Properties

Label 624.4.a.f.1.1
Level $624$
Weight $4$
Character 624.1
Self dual yes
Analytic conductor $36.817$
Analytic rank $0$
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] = [624,4,Mod(1,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("624.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 624.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -16.0000 q^{5} -28.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -16.0000 q^{5} -28.0000 q^{7} +9.00000 q^{9} -34.0000 q^{11} -13.0000 q^{13} -48.0000 q^{15} +138.000 q^{17} -108.000 q^{19} -84.0000 q^{21} +52.0000 q^{23} +131.000 q^{25} +27.0000 q^{27} -190.000 q^{29} +176.000 q^{31} -102.000 q^{33} +448.000 q^{35} +342.000 q^{37} -39.0000 q^{39} +240.000 q^{41} +140.000 q^{43} -144.000 q^{45} -454.000 q^{47} +441.000 q^{49} +414.000 q^{51} +198.000 q^{53} +544.000 q^{55} -324.000 q^{57} +154.000 q^{59} +34.0000 q^{61} -252.000 q^{63} +208.000 q^{65} +656.000 q^{67} +156.000 q^{69} -550.000 q^{71} +614.000 q^{73} +393.000 q^{75} +952.000 q^{77} -8.00000 q^{79} +81.0000 q^{81} -762.000 q^{83} -2208.00 q^{85} -570.000 q^{87} -444.000 q^{89} +364.000 q^{91} +528.000 q^{93} +1728.00 q^{95} +1022.00 q^{97} -306.000 q^{99} +O(q^{100})\)

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −16.0000 −1.43108 −0.715542 0.698570i \(-0.753820\pi\)
−0.715542 + 0.698570i \(0.753820\pi\)
\(6\) 0 0
\(7\) −28.0000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −34.0000 −0.931944 −0.465972 0.884799i \(-0.654295\pi\)
−0.465972 + 0.884799i \(0.654295\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) −48.0000 −0.826236
\(16\) 0 0
\(17\) 138.000 1.96882 0.984409 0.175893i \(-0.0562813\pi\)
0.984409 + 0.175893i \(0.0562813\pi\)
\(18\) 0 0
\(19\) −108.000 −1.30405 −0.652024 0.758199i \(-0.726080\pi\)
−0.652024 + 0.758199i \(0.726080\pi\)
\(20\) 0 0
\(21\) −84.0000 −0.872872
\(22\) 0 0
\(23\) 52.0000 0.471424 0.235712 0.971823i \(-0.424258\pi\)
0.235712 + 0.971823i \(0.424258\pi\)
\(24\) 0 0
\(25\) 131.000 1.04800
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −190.000 −1.21662 −0.608312 0.793698i \(-0.708153\pi\)
−0.608312 + 0.793698i \(0.708153\pi\)
\(30\) 0 0
\(31\) 176.000 1.01969 0.509847 0.860265i \(-0.329702\pi\)
0.509847 + 0.860265i \(0.329702\pi\)
\(32\) 0 0
\(33\) −102.000 −0.538058
\(34\) 0 0
\(35\) 448.000 2.16359
\(36\) 0 0
\(37\) 342.000 1.51958 0.759790 0.650169i \(-0.225302\pi\)
0.759790 + 0.650169i \(0.225302\pi\)
\(38\) 0 0
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) 240.000 0.914188 0.457094 0.889418i \(-0.348890\pi\)
0.457094 + 0.889418i \(0.348890\pi\)
\(42\) 0 0
\(43\) 140.000 0.496507 0.248253 0.968695i \(-0.420143\pi\)
0.248253 + 0.968695i \(0.420143\pi\)
\(44\) 0 0
\(45\) −144.000 −0.477028
\(46\) 0 0
\(47\) −454.000 −1.40899 −0.704497 0.709707i \(-0.748827\pi\)
−0.704497 + 0.709707i \(0.748827\pi\)
\(48\) 0 0
\(49\) 441.000 1.28571
\(50\) 0 0
\(51\) 414.000 1.13670
\(52\) 0 0
\(53\) 198.000 0.513158 0.256579 0.966523i \(-0.417405\pi\)
0.256579 + 0.966523i \(0.417405\pi\)
\(54\) 0 0
\(55\) 544.000 1.33369
\(56\) 0 0
\(57\) −324.000 −0.752892
\(58\) 0 0
\(59\) 154.000 0.339815 0.169908 0.985460i \(-0.445653\pi\)
0.169908 + 0.985460i \(0.445653\pi\)
\(60\) 0 0
\(61\) 34.0000 0.0713648 0.0356824 0.999363i \(-0.488640\pi\)
0.0356824 + 0.999363i \(0.488640\pi\)
\(62\) 0 0
\(63\) −252.000 −0.503953
\(64\) 0 0
\(65\) 208.000 0.396911
\(66\) 0 0
\(67\) 656.000 1.19617 0.598083 0.801434i \(-0.295929\pi\)
0.598083 + 0.801434i \(0.295929\pi\)
\(68\) 0 0
\(69\) 156.000 0.272177
\(70\) 0 0
\(71\) −550.000 −0.919338 −0.459669 0.888090i \(-0.652032\pi\)
−0.459669 + 0.888090i \(0.652032\pi\)
\(72\) 0 0
\(73\) 614.000 0.984428 0.492214 0.870474i \(-0.336188\pi\)
0.492214 + 0.870474i \(0.336188\pi\)
\(74\) 0 0
\(75\) 393.000 0.605063
\(76\) 0 0
\(77\) 952.000 1.40897
\(78\) 0 0
\(79\) −8.00000 −0.0113933 −0.00569665 0.999984i \(-0.501813\pi\)
−0.00569665 + 0.999984i \(0.501813\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −762.000 −1.00772 −0.503858 0.863787i \(-0.668086\pi\)
−0.503858 + 0.863787i \(0.668086\pi\)
\(84\) 0 0
\(85\) −2208.00 −2.81754
\(86\) 0 0
\(87\) −570.000 −0.702419
\(88\) 0 0
\(89\) −444.000 −0.528808 −0.264404 0.964412i \(-0.585175\pi\)
−0.264404 + 0.964412i \(0.585175\pi\)
\(90\) 0 0
\(91\) 364.000 0.419314
\(92\) 0 0
\(93\) 528.000 0.588721
\(94\) 0 0
\(95\) 1728.00 1.86620
\(96\) 0 0
\(97\) 1022.00 1.06978 0.534889 0.844923i \(-0.320354\pi\)
0.534889 + 0.844923i \(0.320354\pi\)
\(98\) 0 0
\(99\) −306.000 −0.310648
\(100\) 0 0
\(101\) −1190.00 −1.17237 −0.586185 0.810177i \(-0.699371\pi\)
−0.586185 + 0.810177i \(0.699371\pi\)
\(102\) 0 0
\(103\) 224.000 0.214285 0.107143 0.994244i \(-0.465830\pi\)
0.107143 + 0.994244i \(0.465830\pi\)
\(104\) 0 0
\(105\) 1344.00 1.24915
\(106\) 0 0
\(107\) 640.000 0.578235 0.289117 0.957294i \(-0.406638\pi\)
0.289117 + 0.957294i \(0.406638\pi\)
\(108\) 0 0
\(109\) −1934.00 −1.69948 −0.849741 0.527200i \(-0.823242\pi\)
−0.849741 + 0.527200i \(0.823242\pi\)
\(110\) 0 0
\(111\) 1026.00 0.877330
\(112\) 0 0
\(113\) −418.000 −0.347983 −0.173992 0.984747i \(-0.555667\pi\)
−0.173992 + 0.984747i \(0.555667\pi\)
\(114\) 0 0
\(115\) −832.000 −0.674647
\(116\) 0 0
\(117\) −117.000 −0.0924500
\(118\) 0 0
\(119\) −3864.00 −2.97657
\(120\) 0 0
\(121\) −175.000 −0.131480
\(122\) 0 0
\(123\) 720.000 0.527807
\(124\) 0 0
\(125\) −96.0000 −0.0686920
\(126\) 0 0
\(127\) 1040.00 0.726654 0.363327 0.931662i \(-0.381641\pi\)
0.363327 + 0.931662i \(0.381641\pi\)
\(128\) 0 0
\(129\) 420.000 0.286658
\(130\) 0 0
\(131\) 568.000 0.378827 0.189414 0.981897i \(-0.439341\pi\)
0.189414 + 0.981897i \(0.439341\pi\)
\(132\) 0 0
\(133\) 3024.00 1.97153
\(134\) 0 0
\(135\) −432.000 −0.275412
\(136\) 0 0
\(137\) 528.000 0.329271 0.164635 0.986355i \(-0.447355\pi\)
0.164635 + 0.986355i \(0.447355\pi\)
\(138\) 0 0
\(139\) 1556.00 0.949483 0.474742 0.880125i \(-0.342541\pi\)
0.474742 + 0.880125i \(0.342541\pi\)
\(140\) 0 0
\(141\) −1362.00 −0.813483
\(142\) 0 0
\(143\) 442.000 0.258475
\(144\) 0 0
\(145\) 3040.00 1.74109
\(146\) 0 0
\(147\) 1323.00 0.742307
\(148\) 0 0
\(149\) 1524.00 0.837926 0.418963 0.908003i \(-0.362394\pi\)
0.418963 + 0.908003i \(0.362394\pi\)
\(150\) 0 0
\(151\) 3024.00 1.62973 0.814866 0.579649i \(-0.196810\pi\)
0.814866 + 0.579649i \(0.196810\pi\)
\(152\) 0 0
\(153\) 1242.00 0.656273
\(154\) 0 0
\(155\) −2816.00 −1.45927
\(156\) 0 0
\(157\) 2198.00 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 594.000 0.296272
\(160\) 0 0
\(161\) −1456.00 −0.712726
\(162\) 0 0
\(163\) 268.000 0.128781 0.0643907 0.997925i \(-0.479490\pi\)
0.0643907 + 0.997925i \(0.479490\pi\)
\(164\) 0 0
\(165\) 1632.00 0.770006
\(166\) 0 0
\(167\) −702.000 −0.325284 −0.162642 0.986685i \(-0.552002\pi\)
−0.162642 + 0.986685i \(0.552002\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) −972.000 −0.434682
\(172\) 0 0
\(173\) 2066.00 0.907948 0.453974 0.891015i \(-0.350006\pi\)
0.453974 + 0.891015i \(0.350006\pi\)
\(174\) 0 0
\(175\) −3668.00 −1.58443
\(176\) 0 0
\(177\) 462.000 0.196192
\(178\) 0 0
\(179\) 276.000 0.115247 0.0576235 0.998338i \(-0.481648\pi\)
0.0576235 + 0.998338i \(0.481648\pi\)
\(180\) 0 0
\(181\) −3474.00 −1.42663 −0.713316 0.700843i \(-0.752808\pi\)
−0.713316 + 0.700843i \(0.752808\pi\)
\(182\) 0 0
\(183\) 102.000 0.0412025
\(184\) 0 0
\(185\) −5472.00 −2.17465
\(186\) 0 0
\(187\) −4692.00 −1.83483
\(188\) 0 0
\(189\) −756.000 −0.290957
\(190\) 0 0
\(191\) 3920.00 1.48503 0.742516 0.669828i \(-0.233632\pi\)
0.742516 + 0.669828i \(0.233632\pi\)
\(192\) 0 0
\(193\) 2186.00 0.815294 0.407647 0.913140i \(-0.366349\pi\)
0.407647 + 0.913140i \(0.366349\pi\)
\(194\) 0 0
\(195\) 624.000 0.229157
\(196\) 0 0
\(197\) −1368.00 −0.494751 −0.247376 0.968920i \(-0.579568\pi\)
−0.247376 + 0.968920i \(0.579568\pi\)
\(198\) 0 0
\(199\) 1072.00 0.381870 0.190935 0.981603i \(-0.438848\pi\)
0.190935 + 0.981603i \(0.438848\pi\)
\(200\) 0 0
\(201\) 1968.00 0.690607
\(202\) 0 0
\(203\) 5320.00 1.83936
\(204\) 0 0
\(205\) −3840.00 −1.30828
\(206\) 0 0
\(207\) 468.000 0.157141
\(208\) 0 0
\(209\) 3672.00 1.21530
\(210\) 0 0
\(211\) −5444.00 −1.77621 −0.888105 0.459640i \(-0.847978\pi\)
−0.888105 + 0.459640i \(0.847978\pi\)
\(212\) 0 0
\(213\) −1650.00 −0.530780
\(214\) 0 0
\(215\) −2240.00 −0.710543
\(216\) 0 0
\(217\) −4928.00 −1.54163
\(218\) 0 0
\(219\) 1842.00 0.568360
\(220\) 0 0
\(221\) −1794.00 −0.546052
\(222\) 0 0
\(223\) −96.0000 −0.0288280 −0.0144140 0.999896i \(-0.504588\pi\)
−0.0144140 + 0.999896i \(0.504588\pi\)
\(224\) 0 0
\(225\) 1179.00 0.349333
\(226\) 0 0
\(227\) −198.000 −0.0578930 −0.0289465 0.999581i \(-0.509215\pi\)
−0.0289465 + 0.999581i \(0.509215\pi\)
\(228\) 0 0
\(229\) 5922.00 1.70889 0.854447 0.519538i \(-0.173896\pi\)
0.854447 + 0.519538i \(0.173896\pi\)
\(230\) 0 0
\(231\) 2856.00 0.813468
\(232\) 0 0
\(233\) −5114.00 −1.43789 −0.718947 0.695065i \(-0.755376\pi\)
−0.718947 + 0.695065i \(0.755376\pi\)
\(234\) 0 0
\(235\) 7264.00 2.01639
\(236\) 0 0
\(237\) −24.0000 −0.00657792
\(238\) 0 0
\(239\) 5226.00 1.41440 0.707200 0.707013i \(-0.249958\pi\)
0.707200 + 0.707013i \(0.249958\pi\)
\(240\) 0 0
\(241\) −762.000 −0.203671 −0.101836 0.994801i \(-0.532472\pi\)
−0.101836 + 0.994801i \(0.532472\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −7056.00 −1.83996
\(246\) 0 0
\(247\) 1404.00 0.361678
\(248\) 0 0
\(249\) −2286.00 −0.581805
\(250\) 0 0
\(251\) −3240.00 −0.814769 −0.407384 0.913257i \(-0.633559\pi\)
−0.407384 + 0.913257i \(0.633559\pi\)
\(252\) 0 0
\(253\) −1768.00 −0.439341
\(254\) 0 0
\(255\) −6624.00 −1.62671
\(256\) 0 0
\(257\) −1386.00 −0.336406 −0.168203 0.985752i \(-0.553796\pi\)
−0.168203 + 0.985752i \(0.553796\pi\)
\(258\) 0 0
\(259\) −9576.00 −2.29739
\(260\) 0 0
\(261\) −1710.00 −0.405542
\(262\) 0 0
\(263\) −3300.00 −0.773714 −0.386857 0.922140i \(-0.626439\pi\)
−0.386857 + 0.922140i \(0.626439\pi\)
\(264\) 0 0
\(265\) −3168.00 −0.734372
\(266\) 0 0
\(267\) −1332.00 −0.305307
\(268\) 0 0
\(269\) 4290.00 0.972364 0.486182 0.873858i \(-0.338389\pi\)
0.486182 + 0.873858i \(0.338389\pi\)
\(270\) 0 0
\(271\) −2452.00 −0.549625 −0.274813 0.961498i \(-0.588616\pi\)
−0.274813 + 0.961498i \(0.588616\pi\)
\(272\) 0 0
\(273\) 1092.00 0.242091
\(274\) 0 0
\(275\) −4454.00 −0.976677
\(276\) 0 0
\(277\) −42.0000 −0.00911024 −0.00455512 0.999990i \(-0.501450\pi\)
−0.00455512 + 0.999990i \(0.501450\pi\)
\(278\) 0 0
\(279\) 1584.00 0.339898
\(280\) 0 0
\(281\) −2288.00 −0.485732 −0.242866 0.970060i \(-0.578088\pi\)
−0.242866 + 0.970060i \(0.578088\pi\)
\(282\) 0 0
\(283\) −1156.00 −0.242816 −0.121408 0.992603i \(-0.538741\pi\)
−0.121408 + 0.992603i \(0.538741\pi\)
\(284\) 0 0
\(285\) 5184.00 1.07745
\(286\) 0 0
\(287\) −6720.00 −1.38212
\(288\) 0 0
\(289\) 14131.0 2.87625
\(290\) 0 0
\(291\) 3066.00 0.617636
\(292\) 0 0
\(293\) −8684.00 −1.73148 −0.865742 0.500491i \(-0.833153\pi\)
−0.865742 + 0.500491i \(0.833153\pi\)
\(294\) 0 0
\(295\) −2464.00 −0.486304
\(296\) 0 0
\(297\) −918.000 −0.179353
\(298\) 0 0
\(299\) −676.000 −0.130749
\(300\) 0 0
\(301\) −3920.00 −0.750648
\(302\) 0 0
\(303\) −3570.00 −0.676868
\(304\) 0 0
\(305\) −544.000 −0.102129
\(306\) 0 0
\(307\) 7552.00 1.40396 0.701979 0.712197i \(-0.252300\pi\)
0.701979 + 0.712197i \(0.252300\pi\)
\(308\) 0 0
\(309\) 672.000 0.123718
\(310\) 0 0
\(311\) −2652.00 −0.483541 −0.241770 0.970334i \(-0.577728\pi\)
−0.241770 + 0.970334i \(0.577728\pi\)
\(312\) 0 0
\(313\) −4426.00 −0.799273 −0.399636 0.916674i \(-0.630864\pi\)
−0.399636 + 0.916674i \(0.630864\pi\)
\(314\) 0 0
\(315\) 4032.00 0.721198
\(316\) 0 0
\(317\) −4944.00 −0.875971 −0.437985 0.898982i \(-0.644308\pi\)
−0.437985 + 0.898982i \(0.644308\pi\)
\(318\) 0 0
\(319\) 6460.00 1.13383
\(320\) 0 0
\(321\) 1920.00 0.333844
\(322\) 0 0
\(323\) −14904.0 −2.56743
\(324\) 0 0
\(325\) −1703.00 −0.290663
\(326\) 0 0
\(327\) −5802.00 −0.981197
\(328\) 0 0
\(329\) 12712.0 2.13020
\(330\) 0 0
\(331\) 6088.00 1.01096 0.505478 0.862839i \(-0.331316\pi\)
0.505478 + 0.862839i \(0.331316\pi\)
\(332\) 0 0
\(333\) 3078.00 0.506527
\(334\) 0 0
\(335\) −10496.0 −1.71181
\(336\) 0 0
\(337\) 6638.00 1.07298 0.536491 0.843906i \(-0.319750\pi\)
0.536491 + 0.843906i \(0.319750\pi\)
\(338\) 0 0
\(339\) −1254.00 −0.200908
\(340\) 0 0
\(341\) −5984.00 −0.950298
\(342\) 0 0
\(343\) −2744.00 −0.431959
\(344\) 0 0
\(345\) −2496.00 −0.389508
\(346\) 0 0
\(347\) 2292.00 0.354585 0.177293 0.984158i \(-0.443266\pi\)
0.177293 + 0.984158i \(0.443266\pi\)
\(348\) 0 0
\(349\) −9866.00 −1.51322 −0.756612 0.653865i \(-0.773147\pi\)
−0.756612 + 0.653865i \(0.773147\pi\)
\(350\) 0 0
\(351\) −351.000 −0.0533761
\(352\) 0 0
\(353\) 2368.00 0.357042 0.178521 0.983936i \(-0.442869\pi\)
0.178521 + 0.983936i \(0.442869\pi\)
\(354\) 0 0
\(355\) 8800.00 1.31565
\(356\) 0 0
\(357\) −11592.0 −1.71853
\(358\) 0 0
\(359\) −5070.00 −0.745360 −0.372680 0.927960i \(-0.621561\pi\)
−0.372680 + 0.927960i \(0.621561\pi\)
\(360\) 0 0
\(361\) 4805.00 0.700539
\(362\) 0 0
\(363\) −525.000 −0.0759101
\(364\) 0 0
\(365\) −9824.00 −1.40880
\(366\) 0 0
\(367\) 8584.00 1.22093 0.610465 0.792043i \(-0.290983\pi\)
0.610465 + 0.792043i \(0.290983\pi\)
\(368\) 0 0
\(369\) 2160.00 0.304729
\(370\) 0 0
\(371\) −5544.00 −0.775822
\(372\) 0 0
\(373\) 4994.00 0.693243 0.346621 0.938005i \(-0.387329\pi\)
0.346621 + 0.938005i \(0.387329\pi\)
\(374\) 0 0
\(375\) −288.000 −0.0396593
\(376\) 0 0
\(377\) 2470.00 0.337431
\(378\) 0 0
\(379\) −1300.00 −0.176191 −0.0880957 0.996112i \(-0.528078\pi\)
−0.0880957 + 0.996112i \(0.528078\pi\)
\(380\) 0 0
\(381\) 3120.00 0.419534
\(382\) 0 0
\(383\) 4590.00 0.612371 0.306185 0.951972i \(-0.400947\pi\)
0.306185 + 0.951972i \(0.400947\pi\)
\(384\) 0 0
\(385\) −15232.0 −2.01635
\(386\) 0 0
\(387\) 1260.00 0.165502
\(388\) 0 0
\(389\) 3510.00 0.457491 0.228746 0.973486i \(-0.426538\pi\)
0.228746 + 0.973486i \(0.426538\pi\)
\(390\) 0 0
\(391\) 7176.00 0.928148
\(392\) 0 0
\(393\) 1704.00 0.218716
\(394\) 0 0
\(395\) 128.000 0.0163048
\(396\) 0 0
\(397\) 6230.00 0.787594 0.393797 0.919197i \(-0.371161\pi\)
0.393797 + 0.919197i \(0.371161\pi\)
\(398\) 0 0
\(399\) 9072.00 1.13827
\(400\) 0 0
\(401\) −7500.00 −0.933995 −0.466998 0.884259i \(-0.654664\pi\)
−0.466998 + 0.884259i \(0.654664\pi\)
\(402\) 0 0
\(403\) −2288.00 −0.282812
\(404\) 0 0
\(405\) −1296.00 −0.159009
\(406\) 0 0
\(407\) −11628.0 −1.41616
\(408\) 0 0
\(409\) 8254.00 0.997883 0.498941 0.866636i \(-0.333722\pi\)
0.498941 + 0.866636i \(0.333722\pi\)
\(410\) 0 0
\(411\) 1584.00 0.190105
\(412\) 0 0
\(413\) −4312.00 −0.513752
\(414\) 0 0
\(415\) 12192.0 1.44212
\(416\) 0 0
\(417\) 4668.00 0.548185
\(418\) 0 0
\(419\) 14808.0 1.72653 0.863267 0.504747i \(-0.168414\pi\)
0.863267 + 0.504747i \(0.168414\pi\)
\(420\) 0 0
\(421\) 10354.0 1.19863 0.599315 0.800513i \(-0.295440\pi\)
0.599315 + 0.800513i \(0.295440\pi\)
\(422\) 0 0
\(423\) −4086.00 −0.469665
\(424\) 0 0
\(425\) 18078.0 2.06332
\(426\) 0 0
\(427\) −952.000 −0.107893
\(428\) 0 0
\(429\) 1326.00 0.149230
\(430\) 0 0
\(431\) 15486.0 1.73071 0.865353 0.501163i \(-0.167094\pi\)
0.865353 + 0.501163i \(0.167094\pi\)
\(432\) 0 0
\(433\) −2018.00 −0.223970 −0.111985 0.993710i \(-0.535721\pi\)
−0.111985 + 0.993710i \(0.535721\pi\)
\(434\) 0 0
\(435\) 9120.00 1.00522
\(436\) 0 0
\(437\) −5616.00 −0.614759
\(438\) 0 0
\(439\) −8792.00 −0.955853 −0.477926 0.878400i \(-0.658611\pi\)
−0.477926 + 0.878400i \(0.658611\pi\)
\(440\) 0 0
\(441\) 3969.00 0.428571
\(442\) 0 0
\(443\) 2760.00 0.296008 0.148004 0.988987i \(-0.452715\pi\)
0.148004 + 0.988987i \(0.452715\pi\)
\(444\) 0 0
\(445\) 7104.00 0.756768
\(446\) 0 0
\(447\) 4572.00 0.483777
\(448\) 0 0
\(449\) 9532.00 1.00188 0.500939 0.865483i \(-0.332988\pi\)
0.500939 + 0.865483i \(0.332988\pi\)
\(450\) 0 0
\(451\) −8160.00 −0.851972
\(452\) 0 0
\(453\) 9072.00 0.940927
\(454\) 0 0
\(455\) −5824.00 −0.600073
\(456\) 0 0
\(457\) 12862.0 1.31654 0.658270 0.752782i \(-0.271288\pi\)
0.658270 + 0.752782i \(0.271288\pi\)
\(458\) 0 0
\(459\) 3726.00 0.378899
\(460\) 0 0
\(461\) −6744.00 −0.681344 −0.340672 0.940182i \(-0.610654\pi\)
−0.340672 + 0.940182i \(0.610654\pi\)
\(462\) 0 0
\(463\) 9572.00 0.960796 0.480398 0.877051i \(-0.340492\pi\)
0.480398 + 0.877051i \(0.340492\pi\)
\(464\) 0 0
\(465\) −8448.00 −0.842509
\(466\) 0 0
\(467\) −9104.00 −0.902105 −0.451052 0.892498i \(-0.648951\pi\)
−0.451052 + 0.892498i \(0.648951\pi\)
\(468\) 0 0
\(469\) −18368.0 −1.80843
\(470\) 0 0
\(471\) 6594.00 0.645086
\(472\) 0 0
\(473\) −4760.00 −0.462717
\(474\) 0 0
\(475\) −14148.0 −1.36664
\(476\) 0 0
\(477\) 1782.00 0.171053
\(478\) 0 0
\(479\) 18870.0 1.79998 0.899992 0.435906i \(-0.143572\pi\)
0.899992 + 0.435906i \(0.143572\pi\)
\(480\) 0 0
\(481\) −4446.00 −0.421456
\(482\) 0 0
\(483\) −4368.00 −0.411493
\(484\) 0 0
\(485\) −16352.0 −1.53094
\(486\) 0 0
\(487\) 1744.00 0.162276 0.0811378 0.996703i \(-0.474145\pi\)
0.0811378 + 0.996703i \(0.474145\pi\)
\(488\) 0 0
\(489\) 804.000 0.0743520
\(490\) 0 0
\(491\) −13360.0 −1.22796 −0.613980 0.789322i \(-0.710432\pi\)
−0.613980 + 0.789322i \(0.710432\pi\)
\(492\) 0 0
\(493\) −26220.0 −2.39531
\(494\) 0 0
\(495\) 4896.00 0.444563
\(496\) 0 0
\(497\) 15400.0 1.38991
\(498\) 0 0
\(499\) −17368.0 −1.55811 −0.779057 0.626954i \(-0.784302\pi\)
−0.779057 + 0.626954i \(0.784302\pi\)
\(500\) 0 0
\(501\) −2106.00 −0.187803
\(502\) 0 0
\(503\) 5828.00 0.516616 0.258308 0.966063i \(-0.416835\pi\)
0.258308 + 0.966063i \(0.416835\pi\)
\(504\) 0 0
\(505\) 19040.0 1.67776
\(506\) 0 0
\(507\) 507.000 0.0444116
\(508\) 0 0
\(509\) 10744.0 0.935598 0.467799 0.883835i \(-0.345047\pi\)
0.467799 + 0.883835i \(0.345047\pi\)
\(510\) 0 0
\(511\) −17192.0 −1.48832
\(512\) 0 0
\(513\) −2916.00 −0.250964
\(514\) 0 0
\(515\) −3584.00 −0.306660
\(516\) 0 0
\(517\) 15436.0 1.31310
\(518\) 0 0
\(519\) 6198.00 0.524204
\(520\) 0 0
\(521\) −12234.0 −1.02875 −0.514377 0.857564i \(-0.671977\pi\)
−0.514377 + 0.857564i \(0.671977\pi\)
\(522\) 0 0
\(523\) −1812.00 −0.151498 −0.0757488 0.997127i \(-0.524135\pi\)
−0.0757488 + 0.997127i \(0.524135\pi\)
\(524\) 0 0
\(525\) −11004.0 −0.914769
\(526\) 0 0
\(527\) 24288.0 2.00759
\(528\) 0 0
\(529\) −9463.00 −0.777760
\(530\) 0 0
\(531\) 1386.00 0.113272
\(532\) 0 0
\(533\) −3120.00 −0.253550
\(534\) 0 0
\(535\) −10240.0 −0.827502
\(536\) 0 0
\(537\) 828.000 0.0665379
\(538\) 0 0
\(539\) −14994.0 −1.19821
\(540\) 0 0
\(541\) 6098.00 0.484609 0.242305 0.970200i \(-0.422097\pi\)
0.242305 + 0.970200i \(0.422097\pi\)
\(542\) 0 0
\(543\) −10422.0 −0.823666
\(544\) 0 0
\(545\) 30944.0 2.43210
\(546\) 0 0
\(547\) 18332.0 1.43294 0.716471 0.697616i \(-0.245756\pi\)
0.716471 + 0.697616i \(0.245756\pi\)
\(548\) 0 0
\(549\) 306.000 0.0237883
\(550\) 0 0
\(551\) 20520.0 1.58654
\(552\) 0 0
\(553\) 224.000 0.0172250
\(554\) 0 0
\(555\) −16416.0 −1.25553
\(556\) 0 0
\(557\) −20004.0 −1.52172 −0.760859 0.648917i \(-0.775222\pi\)
−0.760859 + 0.648917i \(0.775222\pi\)
\(558\) 0 0
\(559\) −1820.00 −0.137706
\(560\) 0 0
\(561\) −14076.0 −1.05934
\(562\) 0 0
\(563\) −10988.0 −0.822538 −0.411269 0.911514i \(-0.634914\pi\)
−0.411269 + 0.911514i \(0.634914\pi\)
\(564\) 0 0
\(565\) 6688.00 0.497993
\(566\) 0 0
\(567\) −2268.00 −0.167984
\(568\) 0 0
\(569\) 11062.0 0.815014 0.407507 0.913202i \(-0.366398\pi\)
0.407507 + 0.913202i \(0.366398\pi\)
\(570\) 0 0
\(571\) 708.000 0.0518895 0.0259447 0.999663i \(-0.491741\pi\)
0.0259447 + 0.999663i \(0.491741\pi\)
\(572\) 0 0
\(573\) 11760.0 0.857384
\(574\) 0 0
\(575\) 6812.00 0.494052
\(576\) 0 0
\(577\) −2094.00 −0.151082 −0.0755410 0.997143i \(-0.524068\pi\)
−0.0755410 + 0.997143i \(0.524068\pi\)
\(578\) 0 0
\(579\) 6558.00 0.470710
\(580\) 0 0
\(581\) 21336.0 1.52352
\(582\) 0 0
\(583\) −6732.00 −0.478235
\(584\) 0 0
\(585\) 1872.00 0.132304
\(586\) 0 0
\(587\) 17854.0 1.25539 0.627695 0.778460i \(-0.283999\pi\)
0.627695 + 0.778460i \(0.283999\pi\)
\(588\) 0 0
\(589\) −19008.0 −1.32973
\(590\) 0 0
\(591\) −4104.00 −0.285645
\(592\) 0 0
\(593\) 23948.0 1.65839 0.829196 0.558958i \(-0.188799\pi\)
0.829196 + 0.558958i \(0.188799\pi\)
\(594\) 0 0
\(595\) 61824.0 4.25973
\(596\) 0 0
\(597\) 3216.00 0.220473
\(598\) 0 0
\(599\) 18068.0 1.23245 0.616226 0.787570i \(-0.288661\pi\)
0.616226 + 0.787570i \(0.288661\pi\)
\(600\) 0 0
\(601\) 19942.0 1.35350 0.676748 0.736215i \(-0.263389\pi\)
0.676748 + 0.736215i \(0.263389\pi\)
\(602\) 0 0
\(603\) 5904.00 0.398722
\(604\) 0 0
\(605\) 2800.00 0.188159
\(606\) 0 0
\(607\) −26376.0 −1.76370 −0.881852 0.471526i \(-0.843704\pi\)
−0.881852 + 0.471526i \(0.843704\pi\)
\(608\) 0 0
\(609\) 15960.0 1.06196
\(610\) 0 0
\(611\) 5902.00 0.390785
\(612\) 0 0
\(613\) −19426.0 −1.27995 −0.639975 0.768396i \(-0.721055\pi\)
−0.639975 + 0.768396i \(0.721055\pi\)
\(614\) 0 0
\(615\) −11520.0 −0.755335
\(616\) 0 0
\(617\) −8024.00 −0.523556 −0.261778 0.965128i \(-0.584309\pi\)
−0.261778 + 0.965128i \(0.584309\pi\)
\(618\) 0 0
\(619\) 20648.0 1.34073 0.670366 0.742031i \(-0.266137\pi\)
0.670366 + 0.742031i \(0.266137\pi\)
\(620\) 0 0
\(621\) 1404.00 0.0907256
\(622\) 0 0
\(623\) 12432.0 0.799482
\(624\) 0 0
\(625\) −14839.0 −0.949696
\(626\) 0 0
\(627\) 11016.0 0.701653
\(628\) 0 0
\(629\) 47196.0 2.99178
\(630\) 0 0
\(631\) −12280.0 −0.774737 −0.387369 0.921925i \(-0.626616\pi\)
−0.387369 + 0.921925i \(0.626616\pi\)
\(632\) 0 0
\(633\) −16332.0 −1.02550
\(634\) 0 0
\(635\) −16640.0 −1.03990
\(636\) 0 0
\(637\) −5733.00 −0.356593
\(638\) 0 0
\(639\) −4950.00 −0.306446
\(640\) 0 0
\(641\) −15878.0 −0.978383 −0.489191 0.872176i \(-0.662708\pi\)
−0.489191 + 0.872176i \(0.662708\pi\)
\(642\) 0 0
\(643\) 21520.0 1.31985 0.659927 0.751330i \(-0.270587\pi\)
0.659927 + 0.751330i \(0.270587\pi\)
\(644\) 0 0
\(645\) −6720.00 −0.410232
\(646\) 0 0
\(647\) −7312.00 −0.444304 −0.222152 0.975012i \(-0.571308\pi\)
−0.222152 + 0.975012i \(0.571308\pi\)
\(648\) 0 0
\(649\) −5236.00 −0.316689
\(650\) 0 0
\(651\) −14784.0 −0.890062
\(652\) 0 0
\(653\) 3090.00 0.185178 0.0925889 0.995704i \(-0.470486\pi\)
0.0925889 + 0.995704i \(0.470486\pi\)
\(654\) 0 0
\(655\) −9088.00 −0.542134
\(656\) 0 0
\(657\) 5526.00 0.328143
\(658\) 0 0
\(659\) 13428.0 0.793749 0.396875 0.917873i \(-0.370095\pi\)
0.396875 + 0.917873i \(0.370095\pi\)
\(660\) 0 0
\(661\) 22598.0 1.32974 0.664872 0.746958i \(-0.268486\pi\)
0.664872 + 0.746958i \(0.268486\pi\)
\(662\) 0 0
\(663\) −5382.00 −0.315263
\(664\) 0 0
\(665\) −48384.0 −2.82143
\(666\) 0 0
\(667\) −9880.00 −0.573546
\(668\) 0 0
\(669\) −288.000 −0.0166438
\(670\) 0 0
\(671\) −1156.00 −0.0665080
\(672\) 0 0
\(673\) 6178.00 0.353855 0.176927 0.984224i \(-0.443384\pi\)
0.176927 + 0.984224i \(0.443384\pi\)
\(674\) 0 0
\(675\) 3537.00 0.201688
\(676\) 0 0
\(677\) 22398.0 1.27153 0.635764 0.771883i \(-0.280685\pi\)
0.635764 + 0.771883i \(0.280685\pi\)
\(678\) 0 0
\(679\) −28616.0 −1.61735
\(680\) 0 0
\(681\) −594.000 −0.0334246
\(682\) 0 0
\(683\) −11410.0 −0.639226 −0.319613 0.947548i \(-0.603553\pi\)
−0.319613 + 0.947548i \(0.603553\pi\)
\(684\) 0 0
\(685\) −8448.00 −0.471214
\(686\) 0 0
\(687\) 17766.0 0.986631
\(688\) 0 0
\(689\) −2574.00 −0.142325
\(690\) 0 0
\(691\) −32488.0 −1.78857 −0.894285 0.447498i \(-0.852315\pi\)
−0.894285 + 0.447498i \(0.852315\pi\)
\(692\) 0 0
\(693\) 8568.00 0.469656
\(694\) 0 0
\(695\) −24896.0 −1.35879
\(696\) 0 0
\(697\) 33120.0 1.79987
\(698\) 0 0
\(699\) −15342.0 −0.830168
\(700\) 0 0
\(701\) 5094.00 0.274462 0.137231 0.990539i \(-0.456180\pi\)
0.137231 + 0.990539i \(0.456180\pi\)
\(702\) 0 0
\(703\) −36936.0 −1.98160
\(704\) 0 0
\(705\) 21792.0 1.16416
\(706\) 0 0
\(707\) 33320.0 1.77246
\(708\) 0 0
\(709\) 25418.0 1.34639 0.673197 0.739463i \(-0.264921\pi\)
0.673197 + 0.739463i \(0.264921\pi\)
\(710\) 0 0
\(711\) −72.0000 −0.00379777
\(712\) 0 0
\(713\) 9152.00 0.480708
\(714\) 0 0
\(715\) −7072.00 −0.369899
\(716\) 0 0
\(717\) 15678.0 0.816605
\(718\) 0 0
\(719\) 20428.0 1.05958 0.529788 0.848130i \(-0.322271\pi\)
0.529788 + 0.848130i \(0.322271\pi\)
\(720\) 0 0
\(721\) −6272.00 −0.323969
\(722\) 0 0
\(723\) −2286.00 −0.117590
\(724\) 0 0
\(725\) −24890.0 −1.27502
\(726\) 0 0
\(727\) 38336.0 1.95571 0.977857 0.209276i \(-0.0671107\pi\)
0.977857 + 0.209276i \(0.0671107\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 19320.0 0.977532
\(732\) 0 0
\(733\) −166.000 −0.00836473 −0.00418237 0.999991i \(-0.501331\pi\)
−0.00418237 + 0.999991i \(0.501331\pi\)
\(734\) 0 0
\(735\) −21168.0 −1.06230
\(736\) 0 0
\(737\) −22304.0 −1.11476
\(738\) 0 0
\(739\) 25248.0 1.25678 0.628392 0.777897i \(-0.283714\pi\)
0.628392 + 0.777897i \(0.283714\pi\)
\(740\) 0 0
\(741\) 4212.00 0.208815
\(742\) 0 0
\(743\) 4442.00 0.219329 0.109664 0.993969i \(-0.465022\pi\)
0.109664 + 0.993969i \(0.465022\pi\)
\(744\) 0 0
\(745\) −24384.0 −1.19914
\(746\) 0 0
\(747\) −6858.00 −0.335905
\(748\) 0 0
\(749\) −17920.0 −0.874209
\(750\) 0 0
\(751\) 19848.0 0.964399 0.482200 0.876061i \(-0.339838\pi\)
0.482200 + 0.876061i \(0.339838\pi\)
\(752\) 0 0
\(753\) −9720.00 −0.470407
\(754\) 0 0
\(755\) −48384.0 −2.33228
\(756\) 0 0
\(757\) −29166.0 −1.40034 −0.700169 0.713977i \(-0.746892\pi\)
−0.700169 + 0.713977i \(0.746892\pi\)
\(758\) 0 0
\(759\) −5304.00 −0.253653
\(760\) 0 0
\(761\) −6240.00 −0.297240 −0.148620 0.988894i \(-0.547483\pi\)
−0.148620 + 0.988894i \(0.547483\pi\)
\(762\) 0 0
\(763\) 54152.0 2.56938
\(764\) 0 0
\(765\) −19872.0 −0.939181
\(766\) 0 0
\(767\) −2002.00 −0.0942478
\(768\) 0 0
\(769\) −39750.0 −1.86401 −0.932004 0.362449i \(-0.881941\pi\)
−0.932004 + 0.362449i \(0.881941\pi\)
\(770\) 0 0
\(771\) −4158.00 −0.194224
\(772\) 0 0
\(773\) 9764.00 0.454317 0.227158 0.973858i \(-0.427057\pi\)
0.227158 + 0.973858i \(0.427057\pi\)
\(774\) 0 0
\(775\) 23056.0 1.06864
\(776\) 0 0
\(777\) −28728.0 −1.32640
\(778\) 0 0
\(779\) −25920.0 −1.19214
\(780\) 0 0
\(781\) 18700.0 0.856772
\(782\) 0 0
\(783\) −5130.00 −0.234140
\(784\) 0 0
\(785\) −35168.0 −1.59898
\(786\) 0 0
\(787\) 36016.0 1.63130 0.815649 0.578547i \(-0.196380\pi\)
0.815649 + 0.578547i \(0.196380\pi\)
\(788\) 0 0
\(789\) −9900.00 −0.446704
\(790\) 0 0
\(791\) 11704.0 0.526102
\(792\) 0 0
\(793\) −442.000 −0.0197930
\(794\) 0 0
\(795\) −9504.00 −0.423990
\(796\) 0 0
\(797\) −22290.0 −0.990655 −0.495328 0.868706i \(-0.664952\pi\)
−0.495328 + 0.868706i \(0.664952\pi\)
\(798\) 0 0
\(799\) −62652.0 −2.77405
\(800\) 0 0
\(801\) −3996.00 −0.176269
\(802\) 0 0
\(803\) −20876.0 −0.917432
\(804\) 0 0
\(805\) 23296.0 1.01997
\(806\) 0 0
\(807\) 12870.0 0.561395
\(808\) 0 0
\(809\) −25578.0 −1.11159 −0.555794 0.831320i \(-0.687586\pi\)
−0.555794 + 0.831320i \(0.687586\pi\)
\(810\) 0 0
\(811\) −29900.0 −1.29461 −0.647306 0.762230i \(-0.724105\pi\)
−0.647306 + 0.762230i \(0.724105\pi\)
\(812\) 0 0
\(813\) −7356.00 −0.317326
\(814\) 0 0
\(815\) −4288.00 −0.184297
\(816\) 0 0
\(817\) −15120.0 −0.647469
\(818\) 0 0
\(819\) 3276.00 0.139771
\(820\) 0 0
\(821\) 16412.0 0.697665 0.348832 0.937185i \(-0.386578\pi\)
0.348832 + 0.937185i \(0.386578\pi\)
\(822\) 0 0
\(823\) −18552.0 −0.785762 −0.392881 0.919589i \(-0.628522\pi\)
−0.392881 + 0.919589i \(0.628522\pi\)
\(824\) 0 0
\(825\) −13362.0 −0.563885
\(826\) 0 0
\(827\) 28662.0 1.20517 0.602585 0.798055i \(-0.294137\pi\)
0.602585 + 0.798055i \(0.294137\pi\)
\(828\) 0 0
\(829\) −3686.00 −0.154427 −0.0772136 0.997015i \(-0.524602\pi\)
−0.0772136 + 0.997015i \(0.524602\pi\)
\(830\) 0 0
\(831\) −126.000 −0.00525980
\(832\) 0 0
\(833\) 60858.0 2.53134
\(834\) 0 0
\(835\) 11232.0 0.465508
\(836\) 0 0
\(837\) 4752.00 0.196240
\(838\) 0 0
\(839\) −13370.0 −0.550159 −0.275080 0.961421i \(-0.588704\pi\)
−0.275080 + 0.961421i \(0.588704\pi\)
\(840\) 0 0
\(841\) 11711.0 0.480175
\(842\) 0 0
\(843\) −6864.00 −0.280437
\(844\) 0 0
\(845\) −2704.00 −0.110083
\(846\) 0 0
\(847\) 4900.00 0.198779
\(848\) 0 0
\(849\) −3468.00 −0.140190
\(850\) 0 0
\(851\) 17784.0 0.716366
\(852\) 0 0
\(853\) 11398.0 0.457515 0.228757 0.973483i \(-0.426534\pi\)
0.228757 + 0.973483i \(0.426534\pi\)
\(854\) 0 0
\(855\) 15552.0 0.622067
\(856\) 0 0
\(857\) 7990.00 0.318475 0.159238 0.987240i \(-0.449096\pi\)
0.159238 + 0.987240i \(0.449096\pi\)
\(858\) 0 0
\(859\) −7652.00 −0.303938 −0.151969 0.988385i \(-0.548561\pi\)
−0.151969 + 0.988385i \(0.548561\pi\)
\(860\) 0 0
\(861\) −20160.0 −0.797969
\(862\) 0 0
\(863\) 1022.00 0.0403120 0.0201560 0.999797i \(-0.493584\pi\)
0.0201560 + 0.999797i \(0.493584\pi\)
\(864\) 0 0
\(865\) −33056.0 −1.29935
\(866\) 0 0
\(867\) 42393.0 1.66060
\(868\) 0 0
\(869\) 272.000 0.0106179
\(870\) 0 0
\(871\) −8528.00 −0.331757
\(872\) 0 0
\(873\) 9198.00 0.356592
\(874\) 0 0
\(875\) 2688.00 0.103853
\(876\) 0 0
\(877\) −15546.0 −0.598576 −0.299288 0.954163i \(-0.596749\pi\)
−0.299288 + 0.954163i \(0.596749\pi\)
\(878\) 0 0
\(879\) −26052.0 −0.999673
\(880\) 0 0
\(881\) 11310.0 0.432513 0.216256 0.976337i \(-0.430615\pi\)
0.216256 + 0.976337i \(0.430615\pi\)
\(882\) 0 0
\(883\) −17260.0 −0.657809 −0.328904 0.944363i \(-0.606679\pi\)
−0.328904 + 0.944363i \(0.606679\pi\)
\(884\) 0 0
\(885\) −7392.00 −0.280768
\(886\) 0 0
\(887\) −832.000 −0.0314947 −0.0157474 0.999876i \(-0.505013\pi\)
−0.0157474 + 0.999876i \(0.505013\pi\)
\(888\) 0 0
\(889\) −29120.0 −1.09860
\(890\) 0 0
\(891\) −2754.00 −0.103549
\(892\) 0 0
\(893\) 49032.0 1.83739
\(894\) 0 0
\(895\) −4416.00 −0.164928
\(896\) 0 0
\(897\) −2028.00 −0.0754882
\(898\) 0 0
\(899\) −33440.0 −1.24059
\(900\) 0 0
\(901\) 27324.0 1.01032
\(902\) 0 0
\(903\) −11760.0 −0.433387
\(904\) 0 0
\(905\) 55584.0 2.04163
\(906\) 0 0
\(907\) −31740.0 −1.16197 −0.580986 0.813913i \(-0.697333\pi\)
−0.580986 + 0.813913i \(0.697333\pi\)
\(908\) 0 0
\(909\) −10710.0 −0.390790
\(910\) 0 0
\(911\) 23568.0 0.857127 0.428563 0.903512i \(-0.359020\pi\)
0.428563 + 0.903512i \(0.359020\pi\)
\(912\) 0 0
\(913\) 25908.0 0.939134
\(914\) 0 0
\(915\) −1632.00 −0.0589642
\(916\) 0 0
\(917\) −15904.0 −0.572733
\(918\) 0 0
\(919\) 18864.0 0.677112 0.338556 0.940946i \(-0.390062\pi\)
0.338556 + 0.940946i \(0.390062\pi\)
\(920\) 0 0
\(921\) 22656.0 0.810576
\(922\) 0 0
\(923\) 7150.00 0.254978
\(924\) 0 0
\(925\) 44802.0 1.59252
\(926\) 0 0
\(927\) 2016.00 0.0714284
\(928\) 0 0
\(929\) −19536.0 −0.689941 −0.344971 0.938613i \(-0.612111\pi\)
−0.344971 + 0.938613i \(0.612111\pi\)
\(930\) 0 0
\(931\) −47628.0 −1.67663
\(932\) 0 0
\(933\) −7956.00 −0.279172
\(934\) 0 0
\(935\) 75072.0 2.62579
\(936\) 0 0
\(937\) 18174.0 0.633638 0.316819 0.948486i \(-0.397385\pi\)
0.316819 + 0.948486i \(0.397385\pi\)
\(938\) 0 0
\(939\) −13278.0 −0.461460
\(940\) 0 0
\(941\) 51172.0 1.77275 0.886376 0.462966i \(-0.153215\pi\)
0.886376 + 0.462966i \(0.153215\pi\)
\(942\) 0 0
\(943\) 12480.0 0.430970
\(944\) 0 0
\(945\) 12096.0 0.416384
\(946\) 0 0
\(947\) −3726.00 −0.127855 −0.0639275 0.997955i \(-0.520363\pi\)
−0.0639275 + 0.997955i \(0.520363\pi\)
\(948\) 0 0
\(949\) −7982.00 −0.273031
\(950\) 0 0
\(951\) −14832.0 −0.505742
\(952\) 0 0
\(953\) −40498.0 −1.37656 −0.688279 0.725447i \(-0.741633\pi\)
−0.688279 + 0.725447i \(0.741633\pi\)
\(954\) 0 0
\(955\) −62720.0 −2.12521
\(956\) 0 0
\(957\) 19380.0 0.654615
\(958\) 0 0
\(959\) −14784.0 −0.497810
\(960\) 0 0
\(961\) 1185.00 0.0397771
\(962\) 0 0
\(963\) 5760.00 0.192745
\(964\) 0 0
\(965\) −34976.0 −1.16675
\(966\) 0 0
\(967\) −28568.0 −0.950036 −0.475018 0.879976i \(-0.657558\pi\)
−0.475018 + 0.879976i \(0.657558\pi\)
\(968\) 0 0
\(969\) −44712.0 −1.48231
\(970\) 0 0
\(971\) 8676.00 0.286742 0.143371 0.989669i \(-0.454206\pi\)
0.143371 + 0.989669i \(0.454206\pi\)
\(972\) 0 0
\(973\) −43568.0 −1.43548
\(974\) 0 0
\(975\) −5109.00 −0.167814
\(976\) 0 0
\(977\) −2796.00 −0.0915578 −0.0457789 0.998952i \(-0.514577\pi\)
−0.0457789 + 0.998952i \(0.514577\pi\)
\(978\) 0 0
\(979\) 15096.0 0.492819
\(980\) 0 0
\(981\) −17406.0 −0.566494
\(982\) 0 0
\(983\) 406.000 0.0131733 0.00658667 0.999978i \(-0.497903\pi\)
0.00658667 + 0.999978i \(0.497903\pi\)
\(984\) 0 0
\(985\) 21888.0 0.708030
\(986\) 0 0
\(987\) 38136.0 1.22987
\(988\) 0 0
\(989\) 7280.00 0.234065
\(990\) 0 0
\(991\) 23232.0 0.744691 0.372346 0.928094i \(-0.378554\pi\)
0.372346 + 0.928094i \(0.378554\pi\)
\(992\) 0 0
\(993\) 18264.0 0.583676
\(994\) 0 0
\(995\) −17152.0 −0.546487
\(996\) 0 0
\(997\) 6110.00 0.194088 0.0970440 0.995280i \(-0.469061\pi\)
0.0970440 + 0.995280i \(0.469061\pi\)
\(998\) 0 0
\(999\) 9234.00 0.292443
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 624.4.a.f.1.1 1
3.2 odd 2 1872.4.a.o.1.1 1
4.3 odd 2 78.4.a.a.1.1 1
8.3 odd 2 2496.4.a.q.1.1 1
8.5 even 2 2496.4.a.g.1.1 1
12.11 even 2 234.4.a.k.1.1 1
20.19 odd 2 1950.4.a.o.1.1 1
52.31 even 4 1014.4.b.a.337.2 2
52.47 even 4 1014.4.b.a.337.1 2
52.51 odd 2 1014.4.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.a.1.1 1 4.3 odd 2
234.4.a.k.1.1 1 12.11 even 2
624.4.a.f.1.1 1 1.1 even 1 trivial
1014.4.a.i.1.1 1 52.51 odd 2
1014.4.b.a.337.1 2 52.47 even 4
1014.4.b.a.337.2 2 52.31 even 4
1872.4.a.o.1.1 1 3.2 odd 2
1950.4.a.o.1.1 1 20.19 odd 2
2496.4.a.g.1.1 1 8.5 even 2
2496.4.a.q.1.1 1 8.3 odd 2