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Properties

Label 3328.2.a.j.1.1
Level $3328$
Weight $2$
Character 3328.1
Self dual yes
Analytic conductor $26.574$
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] = [3328,2,Mod(1,3328)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3328, 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("3328.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3328 = 2^{8} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3328.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: \(26.5742137927\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3328.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.00000 q^{5} -3.00000 q^{7} -2.00000 q^{9} -1.00000 q^{13} +3.00000 q^{15} -7.00000 q^{17} +4.00000 q^{19} -3.00000 q^{21} -4.00000 q^{23} +4.00000 q^{25} -5.00000 q^{27} +4.00000 q^{29} -8.00000 q^{31} -9.00000 q^{35} +7.00000 q^{37} -1.00000 q^{39} -2.00000 q^{41} -1.00000 q^{43} -6.00000 q^{45} -7.00000 q^{47} +2.00000 q^{49} -7.00000 q^{51} +4.00000 q^{53} +4.00000 q^{57} -14.0000 q^{59} +10.0000 q^{61} +6.00000 q^{63} -3.00000 q^{65} -2.00000 q^{67} -4.00000 q^{69} +3.00000 q^{71} -14.0000 q^{73} +4.00000 q^{75} -10.0000 q^{79} +1.00000 q^{81} -14.0000 q^{83} -21.0000 q^{85} +4.00000 q^{87} +3.00000 q^{91} -8.00000 q^{93} +12.0000 q^{95} +8.00000 q^{97} +O(q^{100})\)

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
\(3\) 1.00000 0.577350 0.288675 − 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) 3.00000 1.34164 0.670820 − 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 − 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) −7.00000 −1.69775 −0.848875 − 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9.00000 −1.52128
\(36\) 0 0
\(37\) 7.00000 1.15079 0.575396 − 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 − 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) −6.00000 −0.894427
\(46\) 0 0
\(47\) −7.00000 −1.02105 −0.510527 − 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −7.00000 −0.980196
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) −14.0000 −1.82264 −0.911322 − 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 6.00000 0.755929
\(64\) 0 0
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 − 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.0000 −1.53670 −0.768350 − 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 0 0
\(85\) −21.0000 −2.27777
\(86\) 0 0
\(87\) 4.00000 0.428845
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 3.00000 0.314485
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 12.0000 1.23117
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 − 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) −9.00000 −0.878310
\(106\) 0 0
\(107\) −8.00000 −0.773389 −0.386695 − 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) −1.00000 −0.0957826 −0.0478913 − 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 0 0
\(111\) 7.00000 0.664411
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −12.0000 −1.11901
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 21.0000 1.92507
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) −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
\(133\) −12.0000 −1.04053
\(134\) 0 0
\(135\) −15.0000 −1.29099
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 − 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) 11.0000 0.933008 0.466504 − 0.884519i \(-0.345513\pi\)
0.466504 + 0.884519i \(0.345513\pi\)
\(140\) 0 0
\(141\) −7.00000 −0.589506
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 12.0000 0.996546
\(146\) 0 0
\(147\) 2.00000 0.164957
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −17.0000 −1.38344 −0.691720 − 0.722166i \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) 0 0
\(153\) 14.0000 1.13183
\(154\) 0 0
\(155\) −24.0000 −1.92773
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 4.00000 0.317221
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) −24.0000 −1.87983 −0.939913 − 0.341415i \(-0.889094\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −12.0000 −0.907115
\(176\) 0 0
\(177\) −14.0000 −1.05230
\(178\) 0 0
\(179\) −21.0000 −1.56961 −0.784807 − 0.619740i \(-0.787238\pi\)
−0.784807 + 0.619740i \(0.787238\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 21.0000 1.54395
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 15.0000 1.09109
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 − 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 − 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 0 0
\(195\) −3.00000 −0.214834
\(196\) 0 0
\(197\) 17.0000 1.21120 0.605600 − 0.795769i \(-0.292933\pi\)
0.605600 + 0.795769i \(0.292933\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 − 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) 0 0
\(203\) −12.0000 −0.842235
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) 8.00000 0.556038
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 15.0000 1.03264 0.516321 − 0.856395i \(-0.327301\pi\)
0.516321 + 0.856395i \(0.327301\pi\)
\(212\) 0 0
\(213\) 3.00000 0.205557
\(214\) 0 0
\(215\) −3.00000 −0.204598
\(216\) 0 0
\(217\) 24.0000 1.62923
\(218\) 0 0
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) 7.00000 0.470871
\(222\) 0 0
\(223\) −1.00000 −0.0669650 −0.0334825 − 0.999439i \(-0.510660\pi\)
−0.0334825 + 0.999439i \(0.510660\pi\)
\(224\) 0 0
\(225\) −8.00000 −0.533333
\(226\) 0 0
\(227\) 18.0000 1.19470 0.597351 − 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) 21.0000 1.38772 0.693860 − 0.720110i \(-0.255909\pi\)
0.693860 + 0.720110i \(0.255909\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −19.0000 −1.24473 −0.622366 − 0.782727i \(-0.713828\pi\)
−0.622366 + 0.782727i \(0.713828\pi\)
\(234\) 0 0
\(235\) −21.0000 −1.36989
\(236\) 0 0
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) 5.00000 0.323423 0.161712 − 0.986838i \(-0.448299\pi\)
0.161712 + 0.986838i \(0.448299\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 − 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) −14.0000 −0.887214
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 − 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −21.0000 −1.31507
\(256\) 0 0
\(257\) −7.00000 −0.436648 −0.218324 − 0.975876i \(-0.570059\pi\)
−0.218324 + 0.975876i \(0.570059\pi\)
\(258\) 0 0
\(259\) −21.0000 −1.30488
\(260\) 0 0
\(261\) −8.00000 −0.495188
\(262\) 0 0
\(263\) −14.0000 −0.863277 −0.431638 − 0.902047i \(-0.642064\pi\)
−0.431638 + 0.902047i \(0.642064\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −3.00000 −0.182237 −0.0911185 − 0.995840i \(-0.529044\pi\)
−0.0911185 + 0.995840i \(0.529044\pi\)
\(272\) 0 0
\(273\) 3.00000 0.181568
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −18.0000 −1.08152 −0.540758 − 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 0 0
\(279\) 16.0000 0.957895
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 − 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 12.0000 0.710819
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) 0 0
\(293\) −1.00000 −0.0584206 −0.0292103 − 0.999573i \(-0.509299\pi\)
−0.0292103 + 0.999573i \(0.509299\pi\)
\(294\) 0 0
\(295\) −42.0000 −2.44533
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 3.00000 0.172917
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 30.0000 1.71780
\(306\) 0 0
\(307\) −2.00000 −0.114146 −0.0570730 − 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 0 0
\(309\) 6.00000 0.341328
\(310\) 0 0
\(311\) −32.0000 −1.81455 −0.907277 − 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) 0 0
\(313\) −29.0000 −1.63918 −0.819588 − 0.572953i \(-0.805798\pi\)
−0.819588 + 0.572953i \(0.805798\pi\)
\(314\) 0 0
\(315\) 18.0000 1.01419
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) −28.0000 −1.55796
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) −1.00000 −0.0553001
\(328\) 0 0
\(329\) 21.0000 1.15777
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 − 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) 0 0
\(333\) −14.0000 −0.767195
\(334\) 0 0
\(335\) −6.00000 −0.327815
\(336\) 0 0
\(337\) −17.0000 −0.926049 −0.463025 − 0.886345i \(-0.653236\pi\)
−0.463025 + 0.886345i \(0.653236\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) −12.0000 −0.646058
\(346\) 0 0
\(347\) 7.00000 0.375780 0.187890 − 0.982190i \(-0.439835\pi\)
0.187890 + 0.982190i \(0.439835\pi\)
\(348\) 0 0
\(349\) 19.0000 1.01705 0.508523 − 0.861048i \(-0.330192\pi\)
0.508523 + 0.861048i \(0.330192\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 − 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 9.00000 0.477670
\(356\) 0 0
\(357\) 21.0000 1.11144
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) −42.0000 −2.19838
\(366\) 0 0
\(367\) −22.0000 −1.14839 −0.574195 − 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) 0 0
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −36.0000 −1.86401 −0.932005 − 0.362446i \(-0.881942\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) 0 0
\(375\) −3.00000 −0.154919
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 − 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 0 0
\(383\) −1.00000 −0.0510976 −0.0255488 − 0.999674i \(-0.508133\pi\)
−0.0255488 + 0.999674i \(0.508133\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.00000 0.101666
\(388\) 0 0
\(389\) −34.0000 −1.72387 −0.861934 − 0.507020i \(-0.830747\pi\)
−0.861934 + 0.507020i \(0.830747\pi\)
\(390\) 0 0
\(391\) 28.0000 1.41602
\(392\) 0 0
\(393\) 15.0000 0.756650
\(394\) 0 0
\(395\) −30.0000 −1.50946
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 − 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) −12.0000 −0.600751
\(400\) 0 0
\(401\) 32.0000 1.59800 0.799002 − 0.601329i \(-0.205362\pi\)
0.799002 + 0.601329i \(0.205362\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) 0 0
\(405\) 3.00000 0.149071
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) 42.0000 2.06668
\(414\) 0 0
\(415\) −42.0000 −2.06170
\(416\) 0 0
\(417\) 11.0000 0.538672
\(418\) 0 0
\(419\) −11.0000 −0.537385 −0.268693 − 0.963226i \(-0.586592\pi\)
−0.268693 + 0.963226i \(0.586592\pi\)
\(420\) 0 0
\(421\) 15.0000 0.731055 0.365528 − 0.930800i \(-0.380889\pi\)
0.365528 + 0.930800i \(0.380889\pi\)
\(422\) 0 0
\(423\) 14.0000 0.680703
\(424\) 0 0
\(425\) −28.0000 −1.35820
\(426\) 0 0
\(427\) −30.0000 −1.45180
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −3.00000 −0.144505 −0.0722525 − 0.997386i \(-0.523019\pi\)
−0.0722525 + 0.997386i \(0.523019\pi\)
\(432\) 0 0
\(433\) −1.00000 −0.0480569 −0.0240285 − 0.999711i \(-0.507649\pi\)
−0.0240285 + 0.999711i \(0.507649\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) 0 0
\(437\) −16.0000 −0.765384
\(438\) 0 0
\(439\) 10.0000 0.477274 0.238637 − 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 0 0
\(443\) 9.00000 0.427603 0.213801 − 0.976877i \(-0.431415\pi\)
0.213801 + 0.976877i \(0.431415\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) −20.0000 −0.943858 −0.471929 − 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −17.0000 −0.798730
\(454\) 0 0
\(455\) 9.00000 0.421927
\(456\) 0 0
\(457\) 32.0000 1.49690 0.748448 − 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) 0 0
\(459\) 35.0000 1.63366
\(460\) 0 0
\(461\) −5.00000 −0.232873 −0.116437 − 0.993198i \(-0.537147\pi\)
−0.116437 + 0.993198i \(0.537147\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 − 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) −24.0000 −1.11297
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 − 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 6.00000 0.277054
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 16.0000 0.734130
\(476\) 0 0
\(477\) −8.00000 −0.366295
\(478\) 0 0
\(479\) 5.00000 0.228456 0.114228 − 0.993455i \(-0.463561\pi\)
0.114228 + 0.993455i \(0.463561\pi\)
\(480\) 0 0
\(481\) −7.00000 −0.319173
\(482\) 0 0
\(483\) 12.0000 0.546019
\(484\) 0 0
\(485\) 24.0000 1.08978
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 − 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 0 0
\(489\) −24.0000 −1.08532
\(490\) 0 0
\(491\) 15.0000 0.676941 0.338470 − 0.940977i \(-0.390091\pi\)
0.338470 + 0.940977i \(0.390091\pi\)
\(492\) 0 0
\(493\) −28.0000 −1.26106
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.00000 −0.403705
\(498\) 0 0
\(499\) −6.00000 −0.268597 −0.134298 − 0.990941i \(-0.542878\pi\)
−0.134298 + 0.990941i \(0.542878\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) −4.00000 −0.178351 −0.0891756 − 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 14.0000 0.620539 0.310270 − 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 42.0000 1.85797
\(512\) 0 0
\(513\) −20.0000 −0.883022
\(514\) 0 0
\(515\) 18.0000 0.793175
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 − 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 0 0
\(523\) 44.0000 1.92399 0.961993 − 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) 0 0
\(525\) −12.0000 −0.523723
\(526\) 0 0
\(527\) 56.0000 2.43940
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 28.0000 1.21510
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) −24.0000 −1.03761
\(536\) 0 0
\(537\) −21.0000 −0.906217
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 25.0000 1.07483 0.537417 − 0.843317i \(-0.319400\pi\)
0.537417 + 0.843317i \(0.319400\pi\)
\(542\) 0 0
\(543\) 10.0000 0.429141
\(544\) 0 0
\(545\) −3.00000 −0.128506
\(546\) 0 0
\(547\) 13.0000 0.555840 0.277920 − 0.960604i \(-0.410355\pi\)
0.277920 + 0.960604i \(0.410355\pi\)
\(548\) 0 0
\(549\) −20.0000 −0.853579
\(550\) 0 0
\(551\) 16.0000 0.681623
\(552\) 0 0
\(553\) 30.0000 1.27573
\(554\) 0 0
\(555\) 21.0000 0.891400
\(556\) 0 0
\(557\) 13.0000 0.550828 0.275414 − 0.961326i \(-0.411185\pi\)
0.275414 + 0.961326i \(0.411185\pi\)
\(558\) 0 0
\(559\) 1.00000 0.0422955
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 31.0000 1.30649 0.653247 − 0.757145i \(-0.273406\pi\)
0.653247 + 0.757145i \(0.273406\pi\)
\(564\) 0 0
\(565\) −18.0000 −0.757266
\(566\) 0 0
\(567\) −3.00000 −0.125988
\(568\) 0 0
\(569\) −25.0000 −1.04805 −0.524027 − 0.851701i \(-0.675571\pi\)
−0.524027 + 0.851701i \(0.675571\pi\)
\(570\) 0 0
\(571\) 5.00000 0.209243 0.104622 − 0.994512i \(-0.466637\pi\)
0.104622 + 0.994512i \(0.466637\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) −16.0000 −0.667246
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 − 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 0 0
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) 42.0000 1.74245
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 6.00000 0.248069
\(586\) 0 0
\(587\) −28.0000 −1.15568 −0.577842 − 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 17.0000 0.699287
\(592\) 0 0
\(593\) 24.0000 0.985562 0.492781 − 0.870153i \(-0.335980\pi\)
0.492781 + 0.870153i \(0.335980\pi\)
\(594\) 0 0
\(595\) 63.0000 2.58275
\(596\) 0 0
\(597\) 10.0000 0.409273
\(598\) 0 0
\(599\) −10.0000 −0.408589 −0.204294 − 0.978909i \(-0.565490\pi\)
−0.204294 + 0.978909i \(0.565490\pi\)
\(600\) 0 0
\(601\) −27.0000 −1.10135 −0.550676 − 0.834719i \(-0.685630\pi\)
−0.550676 + 0.834719i \(0.685630\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) −33.0000 −1.34164
\(606\) 0 0
\(607\) −22.0000 −0.892952 −0.446476 − 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 0 0
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) 7.00000 0.283190
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 − 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 0 0
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 12.0000 0.483102 0.241551 − 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 0 0
\(619\) 6.00000 0.241160 0.120580 − 0.992704i \(-0.461525\pi\)
0.120580 + 0.992704i \(0.461525\pi\)
\(620\) 0 0
\(621\) 20.0000 0.802572
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −49.0000 −1.95376
\(630\) 0 0
\(631\) 23.0000 0.915616 0.457808 − 0.889051i \(-0.348635\pi\)
0.457808 + 0.889051i \(0.348635\pi\)
\(632\) 0 0
\(633\) 15.0000 0.596196
\(634\) 0 0
\(635\) 24.0000 0.952411
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 − 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 26.0000 1.02534 0.512670 − 0.858586i \(-0.328656\pi\)
0.512670 + 0.858586i \(0.328656\pi\)
\(644\) 0 0
\(645\) −3.00000 −0.118125
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 − 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 24.0000 0.940634
\(652\) 0 0
\(653\) −44.0000 −1.72185 −0.860927 − 0.508729i \(-0.830115\pi\)
−0.860927 + 0.508729i \(0.830115\pi\)
\(654\) 0 0
\(655\) 45.0000 1.75830
\(656\) 0 0
\(657\) 28.0000 1.09238
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 − 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) 30.0000 1.16686 0.583432 − 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) 0 0
\(663\) 7.00000 0.271857
\(664\) 0 0
\(665\) −36.0000 −1.39602
\(666\) 0 0
\(667\) −16.0000 −0.619522
\(668\) 0 0
\(669\) −1.00000 −0.0386622
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 29.0000 1.11787 0.558934 − 0.829212i \(-0.311211\pi\)
0.558934 + 0.829212i \(0.311211\pi\)
\(674\) 0 0
\(675\) −20.0000 −0.769800
\(676\) 0 0
\(677\) 22.0000 0.845529 0.422764 − 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) −26.0000 −0.994862 −0.497431 − 0.867503i \(-0.665723\pi\)
−0.497431 + 0.867503i \(0.665723\pi\)
\(684\) 0 0
\(685\) 36.0000 1.37549
\(686\) 0 0
\(687\) 21.0000 0.801200
\(688\) 0 0
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 33.0000 1.25176
\(696\) 0 0
\(697\) 14.0000 0.530288
\(698\) 0 0
\(699\) −19.0000 −0.718646
\(700\) 0 0
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 28.0000 1.05604
\(704\) 0 0
\(705\) −21.0000 −0.790906
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 − 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 0 0
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 5.00000 0.186728
\(718\) 0 0
\(719\) −30.0000 −1.11881 −0.559406 − 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) −18.0000 −0.670355
\(722\) 0 0
\(723\) −8.00000 −0.297523
\(724\) 0 0
\(725\) 16.0000 0.594225
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 − 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 7.00000 0.258904
\(732\) 0 0
\(733\) 1.00000 0.0369358 0.0184679 − 0.999829i \(-0.494121\pi\)
0.0184679 + 0.999829i \(0.494121\pi\)
\(734\) 0 0
\(735\) 6.00000 0.221313
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 34.0000 1.25071 0.625355 − 0.780340i \(-0.284954\pi\)
0.625355 + 0.780340i \(0.284954\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) 21.0000 0.770415 0.385208 − 0.922830i \(-0.374130\pi\)
0.385208 + 0.922830i \(0.374130\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 0 0
\(747\) 28.0000 1.02447
\(748\) 0 0
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) 2.00000 0.0729810 0.0364905 − 0.999334i \(-0.488382\pi\)
0.0364905 + 0.999334i \(0.488382\pi\)
\(752\) 0 0
\(753\) 20.0000 0.728841
\(754\) 0 0
\(755\) −51.0000 −1.85608
\(756\) 0 0
\(757\) 32.0000 1.16306 0.581530 − 0.813525i \(-0.302454\pi\)
0.581530 + 0.813525i \(0.302454\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 − 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 0 0
\(763\) 3.00000 0.108607
\(764\) 0 0
\(765\) 42.0000 1.51851
\(766\) 0 0
\(767\) 14.0000 0.505511
\(768\) 0 0
\(769\) −20.0000 −0.721218 −0.360609 − 0.932717i \(-0.617431\pi\)
−0.360609 + 0.932717i \(0.617431\pi\)
\(770\) 0 0
\(771\) −7.00000 −0.252099
\(772\) 0 0
\(773\) −21.0000 −0.755318 −0.377659 − 0.925945i \(-0.623271\pi\)
−0.377659 + 0.925945i \(0.623271\pi\)
\(774\) 0 0
\(775\) −32.0000 −1.14947
\(776\) 0 0
\(777\) −21.0000 −0.753371
\(778\) 0 0
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −20.0000 −0.714742
\(784\) 0 0
\(785\) −6.00000 −0.214149
\(786\) 0 0
\(787\) −22.0000 −0.784215 −0.392108 − 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 0 0
\(789\) −14.0000 −0.498413
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) 0 0
\(795\) 12.0000 0.425596
\(796\) 0 0
\(797\) −12.0000 −0.425062 −0.212531 − 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) 49.0000 1.73350
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 36.0000 1.26883
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) −25.0000 −0.878953 −0.439477 − 0.898254i \(-0.644836\pi\)
−0.439477 + 0.898254i \(0.644836\pi\)
\(810\) 0 0
\(811\) 40.0000 1.40459 0.702295 − 0.711886i \(-0.252159\pi\)
0.702295 + 0.711886i \(0.252159\pi\)
\(812\) 0 0
\(813\) −3.00000 −0.105215
\(814\) 0 0
\(815\) −72.0000 −2.52205
\(816\) 0 0
\(817\) −4.00000 −0.139942
\(818\) 0 0
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) −45.0000 −1.57051 −0.785255 − 0.619172i \(-0.787468\pi\)
−0.785255 + 0.619172i \(0.787468\pi\)
\(822\) 0 0
\(823\) 46.0000 1.60346 0.801730 − 0.597687i \(-0.203913\pi\)
0.801730 + 0.597687i \(0.203913\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −8.00000 −0.278187 −0.139094 − 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 − 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) −18.0000 −0.624413
\(832\) 0 0
\(833\) −14.0000 −0.485071
\(834\) 0 0
\(835\) 36.0000 1.24583
\(836\) 0 0
\(837\) 40.0000 1.38260
\(838\) 0 0
\(839\) 40.0000 1.38095 0.690477 − 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 8.00000 0.275535
\(844\) 0 0
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) 33.0000 1.13389
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) −28.0000 −0.959828
\(852\) 0 0
\(853\) 9.00000 0.308154 0.154077 − 0.988059i \(-0.450760\pi\)
0.154077 + 0.988059i \(0.450760\pi\)
\(854\) 0 0
\(855\) −24.0000 −0.820783
\(856\) 0 0
\(857\) 42.0000 1.43469 0.717346 − 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 − 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) 0 0
\(863\) −51.0000 −1.73606 −0.868030 − 0.496512i \(-0.834614\pi\)
−0.868030 + 0.496512i \(0.834614\pi\)
\(864\) 0 0
\(865\) 18.0000 0.612018
\(866\) 0 0
\(867\) 32.0000 1.08678
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 2.00000 0.0677674
\(872\) 0 0
\(873\) −16.0000 −0.541518
\(874\) 0 0
\(875\) 9.00000 0.304256
\(876\) 0 0
\(877\) −7.00000 −0.236373 −0.118187 − 0.992991i \(-0.537708\pi\)
−0.118187 + 0.992991i \(0.537708\pi\)
\(878\) 0 0
\(879\) −1.00000 −0.0337292
\(880\) 0 0
\(881\) −3.00000 −0.101073 −0.0505363 − 0.998722i \(-0.516093\pi\)
−0.0505363 + 0.998722i \(0.516093\pi\)
\(882\) 0 0
\(883\) 41.0000 1.37976 0.689880 − 0.723924i \(-0.257663\pi\)
0.689880 + 0.723924i \(0.257663\pi\)
\(884\) 0 0
\(885\) −42.0000 −1.41181
\(886\) 0 0
\(887\) −18.0000 −0.604381 −0.302190 − 0.953248i \(-0.597718\pi\)
−0.302190 + 0.953248i \(0.597718\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −28.0000 −0.936984
\(894\) 0 0
\(895\) −63.0000 −2.10586
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) 0 0
\(899\) −32.0000 −1.06726
\(900\) 0 0
\(901\) −28.0000 −0.932815
\(902\) 0 0
\(903\) 3.00000 0.0998337
\(904\) 0 0
\(905\) 30.0000 0.997234
\(906\) 0 0
\(907\) −13.0000 −0.431658 −0.215829 − 0.976431i \(-0.569245\pi\)
−0.215829 + 0.976431i \(0.569245\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 − 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 30.0000 0.991769
\(916\) 0 0
\(917\) −45.0000 −1.48603
\(918\) 0 0
\(919\) −60.0000 −1.97922 −0.989609 − 0.143787i \(-0.954072\pi\)
−0.989609 + 0.143787i \(0.954072\pi\)
\(920\) 0 0
\(921\) −2.00000 −0.0659022
\(922\) 0 0
\(923\) −3.00000 −0.0987462
\(924\) 0 0
\(925\) 28.0000 0.920634
\(926\) 0 0
\(927\) −12.0000 −0.394132
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 8.00000 0.262189
\(932\) 0 0
\(933\) −32.0000 −1.04763
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 − 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) −29.0000 −0.946379
\(940\) 0 0
\(941\) 25.0000 0.814977 0.407488 − 0.913210i \(-0.366405\pi\)
0.407488 + 0.913210i \(0.366405\pi\)
\(942\) 0 0
\(943\) 8.00000 0.260516
\(944\) 0 0
\(945\) 45.0000 1.46385
\(946\) 0 0
\(947\) −32.0000 −1.03986 −0.519930 − 0.854209i \(-0.674042\pi\)
−0.519930 + 0.854209i \(0.674042\pi\)
\(948\) 0 0
\(949\) 14.0000 0.454459
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −9.00000 −0.291539 −0.145769 − 0.989319i \(-0.546566\pi\)
−0.145769 + 0.989319i \(0.546566\pi\)
\(954\) 0 0
\(955\) 36.0000 1.16493
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 16.0000 0.515593
\(964\) 0 0
\(965\) −18.0000 −0.579441
\(966\) 0 0
\(967\) 7.00000 0.225105 0.112552 − 0.993646i \(-0.464097\pi\)
0.112552 + 0.993646i \(0.464097\pi\)
\(968\) 0 0
\(969\) −28.0000 −0.899490
\(970\) 0 0
\(971\) −15.0000 −0.481373 −0.240686 − 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 0 0
\(973\) −33.0000 −1.05793
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 − 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) −59.0000 −1.88181 −0.940904 − 0.338674i \(-0.890022\pi\)
−0.940904 + 0.338674i \(0.890022\pi\)
\(984\) 0 0
\(985\) 51.0000 1.62500
\(986\) 0 0
\(987\) 21.0000 0.668437
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 − 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 0 0
\(993\) 10.0000 0.317340
\(994\) 0 0
\(995\) 30.0000 0.951064
\(996\) 0 0
\(997\) 52.0000 1.64686 0.823428 − 0.567420i \(-0.192059\pi\)
0.823428 + 0.567420i \(0.192059\pi\)
\(998\) 0 0
\(999\) −35.0000 −1.10735
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 3328.2.a.j.1.1 1
4.3 odd 2 3328.2.a.f.1.1 1
8.3 odd 2 3328.2.a.g.1.1 1
8.5 even 2 3328.2.a.c.1.1 1
16.3 odd 4 416.2.b.a.209.1 2
16.5 even 4 104.2.b.a.53.2 yes 2
16.11 odd 4 416.2.b.a.209.2 2
16.13 even 4 104.2.b.a.53.1 ✓ 2
48.5 odd 4 936.2.g.a.469.1 2
48.11 even 4 3744.2.g.a.1873.1 2
48.29 odd 4 936.2.g.a.469.2 2
48.35 even 4 3744.2.g.a.1873.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
104.2.b.a.53.1 ✓ 2 16.13 even 4
104.2.b.a.53.2 yes 2 16.5 even 4
416.2.b.a.209.1 2 16.3 odd 4
416.2.b.a.209.2 2 16.11 odd 4
936.2.g.a.469.1 2 48.5 odd 4
936.2.g.a.469.2 2 48.29 odd 4
3328.2.a.c.1.1 1 8.5 even 2
3328.2.a.f.1.1 1 4.3 odd 2
3328.2.a.g.1.1 1 8.3 odd 2
3328.2.a.j.1.1 1 1.1 even 1 trivial
3744.2.g.a.1873.1 2 48.11 even 4
3744.2.g.a.1873.2 2 48.35 even 4