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Properties

Label 624.4.a.d.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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +10.0000 q^{5} +8.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} +10.0000 q^{5} +8.00000 q^{7} +9.00000 q^{9} -40.0000 q^{11} +13.0000 q^{13} -30.0000 q^{15} +130.000 q^{17} +20.0000 q^{19} -24.0000 q^{21} -25.0000 q^{25} -27.0000 q^{27} -18.0000 q^{29} +184.000 q^{31} +120.000 q^{33} +80.0000 q^{35} -74.0000 q^{37} -39.0000 q^{39} -362.000 q^{41} -76.0000 q^{43} +90.0000 q^{45} +452.000 q^{47} -279.000 q^{49} -390.000 q^{51} +382.000 q^{53} -400.000 q^{55} -60.0000 q^{57} -464.000 q^{59} +358.000 q^{61} +72.0000 q^{63} +130.000 q^{65} +700.000 q^{67} +748.000 q^{71} +1058.00 q^{73} +75.0000 q^{75} -320.000 q^{77} +976.000 q^{79} +81.0000 q^{81} +1008.00 q^{83} +1300.00 q^{85} +54.0000 q^{87} -386.000 q^{89} +104.000 q^{91} -552.000 q^{93} +200.000 q^{95} -614.000 q^{97} -360.000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 10.0000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 8.00000 0.431959 0.215980 − 0.976398i \(-0.430705\pi\)
0.215980 + 0.976398i \(0.430705\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −40.0000 −1.09640 −0.548202 − 0.836346i \(-0.684688\pi\)
−0.548202 + 0.836346i \(0.684688\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) −30.0000 −0.516398
\(16\) 0 0
\(17\) 130.000 1.85468 0.927342 − 0.374215i \(-0.122088\pi\)
0.927342 + 0.374215i \(0.122088\pi\)
\(18\) 0 0
\(19\) 20.0000 0.241490 0.120745 − 0.992684i \(-0.461472\pi\)
0.120745 + 0.992684i \(0.461472\pi\)
\(20\) 0 0
\(21\) −24.0000 −0.249392
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −25.0000 −0.200000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −18.0000 −0.115259 −0.0576296 − 0.998338i \(-0.518354\pi\)
−0.0576296 + 0.998338i \(0.518354\pi\)
\(30\) 0 0
\(31\) 184.000 1.06604 0.533022 − 0.846101i \(-0.321056\pi\)
0.533022 + 0.846101i \(0.321056\pi\)
\(32\) 0 0
\(33\) 120.000 0.633010
\(34\) 0 0
\(35\) 80.0000 0.386356
\(36\) 0 0
\(37\) −74.0000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) −362.000 −1.37890 −0.689450 − 0.724333i \(-0.742148\pi\)
−0.689450 + 0.724333i \(0.742148\pi\)
\(42\) 0 0
\(43\) −76.0000 −0.269532 −0.134766 − 0.990877i \(-0.543028\pi\)
−0.134766 + 0.990877i \(0.543028\pi\)
\(44\) 0 0
\(45\) 90.0000 0.298142
\(46\) 0 0
\(47\) 452.000 1.40279 0.701393 − 0.712774i \(-0.252562\pi\)
0.701393 + 0.712774i \(0.252562\pi\)
\(48\) 0 0
\(49\) −279.000 −0.813411
\(50\) 0 0
\(51\) −390.000 −1.07080
\(52\) 0 0
\(53\) 382.000 0.990033 0.495016 − 0.868884i \(-0.335162\pi\)
0.495016 + 0.868884i \(0.335162\pi\)
\(54\) 0 0
\(55\) −400.000 −0.980654
\(56\) 0 0
\(57\) −60.0000 −0.139424
\(58\) 0 0
\(59\) −464.000 −1.02386 −0.511929 − 0.859028i \(-0.671069\pi\)
−0.511929 + 0.859028i \(0.671069\pi\)
\(60\) 0 0
\(61\) 358.000 0.751430 0.375715 − 0.926735i \(-0.377397\pi\)
0.375715 + 0.926735i \(0.377397\pi\)
\(62\) 0 0
\(63\) 72.0000 0.143986
\(64\) 0 0
\(65\) 130.000 0.248069
\(66\) 0 0
\(67\) 700.000 1.27640 0.638199 − 0.769872i \(-0.279680\pi\)
0.638199 + 0.769872i \(0.279680\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 748.000 1.25030 0.625150 − 0.780505i \(-0.285038\pi\)
0.625150 + 0.780505i \(0.285038\pi\)
\(72\) 0 0
\(73\) 1058.00 1.69629 0.848147 − 0.529760i \(-0.177718\pi\)
0.848147 + 0.529760i \(0.177718\pi\)
\(74\) 0 0
\(75\) 75.0000 0.115470
\(76\) 0 0
\(77\) −320.000 −0.473602
\(78\) 0 0
\(79\) 976.000 1.38998 0.694991 − 0.719018i \(-0.255408\pi\)
0.694991 + 0.719018i \(0.255408\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1008.00 1.33304 0.666520 − 0.745487i \(-0.267783\pi\)
0.666520 + 0.745487i \(0.267783\pi\)
\(84\) 0 0
\(85\) 1300.00 1.65888
\(86\) 0 0
\(87\) 54.0000 0.0665449
\(88\) 0 0
\(89\) −386.000 −0.459729 −0.229865 − 0.973223i \(-0.573828\pi\)
−0.229865 + 0.973223i \(0.573828\pi\)
\(90\) 0 0
\(91\) 104.000 0.119804
\(92\) 0 0
\(93\) −552.000 −0.615481
\(94\) 0 0
\(95\) 200.000 0.215995
\(96\) 0 0
\(97\) −614.000 −0.642704 −0.321352 − 0.946960i \(-0.604137\pi\)
−0.321352 + 0.946960i \(0.604137\pi\)
\(98\) 0 0
\(99\) −360.000 −0.365468
\(100\) 0 0
\(101\) 518.000 0.510326 0.255163 − 0.966898i \(-0.417871\pi\)
0.255163 + 0.966898i \(0.417871\pi\)
\(102\) 0 0
\(103\) −112.000 −0.107143 −0.0535713 − 0.998564i \(-0.517060\pi\)
−0.0535713 + 0.998564i \(0.517060\pi\)
\(104\) 0 0
\(105\) −240.000 −0.223063
\(106\) 0 0
\(107\) 372.000 0.336099 0.168050 − 0.985779i \(-0.446253\pi\)
0.168050 + 0.985779i \(0.446253\pi\)
\(108\) 0 0
\(109\) 934.000 0.820743 0.410371 − 0.911918i \(-0.365399\pi\)
0.410371 + 0.911918i \(0.365399\pi\)
\(110\) 0 0
\(111\) 222.000 0.189832
\(112\) 0 0
\(113\) 1914.00 1.59340 0.796699 − 0.604376i \(-0.206578\pi\)
0.796699 + 0.604376i \(0.206578\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 117.000 0.0924500
\(118\) 0 0
\(119\) 1040.00 0.801148
\(120\) 0 0
\(121\) 269.000 0.202104
\(122\) 0 0
\(123\) 1086.00 0.796108
\(124\) 0 0
\(125\) −1500.00 −1.07331
\(126\) 0 0
\(127\) −1296.00 −0.905523 −0.452761 − 0.891632i \(-0.649561\pi\)
−0.452761 + 0.891632i \(0.649561\pi\)
\(128\) 0 0
\(129\) 228.000 0.155615
\(130\) 0 0
\(131\) 892.000 0.594919 0.297460 − 0.954734i \(-0.403861\pi\)
0.297460 + 0.954734i \(0.403861\pi\)
\(132\) 0 0
\(133\) 160.000 0.104314
\(134\) 0 0
\(135\) −270.000 −0.172133
\(136\) 0 0
\(137\) 2326.00 1.45054 0.725269 − 0.688466i \(-0.241716\pi\)
0.725269 + 0.688466i \(0.241716\pi\)
\(138\) 0 0
\(139\) −1932.00 −1.17892 −0.589461 − 0.807797i \(-0.700660\pi\)
−0.589461 + 0.807797i \(0.700660\pi\)
\(140\) 0 0
\(141\) −1356.00 −0.809899
\(142\) 0 0
\(143\) −520.000 −0.304088
\(144\) 0 0
\(145\) −180.000 −0.103091
\(146\) 0 0
\(147\) 837.000 0.469623
\(148\) 0 0
\(149\) 882.000 0.484941 0.242471 − 0.970159i \(-0.422042\pi\)
0.242471 + 0.970159i \(0.422042\pi\)
\(150\) 0 0
\(151\) 1776.00 0.957145 0.478572 − 0.878048i \(-0.341154\pi\)
0.478572 + 0.878048i \(0.341154\pi\)
\(152\) 0 0
\(153\) 1170.00 0.618228
\(154\) 0 0
\(155\) 1840.00 0.953499
\(156\) 0 0
\(157\) −2410.00 −1.22509 −0.612544 − 0.790436i \(-0.709854\pi\)
−0.612544 + 0.790436i \(0.709854\pi\)
\(158\) 0 0
\(159\) −1146.00 −0.571596
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3212.00 −1.54346 −0.771728 − 0.635953i \(-0.780607\pi\)
−0.771728 + 0.635953i \(0.780607\pi\)
\(164\) 0 0
\(165\) 1200.00 0.566181
\(166\) 0 0
\(167\) −1668.00 −0.772896 −0.386448 − 0.922311i \(-0.626298\pi\)
−0.386448 + 0.922311i \(0.626298\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) 180.000 0.0804967
\(172\) 0 0
\(173\) 3598.00 1.58122 0.790609 − 0.612321i \(-0.209764\pi\)
0.790609 + 0.612321i \(0.209764\pi\)
\(174\) 0 0
\(175\) −200.000 −0.0863919
\(176\) 0 0
\(177\) 1392.00 0.591125
\(178\) 0 0
\(179\) −1068.00 −0.445956 −0.222978 − 0.974824i \(-0.571578\pi\)
−0.222978 + 0.974824i \(0.571578\pi\)
\(180\) 0 0
\(181\) −4786.00 −1.96542 −0.982709 − 0.185158i \(-0.940720\pi\)
−0.982709 + 0.185158i \(0.940720\pi\)
\(182\) 0 0
\(183\) −1074.00 −0.433838
\(184\) 0 0
\(185\) −740.000 −0.294086
\(186\) 0 0
\(187\) −5200.00 −2.03348
\(188\) 0 0
\(189\) −216.000 −0.0831306
\(190\) 0 0
\(191\) 1312.00 0.497031 0.248516 − 0.968628i \(-0.420057\pi\)
0.248516 + 0.968628i \(0.420057\pi\)
\(192\) 0 0
\(193\) −350.000 −0.130537 −0.0652683 − 0.997868i \(-0.520790\pi\)
−0.0652683 + 0.997868i \(0.520790\pi\)
\(194\) 0 0
\(195\) −390.000 −0.143223
\(196\) 0 0
\(197\) −342.000 −0.123688 −0.0618439 − 0.998086i \(-0.519698\pi\)
−0.0618439 + 0.998086i \(0.519698\pi\)
\(198\) 0 0
\(199\) 3368.00 1.19975 0.599877 − 0.800092i \(-0.295216\pi\)
0.599877 + 0.800092i \(0.295216\pi\)
\(200\) 0 0
\(201\) −2100.00 −0.736928
\(202\) 0 0
\(203\) −144.000 −0.0497873
\(204\) 0 0
\(205\) −3620.00 −1.23333
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −800.000 −0.264771
\(210\) 0 0
\(211\) 2004.00 0.653844 0.326922 − 0.945051i \(-0.393989\pi\)
0.326922 + 0.945051i \(0.393989\pi\)
\(212\) 0 0
\(213\) −2244.00 −0.721861
\(214\) 0 0
\(215\) −760.000 −0.241077
\(216\) 0 0
\(217\) 1472.00 0.460488
\(218\) 0 0
\(219\) −3174.00 −0.979356
\(220\) 0 0
\(221\) 1690.00 0.514397
\(222\) 0 0
\(223\) 5608.00 1.68403 0.842017 − 0.539451i \(-0.181368\pi\)
0.842017 + 0.539451i \(0.181368\pi\)
\(224\) 0 0
\(225\) −225.000 −0.0666667
\(226\) 0 0
\(227\) 1928.00 0.563726 0.281863 − 0.959455i \(-0.409048\pi\)
0.281863 + 0.959455i \(0.409048\pi\)
\(228\) 0 0
\(229\) −3938.00 −1.13638 −0.568189 − 0.822898i \(-0.692356\pi\)
−0.568189 + 0.822898i \(0.692356\pi\)
\(230\) 0 0
\(231\) 960.000 0.273434
\(232\) 0 0
\(233\) 2562.00 0.720353 0.360176 − 0.932884i \(-0.382717\pi\)
0.360176 + 0.932884i \(0.382717\pi\)
\(234\) 0 0
\(235\) 4520.00 1.25469
\(236\) 0 0
\(237\) −2928.00 −0.802506
\(238\) 0 0
\(239\) −7164.00 −1.93891 −0.969457 − 0.245260i \(-0.921127\pi\)
−0.969457 + 0.245260i \(0.921127\pi\)
\(240\) 0 0
\(241\) −6182.00 −1.65236 −0.826178 − 0.563410i \(-0.809489\pi\)
−0.826178 + 0.563410i \(0.809489\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −2790.00 −0.727537
\(246\) 0 0
\(247\) 260.000 0.0669773
\(248\) 0 0
\(249\) −3024.00 −0.769631
\(250\) 0 0
\(251\) 1396.00 0.351055 0.175527 − 0.984475i \(-0.443837\pi\)
0.175527 + 0.984475i \(0.443837\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −3900.00 −0.957755
\(256\) 0 0
\(257\) 6906.00 1.67620 0.838102 − 0.545514i \(-0.183665\pi\)
0.838102 + 0.545514i \(0.183665\pi\)
\(258\) 0 0
\(259\) −592.000 −0.142027
\(260\) 0 0
\(261\) −162.000 −0.0384197
\(262\) 0 0
\(263\) 6848.00 1.60557 0.802787 − 0.596266i \(-0.203350\pi\)
0.802787 + 0.596266i \(0.203350\pi\)
\(264\) 0 0
\(265\) 3820.00 0.885512
\(266\) 0 0
\(267\) 1158.00 0.265425
\(268\) 0 0
\(269\) −6034.00 −1.36766 −0.683828 − 0.729643i \(-0.739686\pi\)
−0.683828 + 0.729643i \(0.739686\pi\)
\(270\) 0 0
\(271\) −4832.00 −1.08311 −0.541556 − 0.840665i \(-0.682164\pi\)
−0.541556 + 0.840665i \(0.682164\pi\)
\(272\) 0 0
\(273\) −312.000 −0.0691689
\(274\) 0 0
\(275\) 1000.00 0.219281
\(276\) 0 0
\(277\) −4082.00 −0.885428 −0.442714 − 0.896663i \(-0.645984\pi\)
−0.442714 + 0.896663i \(0.645984\pi\)
\(278\) 0 0
\(279\) 1656.00 0.355348
\(280\) 0 0
\(281\) 3350.00 0.711189 0.355595 − 0.934640i \(-0.384278\pi\)
0.355595 + 0.934640i \(0.384278\pi\)
\(282\) 0 0
\(283\) −7796.00 −1.63754 −0.818770 − 0.574121i \(-0.805344\pi\)
−0.818770 + 0.574121i \(0.805344\pi\)
\(284\) 0 0
\(285\) −600.000 −0.124705
\(286\) 0 0
\(287\) −2896.00 −0.595629
\(288\) 0 0
\(289\) 11987.0 2.43985
\(290\) 0 0
\(291\) 1842.00 0.371065
\(292\) 0 0
\(293\) 3922.00 0.781999 0.390999 − 0.920391i \(-0.372129\pi\)
0.390999 + 0.920391i \(0.372129\pi\)
\(294\) 0 0
\(295\) −4640.00 −0.915767
\(296\) 0 0
\(297\) 1080.00 0.211003
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −608.000 −0.116427
\(302\) 0 0
\(303\) −1554.00 −0.294637
\(304\) 0 0
\(305\) 3580.00 0.672099
\(306\) 0 0
\(307\) −5956.00 −1.10725 −0.553627 − 0.832765i \(-0.686757\pi\)
−0.553627 + 0.832765i \(0.686757\pi\)
\(308\) 0 0
\(309\) 336.000 0.0618588
\(310\) 0 0
\(311\) −2352.00 −0.428841 −0.214421 − 0.976741i \(-0.568786\pi\)
−0.214421 + 0.976741i \(0.568786\pi\)
\(312\) 0 0
\(313\) 8442.00 1.52450 0.762252 − 0.647280i \(-0.224093\pi\)
0.762252 + 0.647280i \(0.224093\pi\)
\(314\) 0 0
\(315\) 720.000 0.128785
\(316\) 0 0
\(317\) −5550.00 −0.983341 −0.491670 − 0.870781i \(-0.663614\pi\)
−0.491670 + 0.870781i \(0.663614\pi\)
\(318\) 0 0
\(319\) 720.000 0.126371
\(320\) 0 0
\(321\) −1116.00 −0.194047
\(322\) 0 0
\(323\) 2600.00 0.447888
\(324\) 0 0
\(325\) −325.000 −0.0554700
\(326\) 0 0
\(327\) −2802.00 −0.473856
\(328\) 0 0
\(329\) 3616.00 0.605947
\(330\) 0 0
\(331\) −140.000 −0.0232480 −0.0116240 − 0.999932i \(-0.503700\pi\)
−0.0116240 + 0.999932i \(0.503700\pi\)
\(332\) 0 0
\(333\) −666.000 −0.109599
\(334\) 0 0
\(335\) 7000.00 1.14164
\(336\) 0 0
\(337\) −6174.00 −0.997980 −0.498990 − 0.866608i \(-0.666296\pi\)
−0.498990 + 0.866608i \(0.666296\pi\)
\(338\) 0 0
\(339\) −5742.00 −0.919949
\(340\) 0 0
\(341\) −7360.00 −1.16882
\(342\) 0 0
\(343\) −4976.00 −0.783320
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2988.00 0.462260 0.231130 − 0.972923i \(-0.425758\pi\)
0.231130 + 0.972923i \(0.425758\pi\)
\(348\) 0 0
\(349\) −162.000 −0.0248472 −0.0124236 − 0.999923i \(-0.503955\pi\)
−0.0124236 + 0.999923i \(0.503955\pi\)
\(350\) 0 0
\(351\) −351.000 −0.0533761
\(352\) 0 0
\(353\) −10754.0 −1.62147 −0.810733 − 0.585416i \(-0.800931\pi\)
−0.810733 + 0.585416i \(0.800931\pi\)
\(354\) 0 0
\(355\) 7480.00 1.11830
\(356\) 0 0
\(357\) −3120.00 −0.462543
\(358\) 0 0
\(359\) −3588.00 −0.527486 −0.263743 − 0.964593i \(-0.584957\pi\)
−0.263743 + 0.964593i \(0.584957\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) 0 0
\(363\) −807.000 −0.116685
\(364\) 0 0
\(365\) 10580.0 1.51721
\(366\) 0 0
\(367\) −11272.0 −1.60325 −0.801626 − 0.597826i \(-0.796032\pi\)
−0.801626 + 0.597826i \(0.796032\pi\)
\(368\) 0 0
\(369\) −3258.00 −0.459633
\(370\) 0 0
\(371\) 3056.00 0.427654
\(372\) 0 0
\(373\) −10914.0 −1.51503 −0.757514 − 0.652819i \(-0.773586\pi\)
−0.757514 + 0.652819i \(0.773586\pi\)
\(374\) 0 0
\(375\) 4500.00 0.619677
\(376\) 0 0
\(377\) −234.000 −0.0319671
\(378\) 0 0
\(379\) −8100.00 −1.09781 −0.548904 − 0.835886i \(-0.684955\pi\)
−0.548904 + 0.835886i \(0.684955\pi\)
\(380\) 0 0
\(381\) 3888.00 0.522804
\(382\) 0 0
\(383\) −6180.00 −0.824499 −0.412250 − 0.911071i \(-0.635257\pi\)
−0.412250 + 0.911071i \(0.635257\pi\)
\(384\) 0 0
\(385\) −3200.00 −0.423603
\(386\) 0 0
\(387\) −684.000 −0.0898441
\(388\) 0 0
\(389\) −7522.00 −0.980413 −0.490206 − 0.871606i \(-0.663079\pi\)
−0.490206 + 0.871606i \(0.663079\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −2676.00 −0.343477
\(394\) 0 0
\(395\) 9760.00 1.24324
\(396\) 0 0
\(397\) 6078.00 0.768378 0.384189 − 0.923254i \(-0.374481\pi\)
0.384189 + 0.923254i \(0.374481\pi\)
\(398\) 0 0
\(399\) −480.000 −0.0602257
\(400\) 0 0
\(401\) 1830.00 0.227895 0.113947 − 0.993487i \(-0.463650\pi\)
0.113947 + 0.993487i \(0.463650\pi\)
\(402\) 0 0
\(403\) 2392.00 0.295668
\(404\) 0 0
\(405\) 810.000 0.0993808
\(406\) 0 0
\(407\) 2960.00 0.360496
\(408\) 0 0
\(409\) 12434.0 1.50323 0.751616 − 0.659601i \(-0.229275\pi\)
0.751616 + 0.659601i \(0.229275\pi\)
\(410\) 0 0
\(411\) −6978.00 −0.837468
\(412\) 0 0
\(413\) −3712.00 −0.442265
\(414\) 0 0
\(415\) 10080.0 1.19231
\(416\) 0 0
\(417\) 5796.00 0.680651
\(418\) 0 0
\(419\) 14188.0 1.65425 0.827123 − 0.562021i \(-0.189976\pi\)
0.827123 + 0.562021i \(0.189976\pi\)
\(420\) 0 0
\(421\) 8638.00 0.999977 0.499989 − 0.866032i \(-0.333338\pi\)
0.499989 + 0.866032i \(0.333338\pi\)
\(422\) 0 0
\(423\) 4068.00 0.467596
\(424\) 0 0
\(425\) −3250.00 −0.370937
\(426\) 0 0
\(427\) 2864.00 0.324587
\(428\) 0 0
\(429\) 1560.00 0.175565
\(430\) 0 0
\(431\) −4292.00 −0.479671 −0.239836 − 0.970813i \(-0.577094\pi\)
−0.239836 + 0.970813i \(0.577094\pi\)
\(432\) 0 0
\(433\) −5982.00 −0.663918 −0.331959 − 0.943294i \(-0.607710\pi\)
−0.331959 + 0.943294i \(0.607710\pi\)
\(434\) 0 0
\(435\) 540.000 0.0595196
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −256.000 −0.0278319 −0.0139160 − 0.999903i \(-0.504430\pi\)
−0.0139160 + 0.999903i \(0.504430\pi\)
\(440\) 0 0
\(441\) −2511.00 −0.271137
\(442\) 0 0
\(443\) −12556.0 −1.34662 −0.673311 − 0.739359i \(-0.735128\pi\)
−0.673311 + 0.739359i \(0.735128\pi\)
\(444\) 0 0
\(445\) −3860.00 −0.411194
\(446\) 0 0
\(447\) −2646.00 −0.279981
\(448\) 0 0
\(449\) 5574.00 0.585865 0.292932 − 0.956133i \(-0.405369\pi\)
0.292932 + 0.956133i \(0.405369\pi\)
\(450\) 0 0
\(451\) 14480.0 1.51183
\(452\) 0 0
\(453\) −5328.00 −0.552608
\(454\) 0 0
\(455\) 1040.00 0.107156
\(456\) 0 0
\(457\) 1266.00 0.129586 0.0647932 − 0.997899i \(-0.479361\pi\)
0.0647932 + 0.997899i \(0.479361\pi\)
\(458\) 0 0
\(459\) −3510.00 −0.356934
\(460\) 0 0
\(461\) 7554.00 0.763178 0.381589 − 0.924332i \(-0.375377\pi\)
0.381589 + 0.924332i \(0.375377\pi\)
\(462\) 0 0
\(463\) 6752.00 0.677737 0.338868 − 0.940834i \(-0.389956\pi\)
0.338868 + 0.940834i \(0.389956\pi\)
\(464\) 0 0
\(465\) −5520.00 −0.550503
\(466\) 0 0
\(467\) −7924.00 −0.785180 −0.392590 − 0.919714i \(-0.628421\pi\)
−0.392590 + 0.919714i \(0.628421\pi\)
\(468\) 0 0
\(469\) 5600.00 0.551352
\(470\) 0 0
\(471\) 7230.00 0.707305
\(472\) 0 0
\(473\) 3040.00 0.295517
\(474\) 0 0
\(475\) −500.000 −0.0482980
\(476\) 0 0
\(477\) 3438.00 0.330011
\(478\) 0 0
\(479\) 11084.0 1.05729 0.528644 − 0.848844i \(-0.322701\pi\)
0.528644 + 0.848844i \(0.322701\pi\)
\(480\) 0 0
\(481\) −962.000 −0.0911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6140.00 −0.574852
\(486\) 0 0
\(487\) −4432.00 −0.412388 −0.206194 − 0.978511i \(-0.566108\pi\)
−0.206194 + 0.978511i \(0.566108\pi\)
\(488\) 0 0
\(489\) 9636.00 0.891114
\(490\) 0 0
\(491\) 1140.00 0.104781 0.0523905 − 0.998627i \(-0.483316\pi\)
0.0523905 + 0.998627i \(0.483316\pi\)
\(492\) 0 0
\(493\) −2340.00 −0.213769
\(494\) 0 0
\(495\) −3600.00 −0.326885
\(496\) 0 0
\(497\) 5984.00 0.540079
\(498\) 0 0
\(499\) −1764.00 −0.158251 −0.0791257 − 0.996865i \(-0.525213\pi\)
−0.0791257 + 0.996865i \(0.525213\pi\)
\(500\) 0 0
\(501\) 5004.00 0.446232
\(502\) 0 0
\(503\) −16976.0 −1.50482 −0.752408 − 0.658697i \(-0.771108\pi\)
−0.752408 + 0.658697i \(0.771108\pi\)
\(504\) 0 0
\(505\) 5180.00 0.456449
\(506\) 0 0
\(507\) −507.000 −0.0444116
\(508\) 0 0
\(509\) 9474.00 0.825005 0.412503 − 0.910956i \(-0.364655\pi\)
0.412503 + 0.910956i \(0.364655\pi\)
\(510\) 0 0
\(511\) 8464.00 0.732731
\(512\) 0 0
\(513\) −540.000 −0.0464748
\(514\) 0 0
\(515\) −1120.00 −0.0958313
\(516\) 0 0
\(517\) −18080.0 −1.53802
\(518\) 0 0
\(519\) −10794.0 −0.912917
\(520\) 0 0
\(521\) 14114.0 1.18684 0.593422 − 0.804892i \(-0.297777\pi\)
0.593422 + 0.804892i \(0.297777\pi\)
\(522\) 0 0
\(523\) −20284.0 −1.69590 −0.847952 − 0.530074i \(-0.822164\pi\)
−0.847952 + 0.530074i \(0.822164\pi\)
\(524\) 0 0
\(525\) 600.000 0.0498784
\(526\) 0 0
\(527\) 23920.0 1.97718
\(528\) 0 0
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) −4176.00 −0.341286
\(532\) 0 0
\(533\) −4706.00 −0.382438
\(534\) 0 0
\(535\) 3720.00 0.300616
\(536\) 0 0
\(537\) 3204.00 0.257473
\(538\) 0 0
\(539\) 11160.0 0.891828
\(540\) 0 0
\(541\) −14362.0 −1.14135 −0.570675 − 0.821176i \(-0.693318\pi\)
−0.570675 + 0.821176i \(0.693318\pi\)
\(542\) 0 0
\(543\) 14358.0 1.13473
\(544\) 0 0
\(545\) 9340.00 0.734095
\(546\) 0 0
\(547\) 20956.0 1.63805 0.819025 − 0.573757i \(-0.194515\pi\)
0.819025 + 0.573757i \(0.194515\pi\)
\(548\) 0 0
\(549\) 3222.00 0.250477
\(550\) 0 0
\(551\) −360.000 −0.0278340
\(552\) 0 0
\(553\) 7808.00 0.600416
\(554\) 0 0
\(555\) 2220.00 0.169791
\(556\) 0 0
\(557\) −4134.00 −0.314476 −0.157238 − 0.987561i \(-0.550259\pi\)
−0.157238 + 0.987561i \(0.550259\pi\)
\(558\) 0 0
\(559\) −988.000 −0.0747548
\(560\) 0 0
\(561\) 15600.0 1.17403
\(562\) 0 0
\(563\) 16228.0 1.21479 0.607397 − 0.794399i \(-0.292214\pi\)
0.607397 + 0.794399i \(0.292214\pi\)
\(564\) 0 0
\(565\) 19140.0 1.42518
\(566\) 0 0
\(567\) 648.000 0.0479955
\(568\) 0 0
\(569\) 2514.00 0.185224 0.0926119 − 0.995702i \(-0.470478\pi\)
0.0926119 + 0.995702i \(0.470478\pi\)
\(570\) 0 0
\(571\) 11612.0 0.851046 0.425523 − 0.904948i \(-0.360090\pi\)
0.425523 + 0.904948i \(0.360090\pi\)
\(572\) 0 0
\(573\) −3936.00 −0.286961
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 6354.00 0.458441 0.229221 − 0.973375i \(-0.426382\pi\)
0.229221 + 0.973375i \(0.426382\pi\)
\(578\) 0 0
\(579\) 1050.00 0.0753653
\(580\) 0 0
\(581\) 8064.00 0.575819
\(582\) 0 0
\(583\) −15280.0 −1.08548
\(584\) 0 0
\(585\) 1170.00 0.0826898
\(586\) 0 0
\(587\) 13240.0 0.930960 0.465480 − 0.885059i \(-0.345882\pi\)
0.465480 + 0.885059i \(0.345882\pi\)
\(588\) 0 0
\(589\) 3680.00 0.257439
\(590\) 0 0
\(591\) 1026.00 0.0714112
\(592\) 0 0
\(593\) −1146.00 −0.0793602 −0.0396801 − 0.999212i \(-0.512634\pi\)
−0.0396801 + 0.999212i \(0.512634\pi\)
\(594\) 0 0
\(595\) 10400.0 0.716569
\(596\) 0 0
\(597\) −10104.0 −0.692679
\(598\) 0 0
\(599\) −10464.0 −0.713769 −0.356884 − 0.934149i \(-0.616161\pi\)
−0.356884 + 0.934149i \(0.616161\pi\)
\(600\) 0 0
\(601\) 6650.00 0.451346 0.225673 − 0.974203i \(-0.427542\pi\)
0.225673 + 0.974203i \(0.427542\pi\)
\(602\) 0 0
\(603\) 6300.00 0.425466
\(604\) 0 0
\(605\) 2690.00 0.180767
\(606\) 0 0
\(607\) 6664.00 0.445607 0.222803 − 0.974863i \(-0.428479\pi\)
0.222803 + 0.974863i \(0.428479\pi\)
\(608\) 0 0
\(609\) 432.000 0.0287447
\(610\) 0 0
\(611\) 5876.00 0.389063
\(612\) 0 0
\(613\) 2134.00 0.140606 0.0703030 − 0.997526i \(-0.477603\pi\)
0.0703030 + 0.997526i \(0.477603\pi\)
\(614\) 0 0
\(615\) 10860.0 0.712061
\(616\) 0 0
\(617\) −714.000 −0.0465876 −0.0232938 − 0.999729i \(-0.507415\pi\)
−0.0232938 + 0.999729i \(0.507415\pi\)
\(618\) 0 0
\(619\) −29228.0 −1.89786 −0.948928 − 0.315494i \(-0.897830\pi\)
−0.948928 + 0.315494i \(0.897830\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3088.00 −0.198584
\(624\) 0 0
\(625\) −11875.0 −0.760000
\(626\) 0 0
\(627\) 2400.00 0.152866
\(628\) 0 0
\(629\) −9620.00 −0.609816
\(630\) 0 0
\(631\) 13536.0 0.853977 0.426989 − 0.904257i \(-0.359574\pi\)
0.426989 + 0.904257i \(0.359574\pi\)
\(632\) 0 0
\(633\) −6012.00 −0.377497
\(634\) 0 0
\(635\) −12960.0 −0.809924
\(636\) 0 0
\(637\) −3627.00 −0.225600
\(638\) 0 0
\(639\) 6732.00 0.416767
\(640\) 0 0
\(641\) 17218.0 1.06095 0.530476 − 0.847700i \(-0.322013\pi\)
0.530476 + 0.847700i \(0.322013\pi\)
\(642\) 0 0
\(643\) −15044.0 −0.922671 −0.461335 − 0.887226i \(-0.652630\pi\)
−0.461335 + 0.887226i \(0.652630\pi\)
\(644\) 0 0
\(645\) 2280.00 0.139186
\(646\) 0 0
\(647\) −25176.0 −1.52978 −0.764892 − 0.644158i \(-0.777208\pi\)
−0.764892 + 0.644158i \(0.777208\pi\)
\(648\) 0 0
\(649\) 18560.0 1.12256
\(650\) 0 0
\(651\) −4416.00 −0.265863
\(652\) 0 0
\(653\) −16034.0 −0.960887 −0.480443 − 0.877026i \(-0.659524\pi\)
−0.480443 + 0.877026i \(0.659524\pi\)
\(654\) 0 0
\(655\) 8920.00 0.532112
\(656\) 0 0
\(657\) 9522.00 0.565432
\(658\) 0 0
\(659\) −25356.0 −1.49883 −0.749415 − 0.662100i \(-0.769665\pi\)
−0.749415 + 0.662100i \(0.769665\pi\)
\(660\) 0 0
\(661\) 18310.0 1.07742 0.538711 − 0.842490i \(-0.318911\pi\)
0.538711 + 0.842490i \(0.318911\pi\)
\(662\) 0 0
\(663\) −5070.00 −0.296987
\(664\) 0 0
\(665\) 1600.00 0.0933013
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −16824.0 −0.972277
\(670\) 0 0
\(671\) −14320.0 −0.823871
\(672\) 0 0
\(673\) 24802.0 1.42057 0.710287 − 0.703912i \(-0.248565\pi\)
0.710287 + 0.703912i \(0.248565\pi\)
\(674\) 0 0
\(675\) 675.000 0.0384900
\(676\) 0 0
\(677\) −22706.0 −1.28901 −0.644507 − 0.764598i \(-0.722937\pi\)
−0.644507 + 0.764598i \(0.722937\pi\)
\(678\) 0 0
\(679\) −4912.00 −0.277622
\(680\) 0 0
\(681\) −5784.00 −0.325467
\(682\) 0 0
\(683\) 14792.0 0.828697 0.414349 − 0.910118i \(-0.364009\pi\)
0.414349 + 0.910118i \(0.364009\pi\)
\(684\) 0 0
\(685\) 23260.0 1.29740
\(686\) 0 0
\(687\) 11814.0 0.656088
\(688\) 0 0
\(689\) 4966.00 0.274586
\(690\) 0 0
\(691\) 1148.00 0.0632011 0.0316006 − 0.999501i \(-0.489940\pi\)
0.0316006 + 0.999501i \(0.489940\pi\)
\(692\) 0 0
\(693\) −2880.00 −0.157867
\(694\) 0 0
\(695\) −19320.0 −1.05446
\(696\) 0 0
\(697\) −47060.0 −2.55742
\(698\) 0 0
\(699\) −7686.00 −0.415896
\(700\) 0 0
\(701\) 14870.0 0.801187 0.400594 − 0.916256i \(-0.368804\pi\)
0.400594 + 0.916256i \(0.368804\pi\)
\(702\) 0 0
\(703\) −1480.00 −0.0794015
\(704\) 0 0
\(705\) −13560.0 −0.724396
\(706\) 0 0
\(707\) 4144.00 0.220440
\(708\) 0 0
\(709\) −6354.00 −0.336572 −0.168286 − 0.985738i \(-0.553823\pi\)
−0.168286 + 0.985738i \(0.553823\pi\)
\(710\) 0 0
\(711\) 8784.00 0.463327
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −5200.00 −0.271985
\(716\) 0 0
\(717\) 21492.0 1.11943
\(718\) 0 0
\(719\) −9288.00 −0.481758 −0.240879 − 0.970555i \(-0.577436\pi\)
−0.240879 + 0.970555i \(0.577436\pi\)
\(720\) 0 0
\(721\) −896.000 −0.0462813
\(722\) 0 0
\(723\) 18546.0 0.953988
\(724\) 0 0
\(725\) 450.000 0.0230518
\(726\) 0 0
\(727\) 21544.0 1.09907 0.549534 − 0.835471i \(-0.314805\pi\)
0.549534 + 0.835471i \(0.314805\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −9880.00 −0.499897
\(732\) 0 0
\(733\) 19990.0 1.00730 0.503648 − 0.863909i \(-0.331991\pi\)
0.503648 + 0.863909i \(0.331991\pi\)
\(734\) 0 0
\(735\) 8370.00 0.420044
\(736\) 0 0
\(737\) −28000.0 −1.39945
\(738\) 0 0
\(739\) −532.000 −0.0264816 −0.0132408 − 0.999912i \(-0.504215\pi\)
−0.0132408 + 0.999912i \(0.504215\pi\)
\(740\) 0 0
\(741\) −780.000 −0.0386694
\(742\) 0 0
\(743\) 25452.0 1.25672 0.628360 − 0.777922i \(-0.283726\pi\)
0.628360 + 0.777922i \(0.283726\pi\)
\(744\) 0 0
\(745\) 8820.00 0.433745
\(746\) 0 0
\(747\) 9072.00 0.444347
\(748\) 0 0
\(749\) 2976.00 0.145181
\(750\) 0 0
\(751\) −6440.00 −0.312915 −0.156457 − 0.987685i \(-0.550007\pi\)
−0.156457 + 0.987685i \(0.550007\pi\)
\(752\) 0 0
\(753\) −4188.00 −0.202682
\(754\) 0 0
\(755\) 17760.0 0.856096
\(756\) 0 0
\(757\) −786.000 −0.0377380 −0.0188690 − 0.999822i \(-0.506007\pi\)
−0.0188690 + 0.999822i \(0.506007\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1498.00 −0.0713567 −0.0356784 − 0.999363i \(-0.511359\pi\)
−0.0356784 + 0.999363i \(0.511359\pi\)
\(762\) 0 0
\(763\) 7472.00 0.354528
\(764\) 0 0
\(765\) 11700.0 0.552960
\(766\) 0 0
\(767\) −6032.00 −0.283967
\(768\) 0 0
\(769\) 14738.0 0.691113 0.345556 − 0.938398i \(-0.387690\pi\)
0.345556 + 0.938398i \(0.387690\pi\)
\(770\) 0 0
\(771\) −20718.0 −0.967757
\(772\) 0 0
\(773\) −3822.00 −0.177837 −0.0889184 − 0.996039i \(-0.528341\pi\)
−0.0889184 + 0.996039i \(0.528341\pi\)
\(774\) 0 0
\(775\) −4600.00 −0.213209
\(776\) 0 0
\(777\) 1776.00 0.0819995
\(778\) 0 0
\(779\) −7240.00 −0.332991
\(780\) 0 0
\(781\) −29920.0 −1.37083
\(782\) 0 0
\(783\) 486.000 0.0221816
\(784\) 0 0
\(785\) −24100.0 −1.09575
\(786\) 0 0
\(787\) 11900.0 0.538995 0.269498 − 0.963001i \(-0.413142\pi\)
0.269498 + 0.963001i \(0.413142\pi\)
\(788\) 0 0
\(789\) −20544.0 −0.926978
\(790\) 0 0
\(791\) 15312.0 0.688283
\(792\) 0 0
\(793\) 4654.00 0.208409
\(794\) 0 0
\(795\) −11460.0 −0.511251
\(796\) 0 0
\(797\) −21274.0 −0.945500 −0.472750 − 0.881197i \(-0.656739\pi\)
−0.472750 + 0.881197i \(0.656739\pi\)
\(798\) 0 0
\(799\) 58760.0 2.60173
\(800\) 0 0
\(801\) −3474.00 −0.153243
\(802\) 0 0
\(803\) −42320.0 −1.85983
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 18102.0 0.789617
\(808\) 0 0
\(809\) −27566.0 −1.19798 −0.598992 − 0.800755i \(-0.704432\pi\)
−0.598992 + 0.800755i \(0.704432\pi\)
\(810\) 0 0
\(811\) 11244.0 0.486844 0.243422 − 0.969921i \(-0.421730\pi\)
0.243422 + 0.969921i \(0.421730\pi\)
\(812\) 0 0
\(813\) 14496.0 0.625334
\(814\) 0 0
\(815\) −32120.0 −1.38051
\(816\) 0 0
\(817\) −1520.00 −0.0650894
\(818\) 0 0
\(819\) 936.000 0.0399347
\(820\) 0 0
\(821\) 13554.0 0.576173 0.288086 − 0.957604i \(-0.406981\pi\)
0.288086 + 0.957604i \(0.406981\pi\)
\(822\) 0 0
\(823\) −14384.0 −0.609228 −0.304614 − 0.952476i \(-0.598527\pi\)
−0.304614 + 0.952476i \(0.598527\pi\)
\(824\) 0 0
\(825\) −3000.00 −0.126602
\(826\) 0 0
\(827\) 2488.00 0.104615 0.0523073 − 0.998631i \(-0.483342\pi\)
0.0523073 + 0.998631i \(0.483342\pi\)
\(828\) 0 0
\(829\) −20858.0 −0.873858 −0.436929 − 0.899496i \(-0.643934\pi\)
−0.436929 + 0.899496i \(0.643934\pi\)
\(830\) 0 0
\(831\) 12246.0 0.511202
\(832\) 0 0
\(833\) −36270.0 −1.50862
\(834\) 0 0
\(835\) −16680.0 −0.691300
\(836\) 0 0
\(837\) −4968.00 −0.205160
\(838\) 0 0
\(839\) −23116.0 −0.951195 −0.475598 − 0.879663i \(-0.657768\pi\)
−0.475598 + 0.879663i \(0.657768\pi\)
\(840\) 0 0
\(841\) −24065.0 −0.986715
\(842\) 0 0
\(843\) −10050.0 −0.410605
\(844\) 0 0
\(845\) 1690.00 0.0688021
\(846\) 0 0
\(847\) 2152.00 0.0873006
\(848\) 0 0
\(849\) 23388.0 0.945435
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 934.000 0.0374907 0.0187453 − 0.999824i \(-0.494033\pi\)
0.0187453 + 0.999824i \(0.494033\pi\)
\(854\) 0 0
\(855\) 1800.00 0.0719985
\(856\) 0 0
\(857\) 12642.0 0.503900 0.251950 − 0.967740i \(-0.418928\pi\)
0.251950 + 0.967740i \(0.418928\pi\)
\(858\) 0 0
\(859\) 22796.0 0.905459 0.452730 − 0.891648i \(-0.350450\pi\)
0.452730 + 0.891648i \(0.350450\pi\)
\(860\) 0 0
\(861\) 8688.00 0.343886
\(862\) 0 0
\(863\) 76.0000 0.00299776 0.00149888 − 0.999999i \(-0.499523\pi\)
0.00149888 + 0.999999i \(0.499523\pi\)
\(864\) 0 0
\(865\) 35980.0 1.41429
\(866\) 0 0
\(867\) −35961.0 −1.40865
\(868\) 0 0
\(869\) −39040.0 −1.52398
\(870\) 0 0
\(871\) 9100.00 0.354009
\(872\) 0 0
\(873\) −5526.00 −0.214235
\(874\) 0 0
\(875\) −12000.0 −0.463627
\(876\) 0 0
\(877\) −46130.0 −1.77617 −0.888084 − 0.459681i \(-0.847964\pi\)
−0.888084 + 0.459681i \(0.847964\pi\)
\(878\) 0 0
\(879\) −11766.0 −0.451487
\(880\) 0 0
\(881\) 6682.00 0.255530 0.127765 − 0.991804i \(-0.459220\pi\)
0.127765 + 0.991804i \(0.459220\pi\)
\(882\) 0 0
\(883\) −47404.0 −1.80665 −0.903325 − 0.428957i \(-0.858881\pi\)
−0.903325 + 0.428957i \(0.858881\pi\)
\(884\) 0 0
\(885\) 13920.0 0.528718
\(886\) 0 0
\(887\) −33672.0 −1.27463 −0.637314 − 0.770604i \(-0.719955\pi\)
−0.637314 + 0.770604i \(0.719955\pi\)
\(888\) 0 0
\(889\) −10368.0 −0.391149
\(890\) 0 0
\(891\) −3240.00 −0.121823
\(892\) 0 0
\(893\) 9040.00 0.338759
\(894\) 0 0
\(895\) −10680.0 −0.398875
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −3312.00 −0.122871
\(900\) 0 0
\(901\) 49660.0 1.83620
\(902\) 0 0
\(903\) 1824.00 0.0672192
\(904\) 0 0
\(905\) −47860.0 −1.75792
\(906\) 0 0
\(907\) 14540.0 0.532296 0.266148 − 0.963932i \(-0.414249\pi\)
0.266148 + 0.963932i \(0.414249\pi\)
\(908\) 0 0
\(909\) 4662.00 0.170109
\(910\) 0 0
\(911\) 7840.00 0.285127 0.142564 − 0.989786i \(-0.454465\pi\)
0.142564 + 0.989786i \(0.454465\pi\)
\(912\) 0 0
\(913\) −40320.0 −1.46155
\(914\) 0 0
\(915\) −10740.0 −0.388037
\(916\) 0 0
\(917\) 7136.00 0.256981
\(918\) 0 0
\(919\) −47720.0 −1.71288 −0.856440 − 0.516246i \(-0.827329\pi\)
−0.856440 + 0.516246i \(0.827329\pi\)
\(920\) 0 0
\(921\) 17868.0 0.639273
\(922\) 0 0
\(923\) 9724.00 0.346771
\(924\) 0 0
\(925\) 1850.00 0.0657596
\(926\) 0 0
\(927\) −1008.00 −0.0357142
\(928\) 0 0
\(929\) 7502.00 0.264944 0.132472 − 0.991187i \(-0.457709\pi\)
0.132472 + 0.991187i \(0.457709\pi\)
\(930\) 0 0
\(931\) −5580.00 −0.196431
\(932\) 0 0
\(933\) 7056.00 0.247592
\(934\) 0 0
\(935\) −52000.0 −1.81880
\(936\) 0 0
\(937\) 22058.0 0.769054 0.384527 − 0.923114i \(-0.374365\pi\)
0.384527 + 0.923114i \(0.374365\pi\)
\(938\) 0 0
\(939\) −25326.0 −0.880173
\(940\) 0 0
\(941\) 23338.0 0.808498 0.404249 − 0.914649i \(-0.367533\pi\)
0.404249 + 0.914649i \(0.367533\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −2160.00 −0.0743543
\(946\) 0 0
\(947\) 30488.0 1.04617 0.523087 − 0.852279i \(-0.324780\pi\)
0.523087 + 0.852279i \(0.324780\pi\)
\(948\) 0 0
\(949\) 13754.0 0.470468
\(950\) 0 0
\(951\) 16650.0 0.567732
\(952\) 0 0
\(953\) 9522.00 0.323660 0.161830 − 0.986819i \(-0.448260\pi\)
0.161830 + 0.986819i \(0.448260\pi\)
\(954\) 0 0
\(955\) 13120.0 0.444558
\(956\) 0 0
\(957\) −2160.00 −0.0729602
\(958\) 0 0
\(959\) 18608.0 0.626573
\(960\) 0 0
\(961\) 4065.00 0.136451
\(962\) 0 0
\(963\) 3348.00 0.112033
\(964\) 0 0
\(965\) −3500.00 −0.116755
\(966\) 0 0
\(967\) 7616.00 0.253272 0.126636 − 0.991949i \(-0.459582\pi\)
0.126636 + 0.991949i \(0.459582\pi\)
\(968\) 0 0
\(969\) −7800.00 −0.258588
\(970\) 0 0
\(971\) −51316.0 −1.69599 −0.847996 − 0.530002i \(-0.822191\pi\)
−0.847996 + 0.530002i \(0.822191\pi\)
\(972\) 0 0
\(973\) −15456.0 −0.509246
\(974\) 0 0
\(975\) 975.000 0.0320256
\(976\) 0 0
\(977\) −48666.0 −1.59362 −0.796808 − 0.604232i \(-0.793480\pi\)
−0.796808 + 0.604232i \(0.793480\pi\)
\(978\) 0 0
\(979\) 15440.0 0.504050
\(980\) 0 0
\(981\) 8406.00 0.273581
\(982\) 0 0
\(983\) −17388.0 −0.564182 −0.282091 − 0.959388i \(-0.591028\pi\)
−0.282091 + 0.959388i \(0.591028\pi\)
\(984\) 0 0
\(985\) −3420.00 −0.110630
\(986\) 0 0
\(987\) −10848.0 −0.349844
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −11496.0 −0.368499 −0.184249 − 0.982880i \(-0.558985\pi\)
−0.184249 + 0.982880i \(0.558985\pi\)
\(992\) 0 0
\(993\) 420.000 0.0134223
\(994\) 0 0
\(995\) 33680.0 1.07309
\(996\) 0 0
\(997\) 48862.0 1.55213 0.776066 − 0.630652i \(-0.217212\pi\)
0.776066 + 0.630652i \(0.217212\pi\)
\(998\) 0 0
\(999\) 1998.00 0.0632772
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.d.1.1 1
3.2 odd 2 1872.4.a.d.1.1 1
4.3 odd 2 78.4.a.c.1.1 ✓ 1
8.3 odd 2 2496.4.a.a.1.1 1
8.5 even 2 2496.4.a.j.1.1 1
12.11 even 2 234.4.a.h.1.1 1
20.19 odd 2 1950.4.a.l.1.1 1
52.31 even 4 1014.4.b.h.337.2 2
52.47 even 4 1014.4.b.h.337.1 2
52.51 odd 2 1014.4.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.c.1.1 ✓ 1 4.3 odd 2
234.4.a.h.1.1 1 12.11 even 2
624.4.a.d.1.1 1 1.1 even 1 trivial
1014.4.a.j.1.1 1 52.51 odd 2
1014.4.b.h.337.1 2 52.47 even 4
1014.4.b.h.337.2 2 52.31 even 4
1872.4.a.d.1.1 1 3.2 odd 2
1950.4.a.l.1.1 1 20.19 odd 2
2496.4.a.a.1.1 1 8.3 odd 2
2496.4.a.j.1.1 1 8.5 even 2