978\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 30.0000 1.11881 0.559406 − 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.00000 −0.111417
\(726\) 0 0
\(727\) 23.0000 0.853023 0.426511 − 0.904482i \(-0.359742\pi\)
0.426511 + 0.904482i \(0.359742\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 30.0000 1.10959
\(732\) 0 0
\(733\) 2.00000 0.0738717 0.0369358 − 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 42.0000 1.54709
\(738\) 0 0
\(739\) −46.0000 −1.69214 −0.846069 − 0.533074i \(-0.821037\pi\)
−0.846069 + 0.533074i \(0.821037\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 39.0000 1.43077 0.715386 − 0.698730i \(-0.246251\pi\)
0.715386 + 0.698730i \(0.246251\pi\)
\(744\) 0 0
\(745\) 15.0000 0.549557
\(746\) 0 0
\(747\) 0 0
\(748\) 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
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 − 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 − 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 42.0000 1.51064 0.755318 − 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) −31.0000 −1.10503 −0.552515 − 0.833503i \(-0.686332\pi\)
−0.552515 + 0.833503i \(0.686332\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.0000 0.425062 0.212531 − 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 60.0000 2.11735
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 51.0000 1.79306 0.896532 − 0.442978i \(-0.146078\pi\)
0.896532 + 0.442978i \(0.146078\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 − 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −16.0000 −0.560456
\(816\) 0 0
\(817\) 40.0000 1.39942
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 − 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) −49.0000 −1.70803 −0.854016 − 0.520246i \(-0.825840\pi\)
−0.854016 + 0.520246i \(0.825840\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 45.0000 1.56480 0.782402 − 0.622774i \(-0.213994\pi\)
0.782402 + 0.622774i \(0.213994\pi\)
\(828\) 0 0
\(829\) −10.0000 −0.347314 −0.173657 − 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 21.0000 0.726735
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 42.0000 1.45000 0.725001 − 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) 0 0
\(853\) 32.0000 1.09566 0.547830 − 0.836590i \(-0.315454\pi\)
0.547830 + 0.836590i \(0.315454\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 − 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 − 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −33.0000 −1.12333 −0.561667 − 0.827364i \(-0.689840\pi\)
−0.561667 + 0.827364i \(0.689840\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) −14.0000 −0.474372
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −40.0000 −1.35070 −0.675352 − 0.737496i \(-0.736008\pi\)
−0.675352 + 0.737496i \(0.736008\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −9.00000 −0.303218 −0.151609 − 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) 0 0
\(883\) −28.0000 −0.942275 −0.471138 − 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −57.0000 −1.91387 −0.956936 − 0.290298i \(-0.906246\pi\)
−0.956936 + 0.290298i \(0.906246\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 6.00000 0.200558
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −6.00000 −0.200111
\(900\) 0 0
\(901\) −72.0000 −2.39867
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 17.0000 0.565099
\(906\) 0 0
\(907\) 23.0000 0.763702 0.381851 − 0.924224i \(-0.375287\pi\)
0.381851 + 0.924224i \(0.375287\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 − 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) 18.0000 0.595713
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 − 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 45.0000 1.47640 0.738201 − 0.674581i \(-0.235676\pi\)
0.738201 + 0.674581i \(0.235676\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −36.0000 −1.17733
\(936\) 0 0
\(937\) 44.0000 1.43742 0.718709 − 0.695311i \(-0.244734\pi\)
0.718709 + 0.695311i \(0.244734\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −54.0000 −1.76035 −0.880175 − 0.474650i \(-0.842575\pi\)
−0.880175 + 0.474650i \(0.842575\pi\)
\(942\) 0 0
\(943\) −9.00000 −0.293080
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −9.00000 −0.292461 −0.146230 − 0.989251i \(-0.546714\pi\)
−0.146230 + 0.989251i \(0.546714\pi\)
\(948\) 0 0
\(949\) −20.0000 −0.649227
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −36.0000 −1.16615 −0.583077 − 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 0 0
\(955\) 18.0000 0.582466
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) −7.00000 −0.225105 −0.112552 − 0.993646i \(-0.535903\pi\)
−0.112552 + 0.993646i \(0.535903\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 − 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 54.0000 1.72761 0.863807 − 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 39.0000 1.24391 0.621953 − 0.783054i \(-0.286339\pi\)
0.621953 + 0.783054i \(0.286339\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −15.0000 −0.476972
\(990\) 0 0
\(991\) 26.0000 0.825917 0.412959 − 0.910750i \(-0.364495\pi\)
0.412959 + 0.910750i \(0.364495\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 20.0000 0.634043
\(996\) 0 0
\(997\) 32.0000 1.01345 0.506725 − 0.862108i \(-0.330856\pi\)
0.506725 + 0.862108i \(0.330856\pi\)
\(998\) 0 0
\(999\) 0 0
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 8820.2.a.p.1.1 1
3.2 odd 2 980.2.a.g.1.1 1
7.2 even 3 1260.2.s.c.361.1 2
7.4 even 3 1260.2.s.c.541.1 2
7.6 odd 2 8820.2.a.a.1.1 1
12.11 even 2 3920.2.a.k.1.1 1
15.2 even 4 4900.2.e.m.2549.1 2
15.8 even 4 4900.2.e.m.2549.2 2
15.14 odd 2 4900.2.a.i.1.1 1
21.2 odd 6 140.2.i.a.81.1 ✓ 2
21.5 even 6 980.2.i.f.361.1 2
21.11 odd 6 140.2.i.a.121.1 yes 2
21.17 even 6 980.2.i.f.961.1 2
21.20 even 2 980.2.a.e.1.1 1
84.11 even 6 560.2.q.f.401.1 2
84.23 even 6 560.2.q.f.81.1 2
84.83 odd 2 3920.2.a.w.1.1 1
105.2 even 12 700.2.r.a.249.2 4
105.23 even 12 700.2.r.a.249.1 4
105.32 even 12 700.2.r.a.149.1 4
105.44 odd 6 700.2.i.b.501.1 2
105.53 even 12 700.2.r.a.149.2 4
105.62 odd 4 4900.2.e.n.2549.2 2
105.74 odd 6 700.2.i.b.401.1 2
105.83 odd 4 4900.2.e.n.2549.1 2
105.104 even 2 4900.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.i.a.81.1 ✓ 2 21.2 odd 6
140.2.i.a.121.1 yes 2 21.11 odd 6
560.2.q.f.81.1 2 84.23 even 6
560.2.q.f.401.1 2 84.11 even 6
700.2.i.b.401.1 2 105.74 odd 6
700.2.i.b.501.1 2 105.44 odd 6
700.2.r.a.149.1 4 105.32 even 12
700.2.r.a.149.2 4 105.53 even 12
700.2.r.a.249.1 4 105.23 even 12
700.2.r.a.249.2 4 105.2 even 12
980.2.a.e.1.1 1 21.20 even 2
980.2.a.g.1.1 1 3.2 odd 2
980.2.i.f.361.1 2 21.5 even 6
980.2.i.f.961.1 2 21.17 even 6
1260.2.s.c.361.1 2 7.2 even 3
1260.2.s.c.541.1 2 7.4 even 3
3920.2.a.k.1.1 1 12.11 even 2
3920.2.a.w.1.1 1 84.83 odd 2
4900.2.a.i.1.1 1 15.14 odd 2
4900.2.a.q.1.1 1 105.104 even 2
4900.2.e.m.2549.1 2 15.2 even 4
4900.2.e.m.2549.2 2 15.8 even 4
4900.2.e.n.2549.1 2 105.83 odd 4
4900.2.e.n.2549.2 2 105.62 odd 4
8820.2.a.a.1.1 1 7.6 odd 2
8820.2.a.p.1.1 1 1.1 even 1 trivial