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Properties

Label 693.2.a.c.1.1
Level $693$
Weight $2$
Character 693.1
Self dual yes
Analytic conductor $5.534$
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] = [693,2,Mod(1,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, 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("693.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.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: \(5.53363286007\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 693.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{11} -4.00000 q^{13} +4.00000 q^{16} -2.00000 q^{17} -6.00000 q^{19} -2.00000 q^{20} +5.00000 q^{23} -4.00000 q^{25} +2.00000 q^{28} -10.0000 q^{29} +1.00000 q^{31} -1.00000 q^{35} -5.00000 q^{37} +2.00000 q^{41} -8.00000 q^{43} -2.00000 q^{44} -8.00000 q^{47} +1.00000 q^{49} +8.00000 q^{52} +6.00000 q^{53} +1.00000 q^{55} -3.00000 q^{59} -2.00000 q^{61} -8.00000 q^{64} -4.00000 q^{65} -3.00000 q^{67} +4.00000 q^{68} -1.00000 q^{71} +10.0000 q^{73} +12.0000 q^{76} -1.00000 q^{77} +6.00000 q^{79} +4.00000 q^{80} -12.0000 q^{83} -2.00000 q^{85} +15.0000 q^{89} +4.00000 q^{91} -10.0000 q^{92} -6.00000 q^{95} -5.00000 q^{97} +O(q^{100})\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) 5.00000 1.04257 0.521286 − 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −10.0000 −1.85695 −0.928477 − 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 − 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 − 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 8.00000 1.10940
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.00000 −0.390567 −0.195283 − 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −3.00000 −0.366508 −0.183254 − 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 0 0
\(71\) −1.00000 −0.118678 −0.0593391 − 0.998238i \(-0.518899\pi\)
−0.0593391 + 0.998238i \(0.518899\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 − 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 12.0000 1.37649
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 − 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.0000 1.59000 0.794998 − 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −10.0000 −1.04257
\(93\) 0 0
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) −5.00000 −0.507673 −0.253837 − 0.967247i \(-0.581693\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 8.00000 0.800000
\(101\) 12.0000 1.19404 0.597022 − 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) −12.0000 −1.18240 −0.591198 − 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.0000 0.966736 0.483368 − 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 − 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) 19.0000 1.78737 0.893685 − 0.448695i \(-0.148111\pi\)
0.893685 + 0.448695i \(0.148111\pi\)
\(114\) 0 0
\(115\) 5.00000 0.466252
\(116\) 20.0000 1.85695
\(117\) 0 0
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 − 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 − 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 6.00000 0.520266
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.00000 0.256307 0.128154 − 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 − 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) −10.0000 −0.830455
\(146\) 0 0
\(147\) 0 0
\(148\) 10.0000 0.821995
\(149\) 22.0000 1.80231 0.901155 − 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 0 0
\(151\) 6.00000 0.488273 0.244137 − 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) 7.00000 0.558661 0.279330 − 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.00000 −0.394055
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) 0 0
\(167\) 2.00000 0.154765 0.0773823 − 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 16.0000 1.21999
\(173\) −16.0000 −1.21646 −0.608229 − 0.793762i \(-0.708120\pi\)
−0.608229 + 0.793762i \(0.708120\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) 0 0
\(179\) −1.00000 −0.0747435 −0.0373718 − 0.999301i \(-0.511899\pi\)
−0.0373718 + 0.999301i \(0.511899\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 − 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −5.00000 −0.367607
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) 16.0000 1.16692
\(189\) 0 0
\(190\) 0 0
\(191\) −5.00000 −0.361787 −0.180894 − 0.983503i \(-0.557899\pi\)
−0.180894 + 0.983503i \(0.557899\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.00000 −0.142857
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 10.0000 0.701862
\(204\) 0 0
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) 0 0
\(208\) −16.0000 −1.10940
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −2.00000 −0.137686 −0.0688428 − 0.997628i \(-0.521931\pi\)
−0.0688428 + 0.997628i \(0.521931\pi\)
\(212\) −12.0000 −0.824163
\(213\) 0 0
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −1.00000 −0.0678844
\(218\) 0 0
\(219\) 0 0
\(220\) −2.00000 −0.134840
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) 1.00000 0.0669650 0.0334825 − 0.999439i \(-0.489340\pi\)
0.0334825 + 0.999439i \(0.489340\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 − 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) −7.00000 −0.462573 −0.231287 − 0.972886i \(-0.574293\pi\)
−0.231287 + 0.972886i \(0.574293\pi\)
\(230\) 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
\(235\) −8.00000 −0.521862
\(236\) 6.00000 0.390567
\(237\) 0 0
\(238\) 0 0
\(239\) −4.00000 −0.258738 −0.129369 − 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 0 0
\(241\) −12.0000 −0.772988 −0.386494 − 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 4.00000 0.256074
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 24.0000 1.52708
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 21.0000 1.32551 0.662754 − 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) 0 0
\(253\) 5.00000 0.314347
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 5.00000 0.310685
\(260\) 8.00000 0.496139
\(261\) 0 0
\(262\) 0 0
\(263\) −18.0000 −1.10993 −0.554964 − 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) 6.00000 0.366508
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 − 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −8.00000 −0.485071
\(273\) 0 0
\(274\) 0 0
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) 24.0000 1.44202 0.721010 − 0.692925i \(-0.243678\pi\)
0.721010 + 0.692925i \(0.243678\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.00000 0.238620 0.119310 − 0.992857i \(-0.461932\pi\)
0.119310 + 0.992857i \(0.461932\pi\)
\(282\) 0 0
\(283\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) −20.0000 −1.17041
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) −3.00000 −0.174667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −20.0000 −1.15663
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) 0 0
\(304\) −24.0000 −1.37649
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 − 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 2.00000 0.113961
\(309\) 0 0
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 − 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) −23.0000 −1.30004 −0.650018 − 0.759918i \(-0.725239\pi\)
−0.650018 + 0.759918i \(0.725239\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) −9.00000 −0.505490 −0.252745 − 0.967533i \(-0.581333\pi\)
−0.252745 + 0.967533i \(0.581333\pi\)
\(318\) 0 0
\(319\) −10.0000 −0.559893
\(320\) −8.00000 −0.447214
\(321\) 0 0
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) 16.0000 0.887520
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −17.0000 −0.934405 −0.467202 − 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) 24.0000 1.31717
\(333\) 0 0
\(334\) 0 0
\(335\) −3.00000 −0.163908
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 − 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 4.00000 0.216930
\(341\) 1.00000 0.0541530
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −14.0000 −0.751559 −0.375780 − 0.926709i \(-0.622625\pi\)
−0.375780 + 0.926709i \(0.622625\pi\)
\(348\) 0 0
\(349\) −34.0000 −1.81998 −0.909989 − 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −9.00000 −0.479022 −0.239511 − 0.970894i \(-0.576987\pi\)
−0.239511 + 0.970894i \(0.576987\pi\)
\(354\) 0 0
\(355\) −1.00000 −0.0530745
\(356\) −30.0000 −1.59000
\(357\) 0 0
\(358\) 0 0
\(359\) −8.00000 −0.422224 −0.211112 − 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 0 0
\(364\) −8.00000 −0.419314
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) −11.0000 −0.574195 −0.287098 − 0.957901i \(-0.592690\pi\)
−0.287098 + 0.957901i \(0.592690\pi\)
\(368\) 20.0000 1.04257
\(369\) 0 0
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 − 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 40.0000 2.06010
\(378\) 0 0
\(379\) −29.0000 −1.48963 −0.744815 − 0.667271i \(-0.767462\pi\)
−0.744815 + 0.667271i \(0.767462\pi\)
\(380\) 12.0000 0.615587
\(381\) 0 0
\(382\) 0 0
\(383\) −17.0000 −0.868659 −0.434330 − 0.900754i \(-0.643015\pi\)
−0.434330 + 0.900754i \(0.643015\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) 0 0
\(388\) 10.0000 0.507673
\(389\) −9.00000 −0.456318 −0.228159 − 0.973624i \(-0.573271\pi\)
−0.228159 + 0.973624i \(0.573271\pi\)
\(390\) 0 0
\(391\) −10.0000 −0.505722
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6.00000 0.301893
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 − 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −16.0000 −0.800000
\(401\) 6.00000 0.299626 0.149813 − 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) −24.0000 −1.19404
\(405\) 0 0
\(406\) 0 0
\(407\) −5.00000 −0.247841
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 − 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 24.0000 1.18240
\(413\) 3.00000 0.147620
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −16.0000 −0.781651 −0.390826 − 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 − 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 8.00000 0.388057
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) −20.0000 −0.966736
\(429\) 0 0
\(430\) 0 0
\(431\) 20.0000 0.963366 0.481683 − 0.876346i \(-0.340026\pi\)
0.481683 + 0.876346i \(0.340026\pi\)
\(432\) 0 0
\(433\) −25.0000 −1.20142 −0.600712 − 0.799466i \(-0.705116\pi\)
−0.600712 + 0.799466i \(0.705116\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −8.00000 −0.383131
\(437\) −30.0000 −1.43509
\(438\) 0 0
\(439\) −14.0000 −0.668184 −0.334092 − 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 39.0000 1.85295 0.926473 − 0.376361i \(-0.122825\pi\)
0.926473 + 0.376361i \(0.122825\pi\)
\(444\) 0 0
\(445\) 15.0000 0.711068
\(446\) 0 0
\(447\) 0 0
\(448\) 8.00000 0.377964
\(449\) −15.0000 −0.707894 −0.353947 − 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) −38.0000 −1.78737
\(453\) 0 0
\(454\) 0 0
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 − 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −10.0000 −0.466252
\(461\) −18.0000 −0.838344 −0.419172 − 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 13.0000 0.604161 0.302081 − 0.953282i \(-0.402319\pi\)
0.302081 + 0.953282i \(0.402319\pi\)
\(464\) −40.0000 −1.85695
\(465\) 0 0
\(466\) 0 0
\(467\) −3.00000 −0.138823 −0.0694117 − 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) 0 0
\(469\) 3.00000 0.138527
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) 24.0000 1.10120
\(476\) −4.00000 −0.183340
\(477\) 0 0
\(478\) 0 0
\(479\) 28.0000 1.27935 0.639676 − 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) 0 0
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) −5.00000 −0.227038
\(486\) 0 0
\(487\) −13.0000 −0.589086 −0.294543 − 0.955638i \(-0.595167\pi\)
−0.294543 + 0.955638i \(0.595167\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 − 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) 0 0
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 1.00000 0.0448561
\(498\) 0 0
\(499\) 44.0000 1.96971 0.984855 − 0.173379i \(-0.0554684\pi\)
0.984855 + 0.173379i \(0.0554684\pi\)
\(500\) 18.0000 0.804984
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) −4.00000 −0.177471
\(509\) 31.0000 1.37405 0.687025 − 0.726633i \(-0.258916\pi\)
0.687025 + 0.726633i \(0.258916\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −12.0000 −0.528783
\(516\) 0 0
\(517\) −8.00000 −0.351840
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −7.00000 −0.306676 −0.153338 − 0.988174i \(-0.549002\pi\)
−0.153338 + 0.988174i \(0.549002\pi\)
\(522\) 0 0
\(523\) 32.0000 1.39926 0.699631 − 0.714504i \(-0.253348\pi\)
0.699631 + 0.714504i \(0.253348\pi\)
\(524\) 36.0000 1.57267
\(525\) 0 0
\(526\) 0 0
\(527\) −2.00000 −0.0871214
\(528\) 0 0
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) 0 0
\(532\) −12.0000 −0.520266
\(533\) −8.00000 −0.346518
\(534\) 0 0
\(535\) 10.0000 0.432338
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 32.0000 1.37579 0.687894 − 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) −24.0000 −1.02617 −0.513083 − 0.858339i \(-0.671497\pi\)
−0.513083 + 0.858339i \(0.671497\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) 0 0
\(551\) 60.0000 2.55609
\(552\) 0 0
\(553\) −6.00000 −0.255146
\(554\) 0 0
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) −14.0000 −0.593199 −0.296600 − 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 0 0
\(559\) 32.0000 1.35346
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) 0 0
\(563\) −20.0000 −0.842900 −0.421450 − 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 0 0
\(565\) 19.0000 0.799336
\(566\) 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
\(571\) −20.0000 −0.836974 −0.418487 − 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 8.00000 0.334497
\(573\) 0 0
\(574\) 0 0
\(575\) −20.0000 −0.834058
\(576\) 0 0
\(577\) −25.0000 −1.04076 −0.520382 − 0.853934i \(-0.674210\pi\)
−0.520382 + 0.853934i \(0.674210\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 20.0000 0.830455
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 6.00000 0.248495
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −36.0000 −1.48588 −0.742940 − 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 0 0
\(589\) −6.00000 −0.247226
\(590\) 0 0
\(591\) 0 0
\(592\) −20.0000 −0.821995
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) −44.0000 −1.80231
\(597\) 0 0
\(598\) 0 0
\(599\) 48.0000 1.96123 0.980613 − 0.195952i \(-0.0627798\pi\)
0.980613 + 0.195952i \(0.0627798\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 − 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −12.0000 −0.488273
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −10.0000 −0.405887 −0.202944 − 0.979190i \(-0.565051\pi\)
−0.202944 + 0.979190i \(0.565051\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 32.0000 1.29458
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 − 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 − 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) 17.0000 0.683288 0.341644 − 0.939829i \(-0.389016\pi\)
0.341644 + 0.939829i \(0.389016\pi\)
\(620\) −2.00000 −0.0803219
\(621\) 0 0
\(622\) 0 0
\(623\) −15.0000 −0.600962
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) −14.0000 −0.558661
\(629\) 10.0000 0.398726
\(630\) 0 0
\(631\) 27.0000 1.07485 0.537427 − 0.843311i \(-0.319397\pi\)
0.537427 + 0.843311i \(0.319397\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.00000 0.0793676
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −15.0000 −0.592464 −0.296232 − 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) 0 0
\(643\) −29.0000 −1.14365 −0.571824 − 0.820376i \(-0.693764\pi\)
−0.571824 + 0.820376i \(0.693764\pi\)
\(644\) 10.0000 0.394055
\(645\) 0 0
\(646\) 0 0
\(647\) 21.0000 0.825595 0.412798 − 0.910823i \(-0.364552\pi\)
0.412798 + 0.910823i \(0.364552\pi\)
\(648\) 0 0
\(649\) −3.00000 −0.117760
\(650\) 0 0
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) 17.0000 0.665261 0.332631 − 0.943057i \(-0.392064\pi\)
0.332631 + 0.943057i \(0.392064\pi\)
\(654\) 0 0
\(655\) −18.0000 −0.703318
\(656\) 8.00000 0.312348
\(657\) 0 0
\(658\) 0 0
\(659\) 2.00000 0.0779089 0.0389545 − 0.999241i \(-0.487597\pi\)
0.0389545 + 0.999241i \(0.487597\pi\)
\(660\) 0 0
\(661\) 35.0000 1.36134 0.680671 − 0.732589i \(-0.261688\pi\)
0.680671 + 0.732589i \(0.261688\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6.00000 0.232670
\(666\) 0 0
\(667\) −50.0000 −1.93601
\(668\) −4.00000 −0.154765
\(669\) 0 0
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) 4.00000 0.154189 0.0770943 − 0.997024i \(-0.475436\pi\)
0.0770943 + 0.997024i \(0.475436\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −6.00000 −0.230769
\(677\) −38.0000 −1.46046 −0.730229 − 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 0 0
\(679\) 5.00000 0.191882
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 3.00000 0.114624
\(686\) 0 0
\(687\) 0 0
\(688\) −32.0000 −1.21999
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 15.0000 0.570627 0.285313 − 0.958434i \(-0.407902\pi\)
0.285313 + 0.958434i \(0.407902\pi\)
\(692\) 32.0000 1.21646
\(693\) 0 0
\(694\) 0 0
\(695\) −10.0000 −0.379322
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 0 0
\(699\) 0 0
\(700\) −8.00000 −0.302372
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 30.0000 1.13147
\(704\) −8.00000 −0.301511
\(705\) 0 0
\(706\) 0 0
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) 39.0000 1.46468 0.732338 − 0.680941i \(-0.238429\pi\)
0.732338 + 0.680941i \(0.238429\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.00000 0.187251
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 2.00000 0.0747435
\(717\) 0 0
\(718\) 0 0
\(719\) 11.0000 0.410231 0.205115 − 0.978738i \(-0.434243\pi\)
0.205115 + 0.978738i \(0.434243\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 0 0
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) 40.0000 1.48556
\(726\) 0 0
\(727\) −19.0000 −0.704671 −0.352335 − 0.935874i \(-0.614612\pi\)
−0.352335 + 0.935874i \(0.614612\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) 0 0
\(733\) −4.00000 −0.147743 −0.0738717 − 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3.00000 −0.110506
\(738\) 0 0
\(739\) −18.0000 −0.662141 −0.331070 − 0.943606i \(-0.607410\pi\)
−0.331070 + 0.943606i \(0.607410\pi\)
\(740\) 10.0000 0.367607
\(741\) 0 0
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 − 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) 22.0000 0.806018
\(746\) 0 0
\(747\) 0 0
\(748\) 4.00000 0.146254
\(749\) −10.0000 −0.365392
\(750\) 0 0
\(751\) −23.0000 −0.839282 −0.419641 − 0.907690i \(-0.637844\pi\)
−0.419641 + 0.907690i \(0.637844\pi\)
\(752\) −32.0000 −1.16692
\(753\) 0 0
\(754\) 0 0
\(755\) 6.00000 0.218362
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 − 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 48.0000 1.74000 0.869999 − 0.493053i \(-0.164119\pi\)
0.869999 + 0.493053i \(0.164119\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) 10.0000 0.361787
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) 40.0000 1.44244 0.721218 − 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −28.0000 −1.00774
\(773\) 6.00000 0.215805 0.107903 − 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) −1.00000 −0.0357828
\(782\) 0 0
\(783\) 0 0
\(784\) 4.00000 0.142857
\(785\) 7.00000 0.249841
\(786\) 0 0
\(787\) −22.0000 −0.784215 −0.392108 − 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 36.0000 1.28245
\(789\) 0 0
\(790\) 0 0
\(791\) −19.0000 −0.675562
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 0 0
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) −23.0000 −0.814702 −0.407351 − 0.913272i \(-0.633547\pi\)
−0.407351 + 0.913272i \(0.633547\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10.0000 0.352892
\(804\) 0 0
\(805\) −5.00000 −0.176227
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 − 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) −22.0000 −0.772524 −0.386262 − 0.922389i \(-0.626234\pi\)
−0.386262 + 0.922389i \(0.626234\pi\)
\(812\) −20.0000 −0.701862
\(813\) 0 0
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) 48.0000 1.67931
\(818\) 0 0
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) −18.0000 −0.628204 −0.314102 − 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) −25.0000 −0.871445 −0.435723 − 0.900081i \(-0.643507\pi\)
−0.435723 + 0.900081i \(0.643507\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −20.0000 −0.695468 −0.347734 − 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) −29.0000 −1.00721 −0.503606 − 0.863934i \(-0.667994\pi\)
−0.503606 + 0.863934i \(0.667994\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 32.0000 1.10940
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 2.00000 0.0692129
\(836\) 12.0000 0.415029
\(837\) 0 0
\(838\) 0 0
\(839\) −45.0000 −1.55357 −0.776786 − 0.629764i \(-0.783151\pi\)
−0.776786 + 0.629764i \(0.783151\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 24.0000 0.824163
\(849\) 0 0
\(850\) 0 0
\(851\) −25.0000 −0.856989
\(852\) 0 0
\(853\) −34.0000 −1.16414 −0.582069 − 0.813139i \(-0.697757\pi\)
−0.582069 + 0.813139i \(0.697757\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.0000 0.956462 0.478231 − 0.878234i \(-0.341278\pi\)
0.478231 + 0.878234i \(0.341278\pi\)
\(858\) 0 0
\(859\) 55.0000 1.87658 0.938288 − 0.345855i \(-0.112411\pi\)
0.938288 + 0.345855i \(0.112411\pi\)
\(860\) 16.0000 0.545595
\(861\) 0 0
\(862\) 0 0
\(863\) −52.0000 −1.77010 −0.885050 − 0.465495i \(-0.845876\pi\)
−0.885050 + 0.465495i \(0.845876\pi\)
\(864\) 0 0
\(865\) −16.0000 −0.544016
\(866\) 0 0
\(867\) 0 0
\(868\) 2.00000 0.0678844
\(869\) 6.00000 0.203536
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 9.00000 0.304256
\(876\) 0 0
\(877\) −38.0000 −1.28317 −0.641584 − 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) −27.0000 −0.909653 −0.454827 − 0.890580i \(-0.650299\pi\)
−0.454827 + 0.890580i \(0.650299\pi\)
\(882\) 0 0
\(883\) −44.0000 −1.48072 −0.740359 − 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) −16.0000 −0.538138
\(885\) 0 0
\(886\) 0 0
\(887\) 2.00000 0.0671534 0.0335767 − 0.999436i \(-0.489310\pi\)
0.0335767 + 0.999436i \(0.489310\pi\)
\(888\) 0 0
\(889\) −2.00000 −0.0670778
\(890\) 0 0
\(891\) 0 0
\(892\) −2.00000 −0.0669650
\(893\) 48.0000 1.60626
\(894\) 0 0
\(895\) −1.00000 −0.0334263
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −10.0000 −0.333519
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 5.00000 0.166206
\(906\) 0 0
\(907\) −40.0000 −1.32818 −0.664089 − 0.747653i \(-0.731180\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) 8.00000 0.265489
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 18.0000 0.594412
\(918\) 0 0
\(919\) 48.0000 1.58337 0.791687 − 0.610927i \(-0.209203\pi\)
0.791687 + 0.610927i \(0.209203\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 4.00000 0.131662
\(924\) 0 0
\(925\) 20.0000 0.657596
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 − 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 12.0000 0.393073
\(933\) 0 0
\(934\) 0 0
\(935\) −2.00000 −0.0654070
\(936\) 0 0
\(937\) 36.0000 1.17607 0.588034 − 0.808836i \(-0.299902\pi\)
0.588034 + 0.808836i \(0.299902\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 16.0000 0.521862
\(941\) −58.0000 −1.89075 −0.945373 − 0.325991i \(-0.894302\pi\)
−0.945373 + 0.325991i \(0.894302\pi\)
\(942\) 0 0
\(943\) 10.0000 0.325645
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) −5.00000 −0.162478 −0.0812391 − 0.996695i \(-0.525888\pi\)
−0.0812391 + 0.996695i \(0.525888\pi\)
\(948\) 0 0
\(949\) −40.0000 −1.29845
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −44.0000 −1.42530 −0.712650 − 0.701520i \(-0.752505\pi\)
−0.712650 + 0.701520i \(0.752505\pi\)
\(954\) 0 0
\(955\) −5.00000 −0.161796
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) 0 0
\(959\) −3.00000 −0.0968751
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 0 0
\(964\) 24.0000 0.772988
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) −34.0000 −1.09337 −0.546683 − 0.837340i \(-0.684110\pi\)
−0.546683 + 0.837340i \(0.684110\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −29.0000 −0.930654 −0.465327 − 0.885139i \(-0.654063\pi\)
−0.465327 + 0.885139i \(0.654063\pi\)
\(972\) 0 0
\(973\) 10.0000 0.320585
\(974\) 0 0
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) 31.0000 0.991778 0.495889 − 0.868386i \(-0.334842\pi\)
0.495889 + 0.868386i \(0.334842\pi\)
\(978\) 0 0
\(979\) 15.0000 0.479402
\(980\) −2.00000 −0.0638877
\(981\) 0 0
\(982\) 0 0
\(983\) 27.0000 0.861166 0.430583 − 0.902551i \(-0.358308\pi\)
0.430583 + 0.902551i \(0.358308\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) −48.0000 −1.52708
\(989\) −40.0000 −1.27193
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 − 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) 0 0
\(997\) −12.0000 −0.380044 −0.190022 − 0.981780i \(-0.560856\pi\)
−0.190022 + 0.981780i \(0.560856\pi\)
\(998\) 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 693.2.a.c.1.1 1
3.2 odd 2 77.2.a.a.1.1 ✓ 1
7.6 odd 2 4851.2.a.j.1.1 1
11.10 odd 2 7623.2.a.j.1.1 1
12.11 even 2 1232.2.a.l.1.1 1
15.2 even 4 1925.2.b.e.1849.2 2
15.8 even 4 1925.2.b.e.1849.1 2
15.14 odd 2 1925.2.a.h.1.1 1
21.2 odd 6 539.2.e.f.67.1 2
21.5 even 6 539.2.e.c.67.1 2
21.11 odd 6 539.2.e.f.177.1 2
21.17 even 6 539.2.e.c.177.1 2
21.20 even 2 539.2.a.c.1.1 1
24.5 odd 2 4928.2.a.bj.1.1 1
24.11 even 2 4928.2.a.a.1.1 1
33.2 even 10 847.2.f.h.323.1 4
33.5 odd 10 847.2.f.i.729.1 4
33.8 even 10 847.2.f.h.372.1 4
33.14 odd 10 847.2.f.i.372.1 4
33.17 even 10 847.2.f.h.729.1 4
33.20 odd 10 847.2.f.i.323.1 4
33.26 odd 10 847.2.f.i.148.1 4
33.29 even 10 847.2.f.h.148.1 4
33.32 even 2 847.2.a.b.1.1 1
84.83 odd 2 8624.2.a.a.1.1 1
231.230 odd 2 5929.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.a.1.1 ✓ 1 3.2 odd 2
539.2.a.c.1.1 1 21.20 even 2
539.2.e.c.67.1 2 21.5 even 6
539.2.e.c.177.1 2 21.17 even 6
539.2.e.f.67.1 2 21.2 odd 6
539.2.e.f.177.1 2 21.11 odd 6
693.2.a.c.1.1 1 1.1 even 1 trivial
847.2.a.b.1.1 1 33.32 even 2
847.2.f.h.148.1 4 33.29 even 10
847.2.f.h.323.1 4 33.2 even 10
847.2.f.h.372.1 4 33.8 even 10
847.2.f.h.729.1 4 33.17 even 10
847.2.f.i.148.1 4 33.26 odd 10
847.2.f.i.323.1 4 33.20 odd 10
847.2.f.i.372.1 4 33.14 odd 10
847.2.f.i.729.1 4 33.5 odd 10
1232.2.a.l.1.1 1 12.11 even 2
1925.2.a.h.1.1 1 15.14 odd 2
1925.2.b.e.1849.1 2 15.8 even 4
1925.2.b.e.1849.2 2 15.2 even 4
4851.2.a.j.1.1 1 7.6 odd 2
4928.2.a.a.1.1 1 24.11 even 2
4928.2.a.bj.1.1 1 24.5 odd 2
5929.2.a.f.1.1 1 231.230 odd 2
7623.2.a.j.1.1 1 11.10 odd 2
8624.2.a.a.1.1 1 84.83 odd 2