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Properties

Label 6384.2.a.be.1.1
Level $6384$
Weight $2$
Character 6384.1
Self dual yes
Analytic conductor $50.976$
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] = [6384,2,Mod(1,6384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6384.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6384.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: \(50.9764966504\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3192)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6384.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +2.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -6.00000 q^{11} -2.00000 q^{13} +2.00000 q^{15} -8.00000 q^{17} -1.00000 q^{19} +1.00000 q^{21} +8.00000 q^{23} -1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} +6.00000 q^{31} -6.00000 q^{33} +2.00000 q^{35} -4.00000 q^{37} -2.00000 q^{39} -2.00000 q^{41} -4.00000 q^{43} +2.00000 q^{45} +6.00000 q^{47} +1.00000 q^{49} -8.00000 q^{51} +6.00000 q^{53} -12.0000 q^{55} -1.00000 q^{57} -4.00000 q^{59} +2.00000 q^{61} +1.00000 q^{63} -4.00000 q^{65} -14.0000 q^{67} +8.00000 q^{69} +8.00000 q^{71} -14.0000 q^{73} -1.00000 q^{75} -6.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} +8.00000 q^{83} -16.0000 q^{85} -2.00000 q^{87} +6.00000 q^{89} -2.00000 q^{91} +6.00000 q^{93} -2.00000 q^{95} -12.0000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 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 \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) −8.00000 −1.94029 −0.970143 − 0.242536i \(-0.922021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 − 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 0 0
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −14.0000 −1.71037 −0.855186 − 0.518321i \(-0.826557\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.00000 0.878114 0.439057 − 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) −16.0000 −1.73544
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 6.00000 0.622171
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −12.0000 −1.21842 −0.609208 − 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 − 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) −6.00000 −0.591198 −0.295599 − 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −16.0000 −1.53252 −0.766261 − 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 − 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 16.0000 1.49201
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 − 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) 2.00000 0.172133
\(136\) 0 0
\(137\) −14.0000 −1.19610 −0.598050 − 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 − 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −24.0000 −1.96616 −0.983078 − 0.183186i \(-0.941359\pi\)
−0.983078 + 0.183186i \(0.941359\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 − 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) −8.00000 −0.646762
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 − 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 0 0
\(165\) −12.0000 −0.934199
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) −24.0000 −1.79384 −0.896922 − 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 − 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) 48.0000 3.51011
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 − 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −18.0000 −1.29567 −0.647834 − 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 0 0
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) 24.0000 1.70993 0.854965 − 0.518686i \(-0.173579\pi\)
0.854965 + 0.518686i \(0.173579\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) −14.0000 −0.987484
\(202\) 0 0
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 0 0
\(207\) 8.00000 0.556038
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 − 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 0 0
\(213\) 8.00000 0.548151
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 0 0
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) 16.0000 1.07628
\(222\) 0 0
\(223\) 10.0000 0.669650 0.334825 − 0.942280i \(-0.391323\pi\)
0.334825 + 0.942280i \(0.391323\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) 4.00000 0.265489 0.132745 − 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) −6.00000 −0.394771
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 − 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 − 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −48.0000 −3.01773
\(254\) 0 0
\(255\) −16.0000 −1.00196
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 − 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 0 0
\(269\) 30.0000 1.82913 0.914566 − 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 − 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) 6.00000 0.361814
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 − 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 − 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) 47.0000 2.76471
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) 0 0
\(293\) −18.0000 −1.05157 −0.525786 − 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) −6.00000 −0.348155
\(298\) 0 0
\(299\) −16.0000 −0.925304
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) −14.0000 −0.804279
\(304\) 0 0
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) 28.0000 1.59804 0.799022 − 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) 0 0
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) −6.00000 −0.340229 −0.170114 − 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 − 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 0 0
\(315\) 2.00000 0.112687
\(316\) 0 0
\(317\) 34.0000 1.90963 0.954815 − 0.297200i \(-0.0960529\pi\)
0.954815 + 0.297200i \(0.0960529\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 0 0
\(327\) −16.0000 −0.884802
\(328\) 0 0
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 30.0000 1.64895 0.824475 − 0.565899i \(-0.191471\pi\)
0.824475 + 0.565899i \(0.191471\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) −28.0000 −1.52980
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 − 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) −36.0000 −1.94951
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 16.0000 0.861411
\(346\) 0 0
\(347\) −26.0000 −1.39575 −0.697877 − 0.716218i \(-0.745872\pi\)
−0.697877 + 0.716218i \(0.745872\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 − 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) 4.00000 0.212899 0.106449 − 0.994318i \(-0.466052\pi\)
0.106449 + 0.994318i \(0.466052\pi\)
\(354\) 0 0
\(355\) 16.0000 0.849192
\(356\) 0 0
\(357\) −8.00000 −0.423405
\(358\) 0 0
\(359\) −4.00000 −0.211112 −0.105556 − 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 25.0000 1.31216
\(364\) 0 0
\(365\) −28.0000 −1.46559
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) 16.0000 0.828449 0.414224 − 0.910175i \(-0.364053\pi\)
0.414224 + 0.910175i \(0.364053\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) 6.00000 0.308199 0.154100 − 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −12.0000 −0.611577
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) −24.0000 −1.21685 −0.608424 − 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) −64.0000 −3.23662
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) −30.0000 −1.50566 −0.752828 − 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) 0 0
\(399\) −1.00000 −0.0500626
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 − 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) −12.0000 −0.597763
\(404\) 0 0
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) 20.0000 0.988936 0.494468 − 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) 0 0
\(411\) −14.0000 −0.690569
\(412\) 0 0
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 16.0000 0.785409
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) 16.0000 0.781651 0.390826 − 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 0 0
\(421\) −8.00000 −0.389896 −0.194948 − 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) 8.00000 0.388057
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −28.0000 −1.34559 −0.672797 − 0.739827i \(-0.734907\pi\)
−0.672797 + 0.739827i \(0.734907\pi\)
\(434\) 0 0
\(435\) −4.00000 −0.191785
\(436\) 0 0
\(437\) −8.00000 −0.382692
\(438\) 0 0
\(439\) −26.0000 −1.24091 −0.620456 − 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 18.0000 0.855206 0.427603 − 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) 0 0
\(447\) −24.0000 −1.13516
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 − 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) −12.0000 −0.563809
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 − 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) −8.00000 −0.373408
\(460\) 0 0
\(461\) 38.0000 1.76984 0.884918 − 0.465746i \(-0.154214\pi\)
0.884918 + 0.465746i \(0.154214\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 − 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 0 0
\(465\) 12.0000 0.556487
\(466\) 0 0
\(467\) −8.00000 −0.370196 −0.185098 − 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 0 0
\(469\) −14.0000 −0.646460
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 24.0000 1.10352
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 22.0000 1.00521 0.502603 − 0.864517i \(-0.332376\pi\)
0.502603 + 0.864517i \(0.332376\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 0 0
\(483\) 8.00000 0.364013
\(484\) 0 0
\(485\) −24.0000 −1.08978
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 − 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) 18.0000 0.812329 0.406164 − 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(492\) 0 0
\(493\) 16.0000 0.720604
\(494\) 0 0
\(495\) −12.0000 −0.539360
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 18.0000 0.802580 0.401290 − 0.915951i \(-0.368562\pi\)
0.401290 + 0.915951i \(0.368562\pi\)
\(504\) 0 0
\(505\) −28.0000 −1.24598
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 − 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −12.0000 −0.528783
\(516\) 0 0
\(517\) −36.0000 −1.58328
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) 38.0000 1.66481 0.832405 − 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) 0 0
\(523\) −44.0000 −1.92399 −0.961993 − 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) 0 0
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) −48.0000 −2.09091
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) 0 0
\(535\) −24.0000 −1.03761
\(536\) 0 0
\(537\) −24.0000 −1.03568
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 42.0000 1.80572 0.902861 − 0.429934i \(-0.141463\pi\)
0.902861 + 0.429934i \(0.141463\pi\)
\(542\) 0 0
\(543\) −10.0000 −0.429141
\(544\) 0 0
\(545\) −32.0000 −1.37073
\(546\) 0 0
\(547\) 10.0000 0.427569 0.213785 − 0.976881i \(-0.431421\pi\)
0.213785 + 0.976881i \(0.431421\pi\)
\(548\) 0 0
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 0 0
\(555\) −8.00000 −0.339581
\(556\) 0 0
\(557\) 40.0000 1.69485 0.847427 − 0.530912i \(-0.178150\pi\)
0.847427 + 0.530912i \(0.178150\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 48.0000 2.02656
\(562\) 0 0
\(563\) −12.0000 −0.505740 −0.252870 − 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) −4.00000 −0.168281
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 − 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 24.0000 1.00437 0.502184 − 0.864761i \(-0.332530\pi\)
0.502184 + 0.864761i \(0.332530\pi\)
\(572\) 0 0
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) −8.00000 −0.333623
\(576\) 0 0
\(577\) 10.0000 0.416305 0.208153 − 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 0 0
\(579\) −18.0000 −0.748054
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) −36.0000 −1.49097
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) −6.00000 −0.247226
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) 0 0
\(593\) −32.0000 −1.31408 −0.657041 − 0.753855i \(-0.728192\pi\)
−0.657041 + 0.753855i \(0.728192\pi\)
\(594\) 0 0
\(595\) −16.0000 −0.655936
\(596\) 0 0
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 − 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) −14.0000 −0.570124
\(604\) 0 0
\(605\) 50.0000 2.03279
\(606\) 0 0
\(607\) −2.00000 −0.0811775 −0.0405887 − 0.999176i \(-0.512923\pi\)
−0.0405887 + 0.999176i \(0.512923\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) −6.00000 −0.242338 −0.121169 − 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 0 0
\(615\) −4.00000 −0.161296
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 − 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 − 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 6.00000 0.239617
\(628\) 0 0
\(629\) 32.0000 1.27592
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 − 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) 14.0000 0.556450
\(634\) 0 0
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 34.0000 1.34292 0.671460 − 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) 0 0
\(643\) −12.0000 −0.473234 −0.236617 − 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) −34.0000 −1.33668 −0.668339 − 0.743857i \(-0.732994\pi\)
−0.668339 + 0.743857i \(0.732994\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) 0 0
\(653\) 20.0000 0.782660 0.391330 − 0.920250i \(-0.372015\pi\)
0.391330 + 0.920250i \(0.372015\pi\)
\(654\) 0 0
\(655\) −24.0000 −0.937758
\(656\) 0 0
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) −48.0000 −1.86981 −0.934907 − 0.354892i \(-0.884518\pi\)
−0.934907 + 0.354892i \(0.884518\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 − 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 0 0
\(663\) 16.0000 0.621389
\(664\) 0 0
\(665\) −2.00000 −0.0775567
\(666\) 0 0
\(667\) −16.0000 −0.619522
\(668\) 0 0
\(669\) 10.0000 0.386622
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 − 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 2.00000 0.0768662 0.0384331 − 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) 0 0
\(679\) −12.0000 −0.460518
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) −20.0000 −0.765279 −0.382639 − 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 0 0
\(685\) −28.0000 −1.06983
\(686\) 0 0
\(687\) 6.00000 0.228914
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 − 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 0 0
\(693\) −6.00000 −0.227921
\(694\) 0 0
\(695\) −24.0000 −0.910372
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) 0 0
\(707\) −14.0000 −0.526524
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 48.0000 1.79761
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) 0 0
\(717\) −12.0000 −0.448148
\(718\) 0 0
\(719\) 10.0000 0.372937 0.186469 − 0.982461i \(-0.440296\pi\)
0.186469 + 0.982461i \(0.440296\pi\)
\(720\) 0 0
\(721\) −6.00000 −0.223452
\(722\) 0 0
\(723\) −8.00000 −0.297523
\(724\) 0 0
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) −20.0000 −0.741759 −0.370879 − 0.928681i \(-0.620944\pi\)
−0.370879 + 0.928681i \(0.620944\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 32.0000 1.18356
\(732\) 0 0
\(733\) −26.0000 −0.960332 −0.480166 − 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) 0 0
\(735\) 2.00000 0.0737711
\(736\) 0 0
\(737\) 84.0000 3.09418
\(738\) 0 0
\(739\) −4.00000 −0.147142 −0.0735712 − 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −48.0000 −1.75858
\(746\) 0 0
\(747\) 8.00000 0.292705
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −24.0000 −0.875772 −0.437886 − 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) −24.0000 −0.873449
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 − 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 0 0
\(759\) −48.0000 −1.74229
\(760\) 0 0
\(761\) −36.0000 −1.30500 −0.652499 − 0.757789i \(-0.726280\pi\)
−0.652499 + 0.757789i \(0.726280\pi\)
\(762\) 0 0
\(763\) −16.0000 −0.579239
\(764\) 0 0
\(765\) −16.0000 −0.578481
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −30.0000 −1.08183 −0.540914 − 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) 0 0
\(773\) 14.0000 0.503545 0.251773 − 0.967786i \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) 0 0
\(775\) −6.00000 −0.215526
\(776\) 0 0
\(777\) −4.00000 −0.143499
\(778\) 0 0
\(779\) 2.00000 0.0716574
\(780\) 0 0
\(781\) −48.0000 −1.71758
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) −8.00000 −0.285169 −0.142585 − 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) 0 0
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) 0 0
\(795\) 12.0000 0.425596
\(796\) 0 0
\(797\) −30.0000 −1.06265 −0.531327 − 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) −48.0000 −1.69812
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 84.0000 2.96430
\(804\) 0 0
\(805\) 16.0000 0.563926
\(806\) 0 0
\(807\) 30.0000 1.05605
\(808\) 0 0
\(809\) 38.0000 1.33601 0.668004 − 0.744157i \(-0.267149\pi\)
0.668004 + 0.744157i \(0.267149\pi\)
\(810\) 0 0
\(811\) 36.0000 1.26413 0.632065 − 0.774915i \(-0.282207\pi\)
0.632065 + 0.774915i \(0.282207\pi\)
\(812\) 0 0
\(813\) 24.0000 0.841717
\(814\) 0 0
\(815\) 32.0000 1.12091
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 0 0
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) −4.00000 −0.139601 −0.0698005 − 0.997561i \(-0.522236\pi\)
−0.0698005 + 0.997561i \(0.522236\pi\)
\(822\) 0 0
\(823\) 40.0000 1.39431 0.697156 − 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 0 0
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) 32.0000 1.11275 0.556375 − 0.830932i \(-0.312192\pi\)
0.556375 + 0.830932i \(0.312192\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 − 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) −8.00000 −0.277184
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 6.00000 0.207390
\(838\) 0 0
\(839\) 48.0000 1.65714 0.828572 − 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 22.0000 0.757720
\(844\) 0 0
\(845\) −18.0000 −0.619219
\(846\) 0 0
\(847\) 25.0000 0.859010
\(848\) 0 0
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) −14.0000 −0.479351 −0.239675 − 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 0 0
\(855\) −2.00000 −0.0683986
\(856\) 0 0
\(857\) 38.0000 1.29806 0.649028 − 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 − 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) −2.00000 −0.0681598
\(862\) 0 0
\(863\) 48.0000 1.63394 0.816970 − 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 0 0
\(865\) 36.0000 1.22404
\(866\) 0 0
\(867\) 47.0000 1.59620
\(868\) 0 0
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 28.0000 0.948744
\(872\) 0 0
\(873\) −12.0000 −0.406138
\(874\) 0 0
\(875\) −12.0000 −0.405674
\(876\) 0 0
\(877\) 28.0000 0.945493 0.472746 − 0.881199i \(-0.343263\pi\)
0.472746 + 0.881199i \(0.343263\pi\)
\(878\) 0 0
\(879\) −18.0000 −0.607125
\(880\) 0 0
\(881\) 20.0000 0.673817 0.336909 − 0.941537i \(-0.390619\pi\)
0.336909 + 0.941537i \(0.390619\pi\)
\(882\) 0 0
\(883\) −36.0000 −1.21150 −0.605748 − 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) 0 0
\(885\) −8.00000 −0.268917
\(886\) 0 0
\(887\) 20.0000 0.671534 0.335767 − 0.941945i \(-0.391004\pi\)
0.335767 + 0.941945i \(0.391004\pi\)
\(888\) 0 0
\(889\) 4.00000 0.134156
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) 0 0
\(893\) −6.00000 −0.200782
\(894\) 0 0
\(895\) −48.0000 −1.60446
\(896\) 0 0
\(897\) −16.0000 −0.534224
\(898\) 0 0
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) −48.0000 −1.59911
\(902\) 0 0
\(903\) −4.00000 −0.133112
\(904\) 0 0
\(905\) −20.0000 −0.664822
\(906\) 0 0
\(907\) −10.0000 −0.332045 −0.166022 − 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) 0 0
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 − 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 0 0
\(913\) −48.0000 −1.58857
\(914\) 0 0
\(915\) 4.00000 0.132236
\(916\) 0 0
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 − 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) 0 0
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 4.00000 0.131519
\(926\) 0 0
\(927\) −6.00000 −0.197066
\(928\) 0 0
\(929\) 24.0000 0.787414 0.393707 − 0.919236i \(-0.371192\pi\)
0.393707 + 0.919236i \(0.371192\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) 0 0
\(933\) −6.00000 −0.196431
\(934\) 0 0
\(935\) 96.0000 3.13954
\(936\) 0 0
\(937\) −10.0000 −0.326686 −0.163343 − 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 0 0
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) 6.00000 0.195594 0.0977972 − 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 0 0
\(943\) −16.0000 −0.521032
\(944\) 0 0
\(945\) 2.00000 0.0650600
\(946\) 0 0
\(947\) −54.0000 −1.75476 −0.877382 − 0.479792i \(-0.840712\pi\)
−0.877382 + 0.479792i \(0.840712\pi\)
\(948\) 0 0
\(949\) 28.0000 0.908918
\(950\) 0 0
\(951\) 34.0000 1.10253
\(952\) 0 0
\(953\) 38.0000 1.23094 0.615470 − 0.788160i \(-0.288966\pi\)
0.615470 + 0.788160i \(0.288966\pi\)
\(954\) 0 0
\(955\) 32.0000 1.03550
\(956\) 0 0
\(957\) 12.0000 0.387905
\(958\) 0 0
\(959\) −14.0000 −0.452084
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) −36.0000 −1.15888
\(966\) 0 0
\(967\) 24.0000 0.771788 0.385894 − 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 60.0000 1.92549 0.962746 − 0.270408i \(-0.0871586\pi\)
0.962746 + 0.270408i \(0.0871586\pi\)
\(972\) 0 0
\(973\) −12.0000 −0.384702
\(974\) 0 0
\(975\) 2.00000 0.0640513
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 − 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) −36.0000 −1.15056
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) 0 0
\(983\) 44.0000 1.40338 0.701691 − 0.712481i \(-0.252429\pi\)
0.701691 + 0.712481i \(0.252429\pi\)
\(984\) 0 0
\(985\) 48.0000 1.52941
\(986\) 0 0
\(987\) 6.00000 0.190982
\(988\) 0 0
\(989\) −32.0000 −1.01754
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 − 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) 30.0000 0.952021
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) 0 0
\(997\) 46.0000 1.45683 0.728417 − 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) 0 0
\(999\) −4.00000 −0.126554
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 6384.2.a.be.1.1 1
4.3 odd 2 3192.2.a.j.1.1 ✓ 1
12.11 even 2 9576.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.j.1.1 ✓ 1 4.3 odd 2
6384.2.a.be.1.1 1 1.1 even 1 trivial
9576.2.a.b.1.1 1 12.11 even 2