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Properties

Label 4600.2.a.f.1.1
Level $4600$
Weight $2$
Character 4600.1
Self dual yes
Analytic conductor $36.731$
Analytic rank $1$
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] = [4600,2,Mod(1,4600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4600, 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("4600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.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: \(36.7311849298\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4600.1

$q$-expansion

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

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
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 − 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 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
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 − 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 − 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.00000 −1.31278 −0.656392 − 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 − 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 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
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 − 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 − 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −8.00000 −0.878114 −0.439057 − 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 − 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 − 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 5.00000 0.450835
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 − 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) 3.00000 0.262111 0.131056 − 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 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
\(139\) 1.00000 0.0848189 0.0424094 − 0.999100i \(-0.486497\pi\)
0.0424094 + 0.999100i \(0.486497\pi\)
\(140\) 0 0
\(141\) 9.00000 0.757937
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) 18.0000 1.47462 0.737309 − 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) −13.0000 −1.05792 −0.528962 − 0.848645i \(-0.677419\pi\)
−0.528962 + 0.848645i \(0.677419\pi\)
\(152\) 0 0
\(153\) −8.00000 −0.646762
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 − 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) −11.0000 −0.861586 −0.430793 − 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −24.0000 −1.85718 −0.928588 − 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) 0 0
\(173\) −14.0000 −1.06440 −0.532200 − 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.00000 0.672692 0.336346 − 0.941739i \(-0.390809\pi\)
0.336346 + 0.941739i \(0.390809\pi\)
\(180\) 0 0
\(181\) −12.0000 −0.891953 −0.445976 − 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 10.0000 0.727393
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 − 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 0 0
\(193\) 9.00000 0.647834 0.323917 − 0.946085i \(-0.395000\pi\)
0.323917 + 0.946085i \(0.395000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.0000 0.926212 0.463106 − 0.886303i \(-0.346735\pi\)
0.463106 + 0.886303i \(0.346735\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 − 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 − 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) −3.00000 −0.205557
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) 0 0
\(219\) 7.00000 0.473016
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 − 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.0000 −0.663723 −0.331862 − 0.943328i \(-0.607677\pi\)
−0.331862 + 0.943328i \(0.607677\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 − 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 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
\(235\) 0 0
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) −21.0000 −1.35838 −0.679189 − 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 − 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.00000 0.187135 0.0935674 − 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) 0 0
\(259\) 16.0000 0.994192
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) −16.0000 −0.986602 −0.493301 − 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 14.0000 0.856786
\(268\) 0 0
\(269\) 25.0000 1.52428 0.762138 − 0.647414i \(-0.224150\pi\)
0.762138 + 0.647414i \(0.224150\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 − 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 0 0
\(273\) 2.00000 0.121046
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3.00000 −0.180253 −0.0901263 − 0.995930i \(-0.528727\pi\)
−0.0901263 + 0.995930i \(0.528727\pi\)
\(278\) 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
\(283\) 10.0000 0.594438 0.297219 − 0.954809i \(-0.403941\pi\)
0.297219 + 0.954809i \(0.403941\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −10.0000 −0.590281
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) 16.0000 0.934730 0.467365 − 0.884064i \(-0.345203\pi\)
0.467365 + 0.884064i \(0.345203\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.00000 0.0578315
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) −2.00000 −0.114897
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 − 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −9.00000 −0.510343 −0.255172 − 0.966896i \(-0.582132\pi\)
−0.255172 + 0.966896i \(0.582132\pi\)
\(312\) 0 0
\(313\) 4.00000 0.226093 0.113047 − 0.993590i \(-0.463939\pi\)
0.113047 + 0.993590i \(0.463939\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −26.0000 −1.46031 −0.730153 − 0.683284i \(-0.760551\pi\)
−0.730153 + 0.683284i \(0.760551\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) −16.0000 −0.890264
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2.00000 −0.110600
\(328\) 0 0
\(329\) −18.0000 −0.992372
\(330\) 0 0
\(331\) −19.0000 −1.04433 −0.522167 − 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) 0 0
\(333\) −16.0000 −0.876795
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 10.0000 0.544735 0.272367 − 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4.00000 −0.214731 −0.107366 − 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) −15.0000 −0.802932 −0.401466 − 0.915874i \(-0.631499\pi\)
−0.401466 + 0.915874i \(0.631499\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) 0 0
\(353\) −3.00000 −0.159674 −0.0798369 − 0.996808i \(-0.525440\pi\)
−0.0798369 + 0.996808i \(0.525440\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −8.00000 −0.423405
\(358\) 0 0
\(359\) 4.00000 0.211112 0.105556 − 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 36.0000 1.87918 0.939592 − 0.342296i \(-0.111204\pi\)
0.939592 + 0.342296i \(0.111204\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 − 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.00000 0.154508
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 7.00000 0.358621
\(382\) 0 0
\(383\) −2.00000 −0.102195 −0.0510976 − 0.998694i \(-0.516272\pi\)
−0.0510976 + 0.998694i \(0.516272\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −12.0000 −0.609994
\(388\) 0 0
\(389\) −24.0000 −1.21685 −0.608424 − 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 0 0
\(393\) −3.00000 −0.151330
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 7.00000 0.351320 0.175660 − 0.984451i \(-0.443794\pi\)
0.175660 + 0.984451i \(0.443794\pi\)
\(398\) 0 0
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 − 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) 1.00000 0.0498135
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −19.0000 −0.939490 −0.469745 − 0.882802i \(-0.655654\pi\)
−0.469745 + 0.882802i \(0.655654\pi\)
\(410\) 0 0
\(411\) −12.0000 −0.591916
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.00000 −0.0489702
\(418\) 0 0
\(419\) −6.00000 −0.293119 −0.146560 − 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 − 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 18.0000 0.875190
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 − 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) −19.0000 −0.906821 −0.453410 − 0.891302i \(-0.649793\pi\)
−0.453410 + 0.891302i \(0.649793\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) −5.00000 −0.237557 −0.118779 − 0.992921i \(-0.537898\pi\)
−0.118779 + 0.992921i \(0.537898\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −18.0000 −0.851371
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 − 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 13.0000 0.610793
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 40.0000 1.87112 0.935561 − 0.353166i \(-0.114895\pi\)
0.935561 + 0.353166i \(0.114895\pi\)
\(458\) 0 0
\(459\) 20.0000 0.933520
\(460\) 0 0
\(461\) −3.00000 −0.139724 −0.0698620 − 0.997557i \(-0.522256\pi\)
−0.0698620 + 0.997557i \(0.522256\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 − 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −18.0000 −0.832941 −0.416470 − 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 4.00000 0.183147
\(478\) 0 0
\(479\) −4.00000 −0.182765 −0.0913823 − 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 0 0
\(483\) 2.00000 0.0910032
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 5.00000 0.226572 0.113286 − 0.993562i \(-0.463862\pi\)
0.113286 + 0.993562i \(0.463862\pi\)
\(488\) 0 0
\(489\) 11.0000 0.497437
\(490\) 0 0
\(491\) −27.0000 −1.21849 −0.609246 − 0.792981i \(-0.708528\pi\)
−0.609246 + 0.792981i \(0.708528\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) 19.0000 0.850557 0.425278 − 0.905063i \(-0.360176\pi\)
0.425278 + 0.905063i \(0.360176\pi\)
\(500\) 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
\(505\) 0 0
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 0 0
\(509\) −19.0000 −0.842160 −0.421080 − 0.907023i \(-0.638349\pi\)
−0.421080 + 0.907023i \(0.638349\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) 0 0
\(513\) −20.0000 −0.883022
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 − 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 0 0
\(523\) 38.0000 1.66162 0.830812 − 0.556553i \(-0.187876\pi\)
0.830812 + 0.556553i \(0.187876\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 5.00000 0.216574
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −9.00000 −0.388379
\(538\) 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
\(543\) 12.0000 0.514969
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.00000 0.0427569 0.0213785 − 0.999771i \(-0.493195\pi\)
0.0213785 + 0.999771i \(0.493195\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −34.0000 −1.44063 −0.720313 − 0.693649i \(-0.756002\pi\)
−0.720313 + 0.693649i \(0.756002\pi\)
\(558\) 0 0
\(559\) −6.00000 −0.253773
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 − 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) −38.0000 −1.59304 −0.796521 − 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) 0 0
\(571\) 24.0000 1.00437 0.502184 − 0.864761i \(-0.332530\pi\)
0.502184 + 0.864761i \(0.332530\pi\)
\(572\) 0 0
\(573\) 10.0000 0.417756
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −23.0000 −0.957503 −0.478751 − 0.877951i \(-0.658910\pi\)
−0.478751 + 0.877951i \(0.658910\pi\)
\(578\) 0 0
\(579\) −9.00000 −0.374027
\(580\) 0 0
\(581\) −16.0000 −0.663792
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −29.0000 −1.19696 −0.598479 − 0.801138i \(-0.704228\pi\)
−0.598479 + 0.801138i \(0.704228\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) −13.0000 −0.534749
\(592\) 0 0
\(593\) −26.0000 −1.06769 −0.533846 − 0.845582i \(-0.679254\pi\)
−0.533846 + 0.845582i \(0.679254\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 20.0000 0.818546
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 − 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 45.0000 1.83559 0.917794 − 0.397057i \(-0.129968\pi\)
0.917794 + 0.397057i \(0.129968\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 40.0000 1.62355 0.811775 − 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 0 0
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) 9.00000 0.364101
\(612\) 0 0
\(613\) 4.00000 0.161558 0.0807792 − 0.996732i \(-0.474259\pi\)
0.0807792 + 0.996732i \(0.474259\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 28.0000 1.12724 0.563619 − 0.826035i \(-0.309409\pi\)
0.563619 + 0.826035i \(0.309409\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 − 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 0 0
\(623\) −28.0000 −1.12180
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 32.0000 1.27592
\(630\) 0 0
\(631\) 26.0000 1.03504 0.517522 − 0.855670i \(-0.326855\pi\)
0.517522 + 0.855670i \(0.326855\pi\)
\(632\) 0 0
\(633\) 8.00000 0.317971
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 6.00000 0.236986 0.118493 − 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 0 0
\(643\) 2.00000 0.0788723 0.0394362 − 0.999222i \(-0.487444\pi\)
0.0394362 + 0.999222i \(0.487444\pi\)
\(644\) 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
\(651\) 2.00000 0.0783862
\(652\) 0 0
\(653\) 35.0000 1.36966 0.684828 − 0.728705i \(-0.259877\pi\)
0.684828 + 0.728705i \(0.259877\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 − 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 − 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) 0 0
\(663\) 4.00000 0.155347
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.00000 0.116160
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 43.0000 1.65753 0.828764 − 0.559598i \(-0.189045\pi\)
0.828764 + 0.559598i \(0.189045\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −26.0000 −0.999261 −0.499631 − 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) 0 0
\(679\) 28.0000 1.07454
\(680\) 0 0
\(681\) 10.0000 0.383201
\(682\) 0 0
\(683\) 9.00000 0.344375 0.172188 − 0.985064i \(-0.444916\pi\)
0.172188 + 0.985064i \(0.444916\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −10.0000 −0.381524
\(688\) 0 0
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) 32.0000 1.21734 0.608669 − 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −20.0000 −0.757554
\(698\) 0 0
\(699\) 21.0000 0.794293
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 − 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) −32.0000 −1.20690
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.00000 0.150435
\(708\) 0 0
\(709\) 22.0000 0.826227 0.413114 − 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 1.00000 0.0374503
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 21.0000 0.784259
\(718\) 0 0
\(719\) 36.0000 1.34257 0.671287 − 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) −22.0000 −0.818189
\(724\) 0 0
\(725\) 0 0
\(726\) 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
\(733\) 28.0000 1.03420 0.517102 − 0.855924i \(-0.327011\pi\)
0.517102 + 0.855924i \(0.327011\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 37.0000 1.36107 0.680534 − 0.732717i \(-0.261748\pi\)
0.680534 + 0.732717i \(0.261748\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) 8.00000 0.293492 0.146746 − 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 16.0000 0.585409
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 2.00000 0.0729810 0.0364905 − 0.999334i \(-0.488382\pi\)
0.0364905 + 0.999334i \(0.488382\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 40.0000 1.45382 0.726912 − 0.686730i \(-0.240955\pi\)
0.726912 + 0.686730i \(0.240955\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 5.00000 0.181250 0.0906249 − 0.995885i \(-0.471114\pi\)
0.0906249 + 0.995885i \(0.471114\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −10.0000 −0.360609 −0.180305 − 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) −3.00000 −0.108042
\(772\) 0 0
\(773\) −18.0000 −0.647415 −0.323708 − 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −16.0000 −0.573997
\(778\) 0 0
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −15.0000 −0.536056
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −32.0000 −1.14068 −0.570338 − 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) 0 0
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 24.0000 0.850124 0.425062 − 0.905164i \(-0.360252\pi\)
0.425062 + 0.905164i \(0.360252\pi\)
\(798\) 0 0
\(799\) −36.0000 −1.27359
\(800\) 0 0
\(801\) 28.0000 0.989331
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −25.0000 −0.880042
\(808\) 0 0
\(809\) −46.0000 −1.61727 −0.808637 − 0.588308i \(-0.799794\pi\)
−0.808637 + 0.588308i \(0.799794\pi\)
\(810\) 0 0
\(811\) 39.0000 1.36948 0.684738 − 0.728790i \(-0.259917\pi\)
0.684738 + 0.728790i \(0.259917\pi\)
\(812\) 0 0
\(813\) 28.0000 0.982003
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −24.0000 −0.839654
\(818\) 0 0
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 34.0000 1.18661 0.593304 − 0.804978i \(-0.297823\pi\)
0.593304 + 0.804978i \(0.297823\pi\)
\(822\) 0 0
\(823\) 3.00000 0.104573 0.0522867 − 0.998632i \(-0.483349\pi\)
0.0522867 + 0.998632i \(0.483349\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 − 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 − 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 3.00000 0.104069
\(832\) 0 0
\(833\) −12.0000 −0.415775
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −5.00000 −0.172825
\(838\) 0 0
\(839\) −18.0000 −0.621429 −0.310715 − 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) −6.00000 −0.206651
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −22.0000 −0.755929
\(848\) 0 0
\(849\) −10.0000 −0.343199
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) −14.0000 −0.479351 −0.239675 − 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.00000 0.239115 0.119558 − 0.992827i \(-0.461852\pi\)
0.119558 + 0.992827i \(0.461852\pi\)
\(858\) 0 0
\(859\) −45.0000 −1.53538 −0.767690 − 0.640821i \(-0.778594\pi\)
−0.767690 + 0.640821i \(0.778594\pi\)
\(860\) 0 0
\(861\) 10.0000 0.340799
\(862\) 0 0
\(863\) −11.0000 −0.374444 −0.187222 − 0.982318i \(-0.559948\pi\)
−0.187222 + 0.982318i \(0.559948\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 0 0
\(873\) −28.0000 −0.947656
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −2.00000 −0.0675352 −0.0337676 − 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 0 0
\(879\) −16.0000 −0.539667
\(880\) 0 0
\(881\) 56.0000 1.88669 0.943344 − 0.331816i \(-0.107661\pi\)
0.943344 + 0.331816i \(0.107661\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 − 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.00000 0.100730 0.0503651 − 0.998731i \(-0.483962\pi\)
0.0503651 + 0.998731i \(0.483962\pi\)
\(888\) 0 0
\(889\) −14.0000 −0.469545
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 36.0000 1.20469
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −1.00000 −0.0333890
\(898\) 0 0
\(899\) 3.00000 0.100056
\(900\) 0 0
\(901\) −8.00000 −0.266519
\(902\) 0 0
\(903\) −12.0000 −0.399335
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 − 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 0 0
\(909\) −4.00000 −0.132672
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 − 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.00000 0.198137
\(918\) 0 0
\(919\) 46.0000 1.51740 0.758700 − 0.651440i \(-0.225835\pi\)
0.758700 + 0.651440i \(0.225835\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 0 0
\(923\) −3.00000 −0.0987462
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 16.0000 0.525509
\(928\) 0 0
\(929\) −45.0000 −1.47640 −0.738201 − 0.674581i \(-0.764324\pi\)
−0.738201 + 0.674581i \(0.764324\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) 0 0
\(933\) 9.00000 0.294647
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −36.0000 −1.17607 −0.588034 − 0.808836i \(-0.700098\pi\)
−0.588034 + 0.808836i \(0.700098\pi\)
\(938\) 0 0
\(939\) −4.00000 −0.130535
\(940\) 0 0
\(941\) 48.0000 1.56476 0.782378 − 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(942\) 0 0
\(943\) 5.00000 0.162822
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −45.0000 −1.46230 −0.731152 − 0.682215i \(-0.761017\pi\)
−0.731152 + 0.682215i \(0.761017\pi\)
\(948\) 0 0
\(949\) 7.00000 0.227230
\(950\) 0 0
\(951\) 26.0000 0.843108
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 8.00000 0.257796
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −23.0000 −0.739630 −0.369815 − 0.929105i \(-0.620579\pi\)
−0.369815 + 0.929105i \(0.620579\pi\)
\(968\) 0 0
\(969\) 16.0000 0.513994
\(970\) 0 0
\(971\) 50.0000 1.60458 0.802288 − 0.596937i \(-0.203616\pi\)
0.802288 + 0.596937i \(0.203616\pi\)
\(972\) 0 0
\(973\) 2.00000 0.0641171
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) 30.0000 0.956851 0.478426 − 0.878128i \(-0.341208\pi\)
0.478426 + 0.878128i \(0.341208\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 18.0000 0.572946
\(988\) 0 0
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 − 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) 19.0000 0.602947
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 − 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 0 0
\(999\) 40.0000 1.26554
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 4600.2.a.f.1.1 1
4.3 odd 2 9200.2.a.z.1.1 1
5.2 odd 4 4600.2.e.g.4049.2 2
5.3 odd 4 4600.2.e.g.4049.1 2
5.4 even 2 920.2.a.d.1.1 ✓ 1
15.14 odd 2 8280.2.a.o.1.1 1
20.19 odd 2 1840.2.a.b.1.1 1
40.19 odd 2 7360.2.a.u.1.1 1
40.29 even 2 7360.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.d.1.1 ✓ 1 5.4 even 2
1840.2.a.b.1.1 1 20.19 odd 2
4600.2.a.f.1.1 1 1.1 even 1 trivial
4600.2.e.g.4049.1 2 5.3 odd 4
4600.2.e.g.4049.2 2 5.2 odd 4
7360.2.a.j.1.1 1 40.29 even 2
7360.2.a.u.1.1 1 40.19 odd 2
8280.2.a.o.1.1 1 15.14 odd 2
9200.2.a.z.1.1 1 4.3 odd 2