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Properties

Label 4368.2.a.l.1.1
Level $4368$
Weight $2$
Character 4368.1
Self dual yes
Analytic conductor $34.879$
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] = [4368,2,Mod(1,4368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4368, 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("4368.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.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: \(34.8786556029\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4368.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} +1.00000 q^{13} -3.00000 q^{15} -3.00000 q^{17} +7.00000 q^{19} +1.00000 q^{21} -9.00000 q^{23} +4.00000 q^{25} -1.00000 q^{27} -9.00000 q^{29} +4.00000 q^{31} +3.00000 q^{33} -3.00000 q^{35} -7.00000 q^{37} -1.00000 q^{39} +12.0000 q^{41} +1.00000 q^{43} +3.00000 q^{45} +1.00000 q^{49} +3.00000 q^{51} -6.00000 q^{53} -9.00000 q^{55} -7.00000 q^{57} -12.0000 q^{59} -1.00000 q^{61} -1.00000 q^{63} +3.00000 q^{65} -14.0000 q^{67} +9.00000 q^{69} -12.0000 q^{71} -7.00000 q^{73} -4.00000 q^{75} +3.00000 q^{77} +10.0000 q^{79} +1.00000 q^{81} +6.00000 q^{83} -9.00000 q^{85} +9.00000 q^{87} -6.00000 q^{89} -1.00000 q^{91} -4.00000 q^{93} +21.0000 q^{95} -10.0000 q^{97} -3.00000 q^{99} +O(q^{100})\)

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
\(5\) 3.00000 1.34164 0.670820 − 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 7.00000 1.60591 0.802955 − 0.596040i \(-0.203260\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −9.00000 −1.87663 −0.938315 − 0.345782i \(-0.887614\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 − 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −9.00000 −1.21356
\(56\) 0 0
\(57\) −7.00000 −0.927173
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 − 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 − 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) −14.0000 −1.71037 −0.855186 − 0.518321i \(-0.826557\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) 0 0
\(69\) 9.00000 1.08347
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 − 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) 0 0
\(87\) 9.00000 0.964901
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) 21.0000 2.15455
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) 12.0000 1.19404 0.597022 − 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) 13.0000 1.28093 0.640464 − 0.767988i \(-0.278742\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) 11.0000 1.05361 0.526804 − 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) 7.00000 0.664411
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 − 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) −27.0000 −2.51776
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −12.0000 −1.08200
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 − 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) 15.0000 1.31056 0.655278 − 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) −7.00000 −0.606977
\(134\) 0 0
\(135\) −3.00000 −0.258199
\(136\) 0 0
\(137\) 9.00000 0.768922 0.384461 − 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.00000 −0.250873
\(144\) 0 0
\(145\) −27.0000 −2.24223
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −12.0000 −0.983078 −0.491539 − 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) −17.0000 −1.38344 −0.691720 − 0.722166i \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 − 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 9.00000 0.709299
\(162\) 0 0
\(163\) −2.00000 −0.156652 −0.0783260 − 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) 0 0
\(165\) 9.00000 0.700649
\(166\) 0 0
\(167\) −3.00000 −0.232147 −0.116073 − 0.993241i \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 7.00000 0.535303
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 1.00000 0.0739221
\(184\) 0 0
\(185\) −21.0000 −1.54395
\(186\) 0 0
\(187\) 9.00000 0.658145
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 15.0000 1.08536 0.542681 − 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 − 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) −3.00000 −0.214834
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 − 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) 7.00000 0.496217 0.248108 − 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) 0 0
\(201\) 14.0000 0.987484
\(202\) 0 0
\(203\) 9.00000 0.631676
\(204\) 0 0
\(205\) 36.0000 2.51435
\(206\) 0 0
\(207\) −9.00000 −0.625543
\(208\) 0 0
\(209\) −21.0000 −1.45260
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 − 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 0 0
\(213\) 12.0000 0.822226
\(214\) 0 0
\(215\) 3.00000 0.204598
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) 7.00000 0.473016
\(220\) 0 0
\(221\) −3.00000 −0.201802
\(222\) 0 0
\(223\) −26.0000 −1.74109 −0.870544 − 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 0 0
\(227\) 18.0000 1.19470 0.597351 − 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 − 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) 24.0000 1.57229 0.786146 − 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 − 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 7.00000 0.445399
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −21.0000 −1.32551 −0.662754 − 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) 0 0
\(253\) 27.0000 1.69748
\(254\) 0 0
\(255\) 9.00000 0.563602
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 7.00000 0.434959
\(260\) 0 0
\(261\) −9.00000 −0.557086
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 − 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) −18.0000 −1.10573
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 − 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 0 0
\(273\) 1.00000 0.0605228
\(274\) 0 0
\(275\) −12.0000 −0.723627
\(276\) 0 0
\(277\) 26.0000 1.56219 0.781094 − 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 − 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) −21.0000 −1.24393
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 0 0
\(293\) −18.0000 −1.05157 −0.525786 − 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) −36.0000 −2.09600
\(296\) 0 0
\(297\) 3.00000 0.174078
\(298\) 0 0
\(299\) −9.00000 −0.520483
\(300\) 0 0
\(301\) −1.00000 −0.0576390
\(302\) 0 0
\(303\) −12.0000 −0.689382
\(304\) 0 0
\(305\) −3.00000 −0.171780
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 − 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) −13.0000 −0.739544
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 − 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 − 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 0 0
\(317\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 27.0000 1.51171
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) −21.0000 −1.16847
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 0 0
\(327\) −11.0000 −0.608301
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 − 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) 0 0
\(333\) −7.00000 −0.383598
\(334\) 0 0
\(335\) −42.0000 −2.29471
\(336\) 0 0
\(337\) −13.0000 −0.708155 −0.354078 − 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) 0 0
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 27.0000 1.45363
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 − 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 − 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −36.0000 −1.91068
\(356\) 0 0
\(357\) −3.00000 −0.158777
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) −21.0000 −1.09919
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 12.0000 0.624695
\(370\) 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
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) −9.00000 −0.463524
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 − 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 0 0
\(383\) 21.0000 1.07305 0.536525 − 0.843884i \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) 0 0
\(385\) 9.00000 0.458682
\(386\) 0 0
\(387\) 1.00000 0.0508329
\(388\) 0 0
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 27.0000 1.36545
\(392\) 0 0
\(393\) −15.0000 −0.756650
\(394\) 0 0
\(395\) 30.0000 1.50946
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 − 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) 7.00000 0.350438
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 − 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) 0 0
\(405\) 3.00000 0.149071
\(406\) 0 0
\(407\) 21.0000 1.04093
\(408\) 0 0
\(409\) −13.0000 −0.642809 −0.321404 − 0.946942i \(-0.604155\pi\)
−0.321404 + 0.946942i \(0.604155\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 18.0000 0.883585
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) −3.00000 −0.146560 −0.0732798 − 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 − 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) 1.00000 0.0483934
\(428\) 0 0
\(429\) 3.00000 0.144841
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 − 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 0 0
\(433\) −16.0000 −0.768911 −0.384455 − 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) 27.0000 1.29455
\(436\) 0 0
\(437\) −63.0000 −3.01370
\(438\) 0 0
\(439\) 1.00000 0.0477274 0.0238637 − 0.999715i \(-0.492403\pi\)
0.0238637 + 0.999715i \(0.492403\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −18.0000 −0.855206 −0.427603 − 0.903967i \(-0.640642\pi\)
−0.427603 + 0.903967i \(0.640642\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 0 0
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) 21.0000 0.991051 0.495526 − 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) 0 0
\(451\) −36.0000 −1.69517
\(452\) 0 0
\(453\) 17.0000 0.798730
\(454\) 0 0
\(455\) −3.00000 −0.140642
\(456\) 0 0
\(457\) −28.0000 −1.30978 −0.654892 − 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) 0 0
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) 33.0000 1.53696 0.768482 − 0.639872i \(-0.221013\pi\)
0.768482 + 0.639872i \(0.221013\pi\)
\(462\) 0 0
\(463\) −41.0000 −1.90543 −0.952716 − 0.303863i \(-0.901724\pi\)
−0.952716 + 0.303863i \(0.901724\pi\)
\(464\) 0 0
\(465\) −12.0000 −0.556487
\(466\) 0 0
\(467\) 3.00000 0.138823 0.0694117 − 0.997588i \(-0.477888\pi\)
0.0694117 + 0.997588i \(0.477888\pi\)
\(468\) 0 0
\(469\) 14.0000 0.646460
\(470\) 0 0
\(471\) 13.0000 0.599008
\(472\) 0 0
\(473\) −3.00000 −0.137940
\(474\) 0 0
\(475\) 28.0000 1.28473
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 15.0000 0.685367 0.342684 − 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) 0 0
\(481\) −7.00000 −0.319173
\(482\) 0 0
\(483\) −9.00000 −0.409514
\(484\) 0 0
\(485\) −30.0000 −1.36223
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 − 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 0 0
\(489\) 2.00000 0.0904431
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 − 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 0 0
\(493\) 27.0000 1.21602
\(494\) 0 0
\(495\) −9.00000 −0.404520
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) −32.0000 −1.43252 −0.716258 − 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) 3.00000 0.134030
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 − 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 36.0000 1.60198
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 9.00000 0.398918 0.199459 − 0.979906i \(-0.436082\pi\)
0.199459 + 0.979906i \(0.436082\pi\)
\(510\) 0 0
\(511\) 7.00000 0.309662
\(512\) 0 0
\(513\) −7.00000 −0.309058
\(514\) 0 0
\(515\) 39.0000 1.71855
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −33.0000 −1.44576 −0.722878 − 0.690976i \(-0.757181\pi\)
−0.722878 + 0.690976i \(0.757181\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 − 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 4.00000 0.174574
\(526\) 0 0
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) 58.0000 2.52174
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 11.0000 0.472927 0.236463 − 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) 0 0
\(543\) −2.00000 −0.0858282
\(544\) 0 0
\(545\) 33.0000 1.41356
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 − 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) −1.00000 −0.0426790
\(550\) 0 0
\(551\) −63.0000 −2.68389
\(552\) 0 0
\(553\) −10.0000 −0.425243
\(554\) 0 0
\(555\) 21.0000 0.891400
\(556\) 0 0
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 1.00000 0.0422955
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 0 0
\(563\) 15.0000 0.632175 0.316087 − 0.948730i \(-0.397631\pi\)
0.316087 + 0.948730i \(0.397631\pi\)
\(564\) 0 0
\(565\) −54.0000 −2.27180
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 − 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 − 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 0 0
\(573\) −15.0000 −0.626634
\(574\) 0 0
\(575\) −36.0000 −1.50130
\(576\) 0 0
\(577\) 38.0000 1.58196 0.790980 − 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 0 0
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) −6.00000 −0.248922
\(582\) 0 0
\(583\) 18.0000 0.745484
\(584\) 0 0
\(585\) 3.00000 0.124035
\(586\) 0 0
\(587\) 18.0000 0.742940 0.371470 − 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) 0 0
\(589\) 28.0000 1.15372
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 − 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 9.00000 0.368964
\(596\) 0 0
\(597\) −7.00000 −0.286491
\(598\) 0 0
\(599\) 9.00000 0.367730 0.183865 − 0.982952i \(-0.441139\pi\)
0.183865 + 0.982952i \(0.441139\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 − 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 0 0
\(603\) −14.0000 −0.570124
\(604\) 0 0
\(605\) −6.00000 −0.243935
\(606\) 0 0
\(607\) −23.0000 −0.933541 −0.466771 − 0.884378i \(-0.654583\pi\)
−0.466771 + 0.884378i \(0.654583\pi\)
\(608\) 0 0
\(609\) −9.00000 −0.364698
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 11.0000 0.444286 0.222143 − 0.975014i \(-0.428695\pi\)
0.222143 + 0.975014i \(0.428695\pi\)
\(614\) 0 0
\(615\) −36.0000 −1.45166
\(616\) 0 0
\(617\) 27.0000 1.08698 0.543490 − 0.839416i \(-0.317103\pi\)
0.543490 + 0.839416i \(0.317103\pi\)
\(618\) 0 0
\(619\) −17.0000 −0.683288 −0.341644 − 0.939829i \(-0.610984\pi\)
−0.341644 + 0.939829i \(0.610984\pi\)
\(620\) 0 0
\(621\) 9.00000 0.361158
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 21.0000 0.838659
\(628\) 0 0
\(629\) 21.0000 0.837325
\(630\) 0 0
\(631\) −29.0000 −1.15447 −0.577236 − 0.816577i \(-0.695869\pi\)
−0.577236 + 0.816577i \(0.695869\pi\)
\(632\) 0 0
\(633\) 5.00000 0.198732
\(634\) 0 0
\(635\) −6.00000 −0.238103
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 49.0000 1.93237 0.966186 − 0.257847i \(-0.0830131\pi\)
0.966186 + 0.257847i \(0.0830131\pi\)
\(644\) 0 0
\(645\) −3.00000 −0.118125
\(646\) 0 0
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) 0 0
\(653\) 9.00000 0.352197 0.176099 − 0.984373i \(-0.443652\pi\)
0.176099 + 0.984373i \(0.443652\pi\)
\(654\) 0 0
\(655\) 45.0000 1.75830
\(656\) 0 0
\(657\) −7.00000 −0.273096
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 − 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 − 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 0 0
\(663\) 3.00000 0.116510
\(664\) 0 0
\(665\) −21.0000 −0.814345
\(666\) 0 0
\(667\) 81.0000 3.13633
\(668\) 0 0
\(669\) 26.0000 1.00522
\(670\) 0 0
\(671\) 3.00000 0.115814
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 − 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 − 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 0 0
\(683\) −33.0000 −1.26271 −0.631355 − 0.775494i \(-0.717501\pi\)
−0.631355 + 0.775494i \(0.717501\pi\)
\(684\) 0 0
\(685\) 27.0000 1.03162
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) 0 0
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 0 0
\(693\) 3.00000 0.113961
\(694\) 0 0
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 0 0
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 − 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) −49.0000 −1.84807
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) 0 0
\(713\) −36.0000 −1.34821
\(714\) 0 0
\(715\) −9.00000 −0.336581
\(716\) 0 0
\(717\) 24.0000 0.896296
\(718\) 0 0
\(719\) −18.0000 −0.671287 −0.335643 − 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) −13.0000 −0.484145
\(722\) 0 0
\(723\) 10.0000 0.371904
\(724\) 0 0
\(725\) −36.0000 −1.33701
\(726\) 0 0
\(727\) 1.00000 0.0370879 0.0185440 − 0.999828i \(-0.494097\pi\)
0.0185440 + 0.999828i \(0.494097\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −3.00000 −0.110959
\(732\) 0 0
\(733\) 32.0000 1.18195 0.590973 − 0.806691i \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) 0 0
\(735\) −3.00000 −0.110657
\(736\) 0 0
\(737\) 42.0000 1.54709
\(738\) 0 0
\(739\) 34.0000 1.25071 0.625355 − 0.780340i \(-0.284954\pi\)
0.625355 + 0.780340i \(0.284954\pi\)
\(740\) 0 0
\(741\) −7.00000 −0.257151
\(742\) 0 0
\(743\) −30.0000 −1.10059 −0.550297 − 0.834969i \(-0.685485\pi\)
−0.550297 + 0.834969i \(0.685485\pi\)
\(744\) 0 0
\(745\) −36.0000 −1.31894
\(746\) 0 0
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 − 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 0 0
\(753\) 21.0000 0.765283
\(754\) 0 0
\(755\) −51.0000 −1.85608
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 − 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 0 0
\(759\) −27.0000 −0.980038
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 − 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 0 0
\(763\) −11.0000 −0.398227
\(764\) 0 0
\(765\) −9.00000 −0.325396
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) −13.0000 −0.468792 −0.234396 − 0.972141i \(-0.575311\pi\)
−0.234396 + 0.972141i \(0.575311\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 0 0
\(773\) 33.0000 1.18693 0.593464 − 0.804861i \(-0.297760\pi\)
0.593464 + 0.804861i \(0.297760\pi\)
\(774\) 0 0
\(775\) 16.0000 0.574737
\(776\) 0 0
\(777\) −7.00000 −0.251124
\(778\) 0 0
\(779\) 84.0000 3.00961
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) 0 0
\(783\) 9.00000 0.321634
\(784\) 0 0
\(785\) −39.0000 −1.39197
\(786\) 0 0
\(787\) 31.0000 1.10503 0.552515 − 0.833503i \(-0.313668\pi\)
0.552515 + 0.833503i \(0.313668\pi\)
\(788\) 0 0
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) −1.00000 −0.0355110
\(794\) 0 0
\(795\) 18.0000 0.638394
\(796\) 0 0
\(797\) 36.0000 1.27519 0.637593 − 0.770374i \(-0.279930\pi\)
0.637593 + 0.770374i \(0.279930\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 21.0000 0.741074
\(804\) 0 0
\(805\) 27.0000 0.951625
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) 54.0000 1.89854 0.949269 − 0.314464i \(-0.101825\pi\)
0.949269 + 0.314464i \(0.101825\pi\)
\(810\) 0 0
\(811\) 43.0000 1.50993 0.754967 − 0.655763i \(-0.227653\pi\)
0.754967 + 0.655763i \(0.227653\pi\)
\(812\) 0 0
\(813\) 2.00000 0.0701431
\(814\) 0 0
\(815\) −6.00000 −0.210171
\(816\) 0 0
\(817\) 7.00000 0.244899
\(818\) 0 0
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 − 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 0 0
\(823\) 40.0000 1.39431 0.697156 − 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 0 0
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) −27.0000 −0.938882 −0.469441 − 0.882964i \(-0.655545\pi\)
−0.469441 + 0.882964i \(0.655545\pi\)
\(828\) 0 0
\(829\) −25.0000 −0.868286 −0.434143 − 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) 0 0
\(831\) −26.0000 −0.901930
\(832\) 0 0
\(833\) −3.00000 −0.103944
\(834\) 0 0
\(835\) −9.00000 −0.311458
\(836\) 0 0
\(837\) −4.00000 −0.138260
\(838\) 0 0
\(839\) 12.0000 0.414286 0.207143 − 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 6.00000 0.206651
\(844\) 0 0
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 63.0000 2.15961
\(852\) 0 0
\(853\) 44.0000 1.50653 0.753266 − 0.657716i \(-0.228477\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(854\) 0 0
\(855\) 21.0000 0.718185
\(856\) 0 0
\(857\) 42.0000 1.43469 0.717346 − 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 − 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) 0 0
\(863\) −6.00000 −0.204242 −0.102121 − 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) −30.0000 −1.01768
\(870\) 0 0
\(871\) −14.0000 −0.474372
\(872\) 0 0
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 − 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 0 0
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) −3.00000 −0.101073 −0.0505363 − 0.998722i \(-0.516093\pi\)
−0.0505363 + 0.998722i \(0.516093\pi\)
\(882\) 0 0
\(883\) 25.0000 0.841317 0.420658 − 0.907219i \(-0.361799\pi\)
0.420658 + 0.907219i \(0.361799\pi\)
\(884\) 0 0
\(885\) 36.0000 1.21013
\(886\) 0 0
\(887\) −36.0000 −1.20876 −0.604381 − 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) 2.00000 0.0670778
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 9.00000 0.300501
\(898\) 0 0
\(899\) −36.0000 −1.20067
\(900\) 0 0
\(901\) 18.0000 0.599667
\(902\) 0 0
\(903\) 1.00000 0.0332779
\(904\) 0 0
\(905\) 6.00000 0.199447
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 − 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 0 0
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) 33.0000 1.09334 0.546669 − 0.837349i \(-0.315895\pi\)
0.546669 + 0.837349i \(0.315895\pi\)
\(912\) 0 0
\(913\) −18.0000 −0.595713
\(914\) 0 0
\(915\) 3.00000 0.0991769
\(916\) 0 0
\(917\) −15.0000 −0.495344
\(918\) 0 0
\(919\) 34.0000 1.12156 0.560778 − 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(920\) 0 0
\(921\) −16.0000 −0.527218
\(922\) 0 0
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) −28.0000 −0.920634
\(926\) 0 0
\(927\) 13.0000 0.426976
\(928\) 0 0
\(929\) 12.0000 0.393707 0.196854 − 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) 0 0
\(933\) 18.0000 0.589294
\(934\) 0 0
\(935\) 27.0000 0.882994
\(936\) 0 0
\(937\) −34.0000 −1.11073 −0.555366 − 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 0 0
\(939\) −8.00000 −0.261070
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 − 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) −108.000 −3.51696
\(944\) 0 0
\(945\) 3.00000 0.0975900
\(946\) 0 0
\(947\) 57.0000 1.85225 0.926126 − 0.377215i \(-0.123118\pi\)
0.926126 + 0.377215i \(0.123118\pi\)
\(948\) 0 0
\(949\) −7.00000 −0.227230
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −24.0000 −0.777436 −0.388718 − 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) 0 0
\(955\) 45.0000 1.45617
\(956\) 0 0
\(957\) −27.0000 −0.872786
\(958\) 0 0
\(959\) −9.00000 −0.290625
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) 0 0
\(965\) −12.0000 −0.386294
\(966\) 0 0
\(967\) 13.0000 0.418052 0.209026 − 0.977910i \(-0.432971\pi\)
0.209026 + 0.977910i \(0.432971\pi\)
\(968\) 0 0
\(969\) 21.0000 0.674617
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −4.00000 −0.128234
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 0 0
\(977\) 15.0000 0.479893 0.239946 − 0.970786i \(-0.422870\pi\)
0.239946 + 0.970786i \(0.422870\pi\)
\(978\) 0 0
\(979\) 18.0000 0.575282
\(980\) 0 0
\(981\) 11.0000 0.351203
\(982\) 0 0
\(983\) 39.0000 1.24391 0.621953 − 0.783054i \(-0.286339\pi\)
0.621953 + 0.783054i \(0.286339\pi\)
\(984\) 0 0
\(985\) −36.0000 −1.14706
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −9.00000 −0.286183
\(990\) 0 0
\(991\) −38.0000 −1.20711 −0.603555 − 0.797321i \(-0.706250\pi\)
−0.603555 + 0.797321i \(0.706250\pi\)
\(992\) 0 0
\(993\) −10.0000 −0.317340
\(994\) 0 0
\(995\) 21.0000 0.665745
\(996\) 0 0
\(997\) −46.0000 −1.45683 −0.728417 − 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) 0 0
\(999\) 7.00000 0.221470
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 4368.2.a.l.1.1 1
4.3 odd 2 546.2.a.d.1.1 ✓ 1
12.11 even 2 1638.2.a.l.1.1 1
28.27 even 2 3822.2.a.a.1.1 1
52.51 odd 2 7098.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.d.1.1 ✓ 1 4.3 odd 2
1638.2.a.l.1.1 1 12.11 even 2
3822.2.a.a.1.1 1 28.27 even 2
4368.2.a.l.1.1 1 1.1 even 1 trivial
7098.2.a.w.1.1 1 52.51 odd 2