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Properties

Label 144.4.a.b.1.1
Level $144$
Weight $4$
Character 144.1
Self dual yes
Analytic conductor $8.496$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,4,Mod(1,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 144.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: \(8.49627504083\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 144.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-14.0000 q^{5} +24.0000 q^{7} +O(q^{10})\) \(q-14.0000 q^{5} +24.0000 q^{7} -28.0000 q^{11} -74.0000 q^{13} -82.0000 q^{17} -92.0000 q^{19} +8.00000 q^{23} +71.0000 q^{25} +138.000 q^{29} -80.0000 q^{31} -336.000 q^{35} +30.0000 q^{37} -282.000 q^{41} -4.00000 q^{43} +240.000 q^{47} +233.000 q^{49} +130.000 q^{53} +392.000 q^{55} +596.000 q^{59} -218.000 q^{61} +1036.00 q^{65} +436.000 q^{67} +856.000 q^{71} -998.000 q^{73} -672.000 q^{77} +32.0000 q^{79} -1508.00 q^{83} +1148.00 q^{85} +246.000 q^{89} -1776.00 q^{91} +1288.00 q^{95} +866.000 q^{97} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −14.0000 −1.25220 −0.626099 − 0.779744i \(-0.715349\pi\)
−0.626099 + 0.779744i \(0.715349\pi\)
\(6\) 0 0
\(7\) 24.0000 1.29588 0.647939 − 0.761692i \(-0.275631\pi\)
0.647939 + 0.761692i \(0.275631\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −28.0000 −0.767483 −0.383742 − 0.923440i \(-0.625365\pi\)
−0.383742 + 0.923440i \(0.625365\pi\)
\(12\) 0 0
\(13\) −74.0000 −1.57876 −0.789381 − 0.613904i \(-0.789598\pi\)
−0.789381 + 0.613904i \(0.789598\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −82.0000 −1.16988 −0.584939 − 0.811077i \(-0.698882\pi\)
−0.584939 + 0.811077i \(0.698882\pi\)
\(18\) 0 0
\(19\) −92.0000 −1.11086 −0.555428 − 0.831565i \(-0.687445\pi\)
−0.555428 + 0.831565i \(0.687445\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000 0.0725268 0.0362634 − 0.999342i \(-0.488454\pi\)
0.0362634 + 0.999342i \(0.488454\pi\)
\(24\) 0 0
\(25\) 71.0000 0.568000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 138.000 0.883654 0.441827 − 0.897100i \(-0.354331\pi\)
0.441827 + 0.897100i \(0.354331\pi\)
\(30\) 0 0
\(31\) −80.0000 −0.463498 −0.231749 − 0.972776i \(-0.574445\pi\)
−0.231749 + 0.972776i \(0.574445\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −336.000 −1.62270
\(36\) 0 0
\(37\) 30.0000 0.133296 0.0666482 − 0.997777i \(-0.478769\pi\)
0.0666482 + 0.997777i \(0.478769\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −282.000 −1.07417 −0.537085 − 0.843528i \(-0.680475\pi\)
−0.537085 + 0.843528i \(0.680475\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.0141859 −0.00709296 − 0.999975i \(-0.502258\pi\)
−0.00709296 + 0.999975i \(0.502258\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 240.000 0.744843 0.372421 − 0.928064i \(-0.378528\pi\)
0.372421 + 0.928064i \(0.378528\pi\)
\(48\) 0 0
\(49\) 233.000 0.679300
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 130.000 0.336922 0.168461 − 0.985708i \(-0.446120\pi\)
0.168461 + 0.985708i \(0.446120\pi\)
\(54\) 0 0
\(55\) 392.000 0.961041
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 596.000 1.31513 0.657564 − 0.753398i \(-0.271587\pi\)
0.657564 + 0.753398i \(0.271587\pi\)
\(60\) 0 0
\(61\) −218.000 −0.457574 −0.228787 − 0.973476i \(-0.573476\pi\)
−0.228787 + 0.973476i \(0.573476\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1036.00 1.97692
\(66\) 0 0
\(67\) 436.000 0.795013 0.397507 − 0.917599i \(-0.369876\pi\)
0.397507 + 0.917599i \(0.369876\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 856.000 1.43082 0.715412 − 0.698703i \(-0.246239\pi\)
0.715412 + 0.698703i \(0.246239\pi\)
\(72\) 0 0
\(73\) −998.000 −1.60010 −0.800048 − 0.599935i \(-0.795193\pi\)
−0.800048 + 0.599935i \(0.795193\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −672.000 −0.994565
\(78\) 0 0
\(79\) 32.0000 0.0455732 0.0227866 − 0.999740i \(-0.492746\pi\)
0.0227866 + 0.999740i \(0.492746\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1508.00 −1.99427 −0.997136 − 0.0756351i \(-0.975902\pi\)
−0.997136 + 0.0756351i \(0.975902\pi\)
\(84\) 0 0
\(85\) 1148.00 1.46492
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 246.000 0.292988 0.146494 − 0.989212i \(-0.453201\pi\)
0.146494 + 0.989212i \(0.453201\pi\)
\(90\) 0 0
\(91\) −1776.00 −2.04588
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1288.00 1.39101
\(96\) 0 0
\(97\) 866.000 0.906484 0.453242 − 0.891387i \(-0.350267\pi\)
0.453242 + 0.891387i \(0.350267\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −270.000 −0.266000 −0.133000 − 0.991116i \(-0.542461\pi\)
−0.133000 + 0.991116i \(0.542461\pi\)
\(102\) 0 0
\(103\) 1496.00 1.43112 0.715560 − 0.698552i \(-0.246172\pi\)
0.715560 + 0.698552i \(0.246172\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1692.00 −1.52871 −0.764354 − 0.644797i \(-0.776942\pi\)
−0.764354 + 0.644797i \(0.776942\pi\)
\(108\) 0 0
\(109\) 406.000 0.356768 0.178384 − 0.983961i \(-0.442913\pi\)
0.178384 + 0.983961i \(0.442913\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −786.000 −0.654342 −0.327171 − 0.944965i \(-0.606095\pi\)
−0.327171 + 0.944965i \(0.606095\pi\)
\(114\) 0 0
\(115\) −112.000 −0.0908179
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1968.00 −1.51602
\(120\) 0 0
\(121\) −547.000 −0.410969
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 756.000 0.540950
\(126\) 0 0
\(127\) −1744.00 −1.21854 −0.609272 − 0.792962i \(-0.708538\pi\)
−0.609272 + 0.792962i \(0.708538\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 652.000 0.434851 0.217426 − 0.976077i \(-0.430234\pi\)
0.217426 + 0.976077i \(0.430234\pi\)
\(132\) 0 0
\(133\) −2208.00 −1.43953
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1530.00 −0.954137 −0.477068 − 0.878866i \(-0.658301\pi\)
−0.477068 + 0.878866i \(0.658301\pi\)
\(138\) 0 0
\(139\) −516.000 −0.314867 −0.157434 − 0.987530i \(-0.550322\pi\)
−0.157434 + 0.987530i \(0.550322\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2072.00 1.21167
\(144\) 0 0
\(145\) −1932.00 −1.10651
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1342.00 −0.737859 −0.368929 − 0.929457i \(-0.620276\pi\)
−0.368929 + 0.929457i \(0.620276\pi\)
\(150\) 0 0
\(151\) 424.000 0.228507 0.114254 − 0.993452i \(-0.463552\pi\)
0.114254 + 0.993452i \(0.463552\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1120.00 0.580391
\(156\) 0 0
\(157\) 262.000 0.133184 0.0665920 − 0.997780i \(-0.478787\pi\)
0.0665920 + 0.997780i \(0.478787\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 192.000 0.0939858
\(162\) 0 0
\(163\) 2292.00 1.10137 0.550685 − 0.834713i \(-0.314367\pi\)
0.550685 + 0.834713i \(0.314367\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1896.00 −0.878544 −0.439272 − 0.898354i \(-0.644764\pi\)
−0.439272 + 0.898354i \(0.644764\pi\)
\(168\) 0 0
\(169\) 3279.00 1.49249
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2874.00 1.26304 0.631521 − 0.775359i \(-0.282431\pi\)
0.631521 + 0.775359i \(0.282431\pi\)
\(174\) 0 0
\(175\) 1704.00 0.736059
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1188.00 −0.496063 −0.248032 − 0.968752i \(-0.579784\pi\)
−0.248032 + 0.968752i \(0.579784\pi\)
\(180\) 0 0
\(181\) −3474.00 −1.42663 −0.713316 − 0.700843i \(-0.752808\pi\)
−0.713316 + 0.700843i \(0.752808\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −420.000 −0.166914
\(186\) 0 0
\(187\) 2296.00 0.897862
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 192.000 0.0727363 0.0363681 − 0.999338i \(-0.488421\pi\)
0.0363681 + 0.999338i \(0.488421\pi\)
\(192\) 0 0
\(193\) 4802.00 1.79096 0.895481 − 0.445100i \(-0.146832\pi\)
0.895481 + 0.445100i \(0.146832\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1518.00 −0.549000 −0.274500 − 0.961587i \(-0.588512\pi\)
−0.274500 + 0.961587i \(0.588512\pi\)
\(198\) 0 0
\(199\) −5128.00 −1.82670 −0.913352 − 0.407170i \(-0.866516\pi\)
−0.913352 + 0.407170i \(0.866516\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3312.00 1.14511
\(204\) 0 0
\(205\) 3948.00 1.34507
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2576.00 0.852563
\(210\) 0 0
\(211\) −1084.00 −0.353676 −0.176838 − 0.984240i \(-0.556587\pi\)
−0.176838 + 0.984240i \(0.556587\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 56.0000 0.0177636
\(216\) 0 0
\(217\) −1920.00 −0.600636
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6068.00 1.84696
\(222\) 0 0
\(223\) −688.000 −0.206600 −0.103300 − 0.994650i \(-0.532940\pi\)
−0.103300 + 0.994650i \(0.532940\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4812.00 1.40698 0.703488 − 0.710707i \(-0.251625\pi\)
0.703488 + 0.710707i \(0.251625\pi\)
\(228\) 0 0
\(229\) 2494.00 0.719686 0.359843 − 0.933013i \(-0.382830\pi\)
0.359843 + 0.933013i \(0.382830\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −698.000 −0.196255 −0.0981277 − 0.995174i \(-0.531285\pi\)
−0.0981277 + 0.995174i \(0.531285\pi\)
\(234\) 0 0
\(235\) −3360.00 −0.932690
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6320.00 −1.71049 −0.855244 − 0.518225i \(-0.826593\pi\)
−0.855244 + 0.518225i \(0.826593\pi\)
\(240\) 0 0
\(241\) −6510.00 −1.74002 −0.870012 − 0.493030i \(-0.835889\pi\)
−0.870012 + 0.493030i \(0.835889\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3262.00 −0.850619
\(246\) 0 0
\(247\) 6808.00 1.75378
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 628.000 0.157924 0.0789622 − 0.996878i \(-0.474839\pi\)
0.0789622 + 0.996878i \(0.474839\pi\)
\(252\) 0 0
\(253\) −224.000 −0.0556631
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4862.00 1.18009 0.590045 − 0.807370i \(-0.299110\pi\)
0.590045 + 0.807370i \(0.299110\pi\)
\(258\) 0 0
\(259\) 720.000 0.172736
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5816.00 1.36361 0.681806 − 0.731533i \(-0.261195\pi\)
0.681806 + 0.731533i \(0.261195\pi\)
\(264\) 0 0
\(265\) −1820.00 −0.421893
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3526.00 −0.799197 −0.399599 − 0.916690i \(-0.630850\pi\)
−0.399599 + 0.916690i \(0.630850\pi\)
\(270\) 0 0
\(271\) 256.000 0.0573834 0.0286917 − 0.999588i \(-0.490866\pi\)
0.0286917 + 0.999588i \(0.490866\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1988.00 −0.435931
\(276\) 0 0
\(277\) 142.000 0.0308013 0.0154006 − 0.999881i \(-0.495098\pi\)
0.0154006 + 0.999881i \(0.495098\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8842.00 −1.87712 −0.938558 − 0.345122i \(-0.887838\pi\)
−0.938558 + 0.345122i \(0.887838\pi\)
\(282\) 0 0
\(283\) 7180.00 1.50815 0.754075 − 0.656788i \(-0.228085\pi\)
0.754075 + 0.656788i \(0.228085\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6768.00 −1.39199
\(288\) 0 0
\(289\) 1811.00 0.368614
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −7374.00 −1.47029 −0.735143 − 0.677912i \(-0.762885\pi\)
−0.735143 + 0.677912i \(0.762885\pi\)
\(294\) 0 0
\(295\) −8344.00 −1.64680
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −592.000 −0.114502
\(300\) 0 0
\(301\) −96.0000 −0.0183832
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3052.00 0.572974
\(306\) 0 0
\(307\) −1500.00 −0.278858 −0.139429 − 0.990232i \(-0.544527\pi\)
−0.139429 + 0.990232i \(0.544527\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −7608.00 −1.38717 −0.693585 − 0.720374i \(-0.743970\pi\)
−0.693585 + 0.720374i \(0.743970\pi\)
\(312\) 0 0
\(313\) −4758.00 −0.859227 −0.429614 − 0.903013i \(-0.641350\pi\)
−0.429614 + 0.903013i \(0.641350\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4374.00 −0.774979 −0.387489 − 0.921874i \(-0.626658\pi\)
−0.387489 + 0.921874i \(0.626658\pi\)
\(318\) 0 0
\(319\) −3864.00 −0.678190
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 7544.00 1.29956
\(324\) 0 0
\(325\) −5254.00 −0.896737
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 5760.00 0.965225
\(330\) 0 0
\(331\) 7804.00 1.29591 0.647956 − 0.761678i \(-0.275624\pi\)
0.647956 + 0.761678i \(0.275624\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6104.00 −0.995514
\(336\) 0 0
\(337\) 5106.00 0.825346 0.412673 − 0.910879i \(-0.364595\pi\)
0.412673 + 0.910879i \(0.364595\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2240.00 0.355727
\(342\) 0 0
\(343\) −2640.00 −0.415588
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4716.00 −0.729591 −0.364796 − 0.931088i \(-0.618861\pi\)
−0.364796 + 0.931088i \(0.618861\pi\)
\(348\) 0 0
\(349\) 7302.00 1.11996 0.559982 − 0.828505i \(-0.310808\pi\)
0.559982 + 0.828505i \(0.310808\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4382.00 0.660709 0.330355 − 0.943857i \(-0.392832\pi\)
0.330355 + 0.943857i \(0.392832\pi\)
\(354\) 0 0
\(355\) −11984.0 −1.79168
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7224.00 1.06203 0.531014 − 0.847363i \(-0.321811\pi\)
0.531014 + 0.847363i \(0.321811\pi\)
\(360\) 0 0
\(361\) 1605.00 0.233999
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 13972.0 2.00364
\(366\) 0 0
\(367\) −1408.00 −0.200264 −0.100132 − 0.994974i \(-0.531927\pi\)
−0.100132 + 0.994974i \(0.531927\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3120.00 0.436610
\(372\) 0 0
\(373\) −1714.00 −0.237929 −0.118965 − 0.992899i \(-0.537957\pi\)
−0.118965 + 0.992899i \(0.537957\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −10212.0 −1.39508
\(378\) 0 0
\(379\) −884.000 −0.119810 −0.0599051 − 0.998204i \(-0.519080\pi\)
−0.0599051 + 0.998204i \(0.519080\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 10368.0 1.38324 0.691619 − 0.722263i \(-0.256898\pi\)
0.691619 + 0.722263i \(0.256898\pi\)
\(384\) 0 0
\(385\) 9408.00 1.24539
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −398.000 −0.0518751 −0.0259375 − 0.999664i \(-0.508257\pi\)
−0.0259375 + 0.999664i \(0.508257\pi\)
\(390\) 0 0
\(391\) −656.000 −0.0848474
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −448.000 −0.0570666
\(396\) 0 0
\(397\) −5098.00 −0.644487 −0.322243 − 0.946657i \(-0.604437\pi\)
−0.322243 + 0.946657i \(0.604437\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10002.0 −1.24558 −0.622788 − 0.782391i \(-0.714000\pi\)
−0.622788 + 0.782391i \(0.714000\pi\)
\(402\) 0 0
\(403\) 5920.00 0.731752
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −840.000 −0.102303
\(408\) 0 0
\(409\) −9270.00 −1.12071 −0.560357 − 0.828251i \(-0.689336\pi\)
−0.560357 + 0.828251i \(0.689336\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 14304.0 1.70425
\(414\) 0 0
\(415\) 21112.0 2.49722
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −6516.00 −0.759731 −0.379866 − 0.925042i \(-0.624030\pi\)
−0.379866 + 0.925042i \(0.624030\pi\)
\(420\) 0 0
\(421\) −2626.00 −0.303999 −0.151999 − 0.988381i \(-0.548571\pi\)
−0.151999 + 0.988381i \(0.548571\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5822.00 −0.664491
\(426\) 0 0
\(427\) −5232.00 −0.592961
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4304.00 −0.481012 −0.240506 − 0.970648i \(-0.577313\pi\)
−0.240506 + 0.970648i \(0.577313\pi\)
\(432\) 0 0
\(433\) 11794.0 1.30897 0.654484 − 0.756076i \(-0.272886\pi\)
0.654484 + 0.756076i \(0.272886\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −736.000 −0.0805667
\(438\) 0 0
\(439\) 5544.00 0.602735 0.301368 − 0.953508i \(-0.402557\pi\)
0.301368 + 0.953508i \(0.402557\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3788.00 −0.406260 −0.203130 − 0.979152i \(-0.565111\pi\)
−0.203130 + 0.979152i \(0.565111\pi\)
\(444\) 0 0
\(445\) −3444.00 −0.366879
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 13342.0 1.40233 0.701167 − 0.712997i \(-0.252663\pi\)
0.701167 + 0.712997i \(0.252663\pi\)
\(450\) 0 0
\(451\) 7896.00 0.824408
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 24864.0 2.56185
\(456\) 0 0
\(457\) −4390.00 −0.449356 −0.224678 − 0.974433i \(-0.572133\pi\)
−0.224678 + 0.974433i \(0.572133\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −5798.00 −0.585770 −0.292885 − 0.956148i \(-0.594615\pi\)
−0.292885 + 0.956148i \(0.594615\pi\)
\(462\) 0 0
\(463\) 14656.0 1.47111 0.735553 − 0.677467i \(-0.236922\pi\)
0.735553 + 0.677467i \(0.236922\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8412.00 0.833535 0.416768 − 0.909013i \(-0.363163\pi\)
0.416768 + 0.909013i \(0.363163\pi\)
\(468\) 0 0
\(469\) 10464.0 1.03024
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 112.000 0.0108875
\(474\) 0 0
\(475\) −6532.00 −0.630966
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 14848.0 1.41633 0.708165 − 0.706047i \(-0.249523\pi\)
0.708165 + 0.706047i \(0.249523\pi\)
\(480\) 0 0
\(481\) −2220.00 −0.210443
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −12124.0 −1.13510
\(486\) 0 0
\(487\) −18568.0 −1.72771 −0.863857 − 0.503738i \(-0.831958\pi\)
−0.863857 + 0.503738i \(0.831958\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −14364.0 −1.32024 −0.660120 − 0.751160i \(-0.729495\pi\)
−0.660120 + 0.751160i \(0.729495\pi\)
\(492\) 0 0
\(493\) −11316.0 −1.03377
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 20544.0 1.85417
\(498\) 0 0
\(499\) −21660.0 −1.94316 −0.971578 − 0.236720i \(-0.923928\pi\)
−0.971578 + 0.236720i \(0.923928\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −17112.0 −1.51687 −0.758436 − 0.651748i \(-0.774036\pi\)
−0.758436 + 0.651748i \(0.774036\pi\)
\(504\) 0 0
\(505\) 3780.00 0.333085
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −11478.0 −0.999516 −0.499758 − 0.866165i \(-0.666578\pi\)
−0.499758 + 0.866165i \(0.666578\pi\)
\(510\) 0 0
\(511\) −23952.0 −2.07353
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −20944.0 −1.79204
\(516\) 0 0
\(517\) −6720.00 −0.571654
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −13114.0 −1.10275 −0.551377 − 0.834256i \(-0.685897\pi\)
−0.551377 + 0.834256i \(0.685897\pi\)
\(522\) 0 0
\(523\) 4508.00 0.376905 0.188452 − 0.982082i \(-0.439653\pi\)
0.188452 + 0.982082i \(0.439653\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6560.00 0.542235
\(528\) 0 0
\(529\) −12103.0 −0.994740
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 20868.0 1.69586
\(534\) 0 0
\(535\) 23688.0 1.91425
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6524.00 −0.521352
\(540\) 0 0
\(541\) 22950.0 1.82384 0.911920 − 0.410368i \(-0.134600\pi\)
0.911920 + 0.410368i \(0.134600\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5684.00 −0.446745
\(546\) 0 0
\(547\) 6580.00 0.514334 0.257167 − 0.966367i \(-0.417211\pi\)
0.257167 + 0.966367i \(0.417211\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −12696.0 −0.981611
\(552\) 0 0
\(553\) 768.000 0.0590573
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7046.00 −0.535994 −0.267997 − 0.963420i \(-0.586362\pi\)
−0.267997 + 0.963420i \(0.586362\pi\)
\(558\) 0 0
\(559\) 296.000 0.0223962
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8252.00 0.617727 0.308864 − 0.951106i \(-0.400051\pi\)
0.308864 + 0.951106i \(0.400051\pi\)
\(564\) 0 0
\(565\) 11004.0 0.819366
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6838.00 0.503803 0.251901 − 0.967753i \(-0.418944\pi\)
0.251901 + 0.967753i \(0.418944\pi\)
\(570\) 0 0
\(571\) −23316.0 −1.70883 −0.854417 − 0.519588i \(-0.826085\pi\)
−0.854417 + 0.519588i \(0.826085\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 568.000 0.0411952
\(576\) 0 0
\(577\) −10558.0 −0.761760 −0.380880 − 0.924625i \(-0.624379\pi\)
−0.380880 + 0.924625i \(0.624379\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −36192.0 −2.58433
\(582\) 0 0
\(583\) −3640.00 −0.258582
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1028.00 0.0722830 0.0361415 − 0.999347i \(-0.488493\pi\)
0.0361415 + 0.999347i \(0.488493\pi\)
\(588\) 0 0
\(589\) 7360.00 0.514879
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1202.00 −0.0832382 −0.0416191 − 0.999134i \(-0.513252\pi\)
−0.0416191 + 0.999134i \(0.513252\pi\)
\(594\) 0 0
\(595\) 27552.0 1.89836
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3576.00 −0.243926 −0.121963 − 0.992535i \(-0.538919\pi\)
−0.121963 + 0.992535i \(0.538919\pi\)
\(600\) 0 0
\(601\) 8650.00 0.587090 0.293545 − 0.955945i \(-0.405165\pi\)
0.293545 + 0.955945i \(0.405165\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7658.00 0.514615
\(606\) 0 0
\(607\) −12656.0 −0.846279 −0.423139 − 0.906065i \(-0.639072\pi\)
−0.423139 + 0.906065i \(0.639072\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −17760.0 −1.17593
\(612\) 0 0
\(613\) −3298.00 −0.217300 −0.108650 − 0.994080i \(-0.534653\pi\)
−0.108650 + 0.994080i \(0.534653\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5370.00 −0.350386 −0.175193 − 0.984534i \(-0.556055\pi\)
−0.175193 + 0.984534i \(0.556055\pi\)
\(618\) 0 0
\(619\) 16220.0 1.05321 0.526605 − 0.850110i \(-0.323465\pi\)
0.526605 + 0.850110i \(0.323465\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5904.00 0.379677
\(624\) 0 0
\(625\) −19459.0 −1.24538
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2460.00 −0.155941
\(630\) 0 0
\(631\) 20360.0 1.28450 0.642249 − 0.766496i \(-0.278001\pi\)
0.642249 + 0.766496i \(0.278001\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 24416.0 1.52586
\(636\) 0 0
\(637\) −17242.0 −1.07245
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −14498.0 −0.893349 −0.446674 − 0.894697i \(-0.647392\pi\)
−0.446674 + 0.894697i \(0.647392\pi\)
\(642\) 0 0
\(643\) −21612.0 −1.32550 −0.662748 − 0.748842i \(-0.730610\pi\)
−0.662748 + 0.748842i \(0.730610\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12184.0 0.740344 0.370172 − 0.928963i \(-0.379299\pi\)
0.370172 + 0.928963i \(0.379299\pi\)
\(648\) 0 0
\(649\) −16688.0 −1.00934
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 28122.0 1.68530 0.842648 − 0.538464i \(-0.180995\pi\)
0.842648 + 0.538464i \(0.180995\pi\)
\(654\) 0 0
\(655\) −9128.00 −0.544520
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −5700.00 −0.336935 −0.168468 − 0.985707i \(-0.553882\pi\)
−0.168468 + 0.985707i \(0.553882\pi\)
\(660\) 0 0
\(661\) −29458.0 −1.73341 −0.866705 − 0.498822i \(-0.833766\pi\)
−0.866705 + 0.498822i \(0.833766\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 30912.0 1.80258
\(666\) 0 0
\(667\) 1104.00 0.0640885
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 6104.00 0.351181
\(672\) 0 0
\(673\) 19810.0 1.13465 0.567325 − 0.823494i \(-0.307978\pi\)
0.567325 + 0.823494i \(0.307978\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10450.0 0.593244 0.296622 − 0.954995i \(-0.404140\pi\)
0.296622 + 0.954995i \(0.404140\pi\)
\(678\) 0 0
\(679\) 20784.0 1.17469
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 23300.0 1.30534 0.652672 − 0.757641i \(-0.273648\pi\)
0.652672 + 0.757641i \(0.273648\pi\)
\(684\) 0 0
\(685\) 21420.0 1.19477
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −9620.00 −0.531920
\(690\) 0 0
\(691\) 14212.0 0.782417 0.391208 − 0.920302i \(-0.372057\pi\)
0.391208 + 0.920302i \(0.372057\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7224.00 0.394276
\(696\) 0 0
\(697\) 23124.0 1.25665
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 15978.0 0.860885 0.430443 − 0.902618i \(-0.358357\pi\)
0.430443 + 0.902618i \(0.358357\pi\)
\(702\) 0 0
\(703\) −2760.00 −0.148073
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6480.00 −0.344704
\(708\) 0 0
\(709\) −8866.00 −0.469633 −0.234816 − 0.972040i \(-0.575449\pi\)
−0.234816 + 0.972040i \(0.575449\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −640.000 −0.0336160
\(714\) 0 0
\(715\) −29008.0 −1.51726
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 7760.00 0.402502 0.201251 − 0.979540i \(-0.435499\pi\)
0.201251 + 0.979540i \(0.435499\pi\)
\(720\) 0 0
\(721\) 35904.0 1.85456
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 9798.00 0.501915
\(726\) 0 0
\(727\) −13080.0 −0.667277 −0.333638 − 0.942701i \(-0.608276\pi\)
−0.333638 + 0.942701i \(0.608276\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 328.000 0.0165958
\(732\) 0 0
\(733\) 16934.0 0.853304 0.426652 − 0.904416i \(-0.359693\pi\)
0.426652 + 0.904416i \(0.359693\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12208.0 −0.610159
\(738\) 0 0
\(739\) 7060.00 0.351429 0.175715 − 0.984441i \(-0.443776\pi\)
0.175715 + 0.984441i \(0.443776\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −12520.0 −0.618189 −0.309094 − 0.951031i \(-0.600026\pi\)
−0.309094 + 0.951031i \(0.600026\pi\)
\(744\) 0 0
\(745\) 18788.0 0.923945
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −40608.0 −1.98102
\(750\) 0 0
\(751\) 9792.00 0.475786 0.237893 − 0.971291i \(-0.423543\pi\)
0.237893 + 0.971291i \(0.423543\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −5936.00 −0.286137
\(756\) 0 0
\(757\) 13166.0 0.632135 0.316068 − 0.948737i \(-0.397637\pi\)
0.316068 + 0.948737i \(0.397637\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 23222.0 1.10617 0.553086 − 0.833124i \(-0.313450\pi\)
0.553086 + 0.833124i \(0.313450\pi\)
\(762\) 0 0
\(763\) 9744.00 0.462328
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −44104.0 −2.07628
\(768\) 0 0
\(769\) −39934.0 −1.87264 −0.936318 − 0.351154i \(-0.885789\pi\)
−0.936318 + 0.351154i \(0.885789\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 17106.0 0.795938 0.397969 − 0.917399i \(-0.369715\pi\)
0.397969 + 0.917399i \(0.369715\pi\)
\(774\) 0 0
\(775\) −5680.00 −0.263267
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 25944.0 1.19325
\(780\) 0 0
\(781\) −23968.0 −1.09813
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3668.00 −0.166773
\(786\) 0 0
\(787\) 9956.00 0.450944 0.225472 − 0.974250i \(-0.427608\pi\)
0.225472 + 0.974250i \(0.427608\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −18864.0 −0.847948
\(792\) 0 0
\(793\) 16132.0 0.722401
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 9130.00 0.405773 0.202887 − 0.979202i \(-0.434968\pi\)
0.202887 + 0.979202i \(0.434968\pi\)
\(798\) 0 0
\(799\) −19680.0 −0.871375
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 27944.0 1.22805
\(804\) 0 0
\(805\) −2688.00 −0.117689
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −11482.0 −0.498993 −0.249497 − 0.968376i \(-0.580265\pi\)
−0.249497 + 0.968376i \(0.580265\pi\)
\(810\) 0 0
\(811\) −4612.00 −0.199691 −0.0998454 − 0.995003i \(-0.531835\pi\)
−0.0998454 + 0.995003i \(0.531835\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −32088.0 −1.37913
\(816\) 0 0
\(817\) 368.000 0.0157585
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 35010.0 1.48826 0.744128 − 0.668038i \(-0.232865\pi\)
0.744128 + 0.668038i \(0.232865\pi\)
\(822\) 0 0
\(823\) −13688.0 −0.579749 −0.289875 − 0.957065i \(-0.593614\pi\)
−0.289875 + 0.957065i \(0.593614\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 11668.0 0.490612 0.245306 − 0.969446i \(-0.421112\pi\)
0.245306 + 0.969446i \(0.421112\pi\)
\(828\) 0 0
\(829\) −29306.0 −1.22779 −0.613896 − 0.789387i \(-0.710399\pi\)
−0.613896 + 0.789387i \(0.710399\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −19106.0 −0.794698
\(834\) 0 0
\(835\) 26544.0 1.10011
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −2664.00 −0.109620 −0.0548102 − 0.998497i \(-0.517455\pi\)
−0.0548102 + 0.998497i \(0.517455\pi\)
\(840\) 0 0
\(841\) −5345.00 −0.219156
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −45906.0 −1.86889
\(846\) 0 0
\(847\) −13128.0 −0.532566
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 240.000 0.00966756
\(852\) 0 0
\(853\) 26030.0 1.04484 0.522421 − 0.852688i \(-0.325029\pi\)
0.522421 + 0.852688i \(0.325029\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −44202.0 −1.76186 −0.880929 − 0.473249i \(-0.843081\pi\)
−0.880929 + 0.473249i \(0.843081\pi\)
\(858\) 0 0
\(859\) 32748.0 1.30075 0.650377 − 0.759612i \(-0.274611\pi\)
0.650377 + 0.759612i \(0.274611\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 45344.0 1.78856 0.894280 − 0.447507i \(-0.147688\pi\)
0.894280 + 0.447507i \(0.147688\pi\)
\(864\) 0 0
\(865\) −40236.0 −1.58158
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −896.000 −0.0349767
\(870\) 0 0
\(871\) −32264.0 −1.25514
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 18144.0 0.701005
\(876\) 0 0
\(877\) −8778.00 −0.337984 −0.168992 − 0.985617i \(-0.554051\pi\)
−0.168992 + 0.985617i \(0.554051\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 4142.00 0.158397 0.0791984 − 0.996859i \(-0.474764\pi\)
0.0791984 + 0.996859i \(0.474764\pi\)
\(882\) 0 0
\(883\) −22076.0 −0.841355 −0.420678 − 0.907210i \(-0.638208\pi\)
−0.420678 + 0.907210i \(0.638208\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −40376.0 −1.52840 −0.764201 − 0.644978i \(-0.776867\pi\)
−0.764201 + 0.644978i \(0.776867\pi\)
\(888\) 0 0
\(889\) −41856.0 −1.57908
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −22080.0 −0.827412
\(894\) 0 0
\(895\) 16632.0 0.621169
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −11040.0 −0.409571
\(900\) 0 0
\(901\) −10660.0 −0.394158
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 48636.0 1.78643
\(906\) 0 0
\(907\) 26396.0 0.966334 0.483167 − 0.875528i \(-0.339486\pi\)
0.483167 + 0.875528i \(0.339486\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 24368.0 0.886222 0.443111 − 0.896467i \(-0.353875\pi\)
0.443111 + 0.896467i \(0.353875\pi\)
\(912\) 0 0
\(913\) 42224.0 1.53057
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 15648.0 0.563514
\(918\) 0 0
\(919\) 5096.00 0.182918 0.0914589 − 0.995809i \(-0.470847\pi\)
0.0914589 + 0.995809i \(0.470847\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −63344.0 −2.25893
\(924\) 0 0
\(925\) 2130.00 0.0757124
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 18494.0 0.653142 0.326571 − 0.945173i \(-0.394107\pi\)
0.326571 + 0.945173i \(0.394107\pi\)
\(930\) 0 0
\(931\) −21436.0 −0.754604
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −32144.0 −1.12430
\(936\) 0 0
\(937\) −33222.0 −1.15829 −0.579144 − 0.815225i \(-0.696613\pi\)
−0.579144 + 0.815225i \(0.696613\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −27846.0 −0.964669 −0.482335 − 0.875987i \(-0.660211\pi\)
−0.482335 + 0.875987i \(0.660211\pi\)
\(942\) 0 0
\(943\) −2256.00 −0.0779061
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 41052.0 1.40867 0.704335 − 0.709868i \(-0.251245\pi\)
0.704335 + 0.709868i \(0.251245\pi\)
\(948\) 0 0
\(949\) 73852.0 2.52617
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −5706.00 −0.193951 −0.0969756 − 0.995287i \(-0.530917\pi\)
−0.0969756 + 0.995287i \(0.530917\pi\)
\(954\) 0 0
\(955\) −2688.00 −0.0910802
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −36720.0 −1.23644
\(960\) 0 0
\(961\) −23391.0 −0.785170
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −67228.0 −2.24264
\(966\) 0 0
\(967\) 39352.0 1.30866 0.654330 − 0.756209i \(-0.272951\pi\)
0.654330 + 0.756209i \(0.272951\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −33180.0 −1.09660 −0.548299 − 0.836282i \(-0.684724\pi\)
−0.548299 + 0.836282i \(0.684724\pi\)
\(972\) 0 0
\(973\) −12384.0 −0.408030
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4014.00 0.131442 0.0657212 − 0.997838i \(-0.479065\pi\)
0.0657212 + 0.997838i \(0.479065\pi\)
\(978\) 0 0
\(979\) −6888.00 −0.224864
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 20328.0 0.659575 0.329788 − 0.944055i \(-0.393023\pi\)
0.329788 + 0.944055i \(0.393023\pi\)
\(984\) 0 0
\(985\) 21252.0 0.687457
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −32.0000 −0.00102886
\(990\) 0 0
\(991\) −11728.0 −0.375936 −0.187968 − 0.982175i \(-0.560190\pi\)
−0.187968 + 0.982175i \(0.560190\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 71792.0 2.28740
\(996\) 0 0
\(997\) 50974.0 1.61922 0.809610 − 0.586968i \(-0.199679\pi\)
0.809610 + 0.586968i \(0.199679\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 144.4.a.b.1.1 1
3.2 odd 2 48.4.a.b.1.1 1
4.3 odd 2 72.4.a.b.1.1 1
8.3 odd 2 576.4.a.u.1.1 1
8.5 even 2 576.4.a.v.1.1 1
12.11 even 2 24.4.a.a.1.1 ✓ 1
15.2 even 4 1200.4.f.p.49.2 2
15.8 even 4 1200.4.f.p.49.1 2
15.14 odd 2 1200.4.a.u.1.1 1
20.3 even 4 1800.4.f.q.649.2 2
20.7 even 4 1800.4.f.q.649.1 2
20.19 odd 2 1800.4.a.bg.1.1 1
21.20 even 2 2352.4.a.w.1.1 1
24.5 odd 2 192.4.a.g.1.1 1
24.11 even 2 192.4.a.a.1.1 1
36.7 odd 6 648.4.i.k.433.1 2
36.11 even 6 648.4.i.b.433.1 2
36.23 even 6 648.4.i.b.217.1 2
36.31 odd 6 648.4.i.k.217.1 2
48.5 odd 4 768.4.d.b.385.2 2
48.11 even 4 768.4.d.o.385.1 2
48.29 odd 4 768.4.d.b.385.1 2
48.35 even 4 768.4.d.o.385.2 2
60.23 odd 4 600.4.f.b.49.2 2
60.47 odd 4 600.4.f.b.49.1 2
60.59 even 2 600.4.a.h.1.1 1
84.83 odd 2 1176.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.4.a.a.1.1 ✓ 1 12.11 even 2
48.4.a.b.1.1 1 3.2 odd 2
72.4.a.b.1.1 1 4.3 odd 2
144.4.a.b.1.1 1 1.1 even 1 trivial
192.4.a.a.1.1 1 24.11 even 2
192.4.a.g.1.1 1 24.5 odd 2
576.4.a.u.1.1 1 8.3 odd 2
576.4.a.v.1.1 1 8.5 even 2
600.4.a.h.1.1 1 60.59 even 2
600.4.f.b.49.1 2 60.47 odd 4
600.4.f.b.49.2 2 60.23 odd 4
648.4.i.b.217.1 2 36.23 even 6
648.4.i.b.433.1 2 36.11 even 6
648.4.i.k.217.1 2 36.31 odd 6
648.4.i.k.433.1 2 36.7 odd 6
768.4.d.b.385.1 2 48.29 odd 4
768.4.d.b.385.2 2 48.5 odd 4
768.4.d.o.385.1 2 48.11 even 4
768.4.d.o.385.2 2 48.35 even 4
1176.4.a.a.1.1 1 84.83 odd 2
1200.4.a.u.1.1 1 15.14 odd 2
1200.4.f.p.49.1 2 15.8 even 4
1200.4.f.p.49.2 2 15.2 even 4
1800.4.a.bg.1.1 1 20.19 odd 2
1800.4.f.q.649.1 2 20.7 even 4
1800.4.f.q.649.2 2 20.3 even 4
2352.4.a.w.1.1 1 21.20 even 2