Properties

Label 7350.2.a.ce.1.1
Level $7350$
Weight $2$
Character 7350.1
Self dual yes
Analytic conductor $58.690$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7350,2,Mod(1,7350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7350, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7350.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7350 = 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7350.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: \(58.6900454856\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7350.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} +1.00000 q^{16} +1.00000 q^{18} -4.00000 q^{19} +3.00000 q^{22} -1.00000 q^{24} +4.00000 q^{26} -1.00000 q^{27} +9.00000 q^{29} -1.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} +1.00000 q^{36} -8.00000 q^{37} -4.00000 q^{38} -4.00000 q^{39} +10.0000 q^{43} +3.00000 q^{44} +6.00000 q^{47} -1.00000 q^{48} +4.00000 q^{52} +3.00000 q^{53} -1.00000 q^{54} +4.00000 q^{57} +9.00000 q^{58} +3.00000 q^{59} -10.0000 q^{61} -1.00000 q^{62} +1.00000 q^{64} -3.00000 q^{66} +10.0000 q^{67} -6.00000 q^{71} +1.00000 q^{72} -2.00000 q^{73} -8.00000 q^{74} -4.00000 q^{76} -4.00000 q^{78} -1.00000 q^{79} +1.00000 q^{81} +9.00000 q^{83} +10.0000 q^{86} -9.00000 q^{87} +3.00000 q^{88} +6.00000 q^{89} +1.00000 q^{93} +6.00000 q^{94} -1.00000 q^{96} +1.00000 q^{97} +3.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −1.00000 −0.288675
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000 0.639602
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) −4.00000 −0.648886
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 9.00000 1.18176
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −1.00000 −0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.00000 −0.369274
\(67\) 10.0000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) −1.00000 −0.112509 −0.0562544 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.00000 0.987878 0.493939 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.0000 1.07833
\(87\) −9.00000 −0.964901
\(88\) 3.00000 0.319801
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.00000 0.103695
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 1.00000 0.101535 0.0507673 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 9.00000 0.835629
\(117\) 4.00000 0.369800
\(118\) 3.00000 0.276172
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) 0 0
\(127\) −5.00000 −0.443678 −0.221839 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(128\) 1.00000 0.0883883
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) −9.00000 −0.786334 −0.393167 0.919467i \(-0.628621\pi\)
−0.393167 + 0.919467i \(0.628621\pi\)
\(132\) −3.00000 −0.261116
\(133\) 0 0
\(134\) 10.0000 0.863868
\(135\) 0 0
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) −6.00000 −0.503509
\(143\) 12.0000 1.00349
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) −8.00000 −0.657596
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) −1.00000 −0.0813788 −0.0406894 0.999172i \(-0.512955\pi\)
−0.0406894 + 0.999172i \(0.512955\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) −1.00000 −0.0795557
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 9.00000 0.698535
\(167\) −6.00000 −0.464294 −0.232147 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 10.0000 0.762493
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −9.00000 −0.682288
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) −3.00000 −0.225494
\(178\) 6.00000 0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 19.0000 1.36765 0.683825 0.729646i \(-0.260315\pi\)
0.683825 + 0.729646i \(0.260315\pi\)
\(194\) 1.00000 0.0717958
\(195\) 0 0
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 3.00000 0.213201
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) −10.0000 −0.705346
\(202\) −18.0000 −1.26648
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 3.00000 0.206041
\(213\) 6.00000 0.411113
\(214\) 3.00000 0.205076
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 14.0000 0.948200
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) 8.00000 0.536925
\(223\) 19.0000 1.27233 0.636167 0.771551i \(-0.280519\pi\)
0.636167 + 0.771551i \(0.280519\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 27.0000 1.79205 0.896026 0.444001i \(-0.146441\pi\)
0.896026 + 0.444001i \(0.146441\pi\)
\(228\) 4.00000 0.264906
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 9.00000 0.590879
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) 3.00000 0.195283
\(237\) 1.00000 0.0649570
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157 −0.0322078 0.999481i \(-0.510254\pi\)
−0.0322078 + 0.999481i \(0.510254\pi\)
\(242\) −2.00000 −0.128565
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) −1.00000 −0.0635001
\(249\) −9.00000 −0.570352
\(250\) 0 0
\(251\) 27.0000 1.70422 0.852112 0.523359i \(-0.175321\pi\)
0.852112 + 0.523359i \(0.175321\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −5.00000 −0.313728
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −10.0000 −0.622573
\(259\) 0 0
\(260\) 0 0
\(261\) 9.00000 0.557086
\(262\) −9.00000 −0.556022
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) −3.00000 −0.184637
\(265\) 0 0
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 10.0000 0.610847
\(269\) 21.0000 1.28039 0.640196 0.768211i \(-0.278853\pi\)
0.640196 + 0.768211i \(0.278853\pi\)
\(270\) 0 0
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 2.00000 0.119952
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −6.00000 −0.357295
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −1.00000 −0.0586210
\(292\) −2.00000 −0.117041
\(293\) −33.0000 −1.92788 −0.963940 0.266119i \(-0.914259\pi\)
−0.963940 + 0.266119i \(0.914259\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) −3.00000 −0.174078
\(298\) 18.0000 1.04271
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −1.00000 −0.0575435
\(303\) 18.0000 1.03407
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 0 0
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −4.00000 −0.226455
\(313\) 31.0000 1.75222 0.876112 0.482108i \(-0.160129\pi\)
0.876112 + 0.482108i \(0.160129\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −1.00000 −0.0562544
\(317\) −9.00000 −0.505490 −0.252745 0.967533i \(-0.581333\pi\)
−0.252745 + 0.967533i \(0.581333\pi\)
\(318\) −3.00000 −0.168232
\(319\) 27.0000 1.51171
\(320\) 0 0
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) −14.0000 −0.774202
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 9.00000 0.493939
\(333\) −8.00000 −0.438397
\(334\) −6.00000 −0.328305
\(335\) 0 0
\(336\) 0 0
\(337\) 7.00000 0.381314 0.190657 0.981657i \(-0.438938\pi\)
0.190657 + 0.981657i \(0.438938\pi\)
\(338\) 3.00000 0.163178
\(339\) 0 0
\(340\) 0 0
\(341\) −3.00000 −0.162459
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) 10.0000 0.539164
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −9.00000 −0.482451
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 3.00000 0.159901
\(353\) −24.0000 −1.27739 −0.638696 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) −3.00000 −0.159448
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 8.00000 0.420471
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) 0 0
\(366\) 10.0000 0.522708
\(367\) 19.0000 0.991792 0.495896 0.868382i \(-0.334840\pi\)
0.495896 + 0.868382i \(0.334840\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 1.00000 0.0518476
\(373\) −8.00000 −0.414224 −0.207112 0.978317i \(-0.566407\pi\)
−0.207112 + 0.978317i \(0.566407\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 36.0000 1.85409
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 5.00000 0.256158
\(382\) 0 0
\(383\) −18.0000 −0.919757 −0.459879 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 19.0000 0.967075
\(387\) 10.0000 0.508329
\(388\) 1.00000 0.0507673
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 9.00000 0.453990
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) 4.00000 0.200754 0.100377 0.994949i \(-0.467995\pi\)
0.100377 + 0.994949i \(0.467995\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) 0 0
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) −10.0000 −0.498755
\(403\) −4.00000 −0.199254
\(404\) −18.0000 −0.895533
\(405\) 0 0
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) −25.0000 −1.23617 −0.618085 0.786111i \(-0.712091\pi\)
−0.618085 + 0.786111i \(0.712091\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) −2.00000 −0.0979404
\(418\) −12.0000 −0.586939
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 14.0000 0.681509
\(423\) 6.00000 0.291730
\(424\) 3.00000 0.145693
\(425\) 0 0
\(426\) 6.00000 0.290701
\(427\) 0 0
\(428\) 3.00000 0.145010
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) 2.00000 0.0955637
\(439\) 35.0000 1.67046 0.835229 0.549902i \(-0.185335\pi\)
0.835229 + 0.549902i \(0.185335\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 33.0000 1.56788 0.783939 0.620838i \(-0.213208\pi\)
0.783939 + 0.620838i \(0.213208\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) 19.0000 0.899676
\(447\) −18.0000 −0.851371
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 1.00000 0.0469841
\(454\) 27.0000 1.26717
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 1.00000 0.0467780 0.0233890 0.999726i \(-0.492554\pi\)
0.0233890 + 0.999726i \(0.492554\pi\)
\(458\) −4.00000 −0.186908
\(459\) 0 0
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 9.00000 0.417815
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 4.00000 0.184900
\(469\) 0 0
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 3.00000 0.138086
\(473\) 30.0000 1.37940
\(474\) 1.00000 0.0459315
\(475\) 0 0
\(476\) 0 0
\(477\) 3.00000 0.137361
\(478\) −24.0000 −1.09773
\(479\) −18.0000 −0.822441 −0.411220 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) 0 0
\(481\) −32.0000 −1.45907
\(482\) −1.00000 −0.0455488
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −41.0000 −1.85789 −0.928944 0.370221i \(-0.879282\pi\)
−0.928944 + 0.370221i \(0.879282\pi\)
\(488\) −10.0000 −0.452679
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) −33.0000 −1.48927 −0.744635 0.667472i \(-0.767376\pi\)
−0.744635 + 0.667472i \(0.767376\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 0 0
\(498\) −9.00000 −0.403300
\(499\) 2.00000 0.0895323 0.0447661 0.998997i \(-0.485746\pi\)
0.0447661 + 0.998997i \(0.485746\pi\)
\(500\) 0 0
\(501\) 6.00000 0.268060
\(502\) 27.0000 1.20507
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −3.00000 −0.133235
\(508\) −5.00000 −0.221839
\(509\) −3.00000 −0.132973 −0.0664863 0.997787i \(-0.521179\pi\)
−0.0664863 + 0.997787i \(0.521179\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 4.00000 0.176604
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) −10.0000 −0.440225
\(517\) 18.0000 0.791639
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 9.00000 0.393919
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −9.00000 −0.393167
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) 0 0
\(528\) −3.00000 −0.130558
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 3.00000 0.130189
\(532\) 0 0
\(533\) 0 0
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) 10.0000 0.431934
\(537\) −12.0000 −0.517838
\(538\) 21.0000 0.905374
\(539\) 0 0
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) 11.0000 0.472490
\(543\) −8.00000 −0.343313
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −18.0000 −0.768922
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −36.0000 −1.53365
\(552\) 0 0
\(553\) 0 0
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) 2.00000 0.0848189
\(557\) −3.00000 −0.127114 −0.0635570 0.997978i \(-0.520244\pi\)
−0.0635570 + 0.997978i \(0.520244\pi\)
\(558\) −1.00000 −0.0423334
\(559\) 40.0000 1.69182
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) −39.0000 −1.64365 −0.821827 0.569737i \(-0.807045\pi\)
−0.821827 + 0.569737i \(0.807045\pi\)
\(564\) −6.00000 −0.252646
\(565\) 0 0
\(566\) −14.0000 −0.588464
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) −36.0000 −1.50920 −0.754599 0.656186i \(-0.772169\pi\)
−0.754599 + 0.656186i \(0.772169\pi\)
\(570\) 0 0
\(571\) −34.0000 −1.42286 −0.711428 0.702759i \(-0.751951\pi\)
−0.711428 + 0.702759i \(0.751951\pi\)
\(572\) 12.0000 0.501745
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −23.0000 −0.957503 −0.478751 0.877951i \(-0.658910\pi\)
−0.478751 + 0.877951i \(0.658910\pi\)
\(578\) −17.0000 −0.707107
\(579\) −19.0000 −0.789613
\(580\) 0 0
\(581\) 0 0
\(582\) −1.00000 −0.0414513
\(583\) 9.00000 0.372742
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −33.0000 −1.36322
\(587\) −21.0000 −0.866763 −0.433381 0.901211i \(-0.642680\pi\)
−0.433381 + 0.901211i \(0.642680\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) −8.00000 −0.328798
\(593\) −24.0000 −0.985562 −0.492781 0.870153i \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) −3.00000 −0.123091
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) −20.0000 −0.818546
\(598\) 0 0
\(599\) −18.0000 −0.735460 −0.367730 0.929933i \(-0.619865\pi\)
−0.367730 + 0.929933i \(0.619865\pi\)
\(600\) 0 0
\(601\) 11.0000 0.448699 0.224350 0.974509i \(-0.427974\pi\)
0.224350 + 0.974509i \(0.427974\pi\)
\(602\) 0 0
\(603\) 10.0000 0.407231
\(604\) −1.00000 −0.0406894
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) 7.00000 0.284121 0.142061 0.989858i \(-0.454627\pi\)
0.142061 + 0.989858i \(0.454627\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 8.00000 0.321807
\(619\) −34.0000 −1.36658 −0.683288 0.730149i \(-0.739451\pi\)
−0.683288 + 0.730149i \(0.739451\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 24.0000 0.962312
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) 31.0000 1.23901
\(627\) 12.0000 0.479234
\(628\) 4.00000 0.159617
\(629\) 0 0
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) −1.00000 −0.0397779
\(633\) −14.0000 −0.556450
\(634\) −9.00000 −0.357436
\(635\) 0 0
\(636\) −3.00000 −0.118958
\(637\) 0 0
\(638\) 27.0000 1.06894
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −3.00000 −0.118401
\(643\) 34.0000 1.34083 0.670415 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 1.00000 0.0392837
\(649\) 9.00000 0.353281
\(650\) 0 0
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) −3.00000 −0.117399 −0.0586995 0.998276i \(-0.518695\pi\)
−0.0586995 + 0.998276i \(0.518695\pi\)
\(654\) −14.0000 −0.547443
\(655\) 0 0
\(656\) 0 0
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 9.00000 0.349268
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) 0 0
\(668\) −6.00000 −0.232147
\(669\) −19.0000 −0.734582
\(670\) 0 0
\(671\) −30.0000 −1.15814
\(672\) 0 0
\(673\) −29.0000 −1.11787 −0.558934 0.829212i \(-0.688789\pi\)
−0.558934 + 0.829212i \(0.688789\pi\)
\(674\) 7.00000 0.269630
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 33.0000 1.26829 0.634147 0.773213i \(-0.281352\pi\)
0.634147 + 0.773213i \(0.281352\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −27.0000 −1.03464
\(682\) −3.00000 −0.114876
\(683\) −33.0000 −1.26271 −0.631355 0.775494i \(-0.717501\pi\)
−0.631355 + 0.775494i \(0.717501\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) 4.00000 0.152610
\(688\) 10.0000 0.381246
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) −9.00000 −0.341144
\(697\) 0 0
\(698\) 26.0000 0.984115
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) −15.0000 −0.566542 −0.283271 0.959040i \(-0.591420\pi\)
−0.283271 + 0.959040i \(0.591420\pi\)
\(702\) −4.00000 −0.150970
\(703\) 32.0000 1.20690
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −24.0000 −0.903252
\(707\) 0 0
\(708\) −3.00000 −0.112747
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) −1.00000 −0.0375029
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 24.0000 0.896296
\(718\) 30.0000 1.11959
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −3.00000 −0.111648
\(723\) 1.00000 0.0371904
\(724\) 8.00000 0.297318
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) 13.0000 0.482143 0.241072 0.970507i \(-0.422501\pi\)
0.241072 + 0.970507i \(0.422501\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 10.0000 0.369611
\(733\) 10.0000 0.369358 0.184679 0.982799i \(-0.440875\pi\)
0.184679 + 0.982799i \(0.440875\pi\)
\(734\) 19.0000 0.701303
\(735\) 0 0
\(736\) 0 0
\(737\) 30.0000 1.10506
\(738\) 0 0
\(739\) 50.0000 1.83928 0.919640 0.392763i \(-0.128481\pi\)
0.919640 + 0.392763i \(0.128481\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) 0 0
\(743\) −42.0000 −1.54083 −0.770415 0.637542i \(-0.779951\pi\)
−0.770415 + 0.637542i \(0.779951\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) −8.00000 −0.292901
\(747\) 9.00000 0.329293
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −7.00000 −0.255434 −0.127717 0.991811i \(-0.540765\pi\)
−0.127717 + 0.991811i \(0.540765\pi\)
\(752\) 6.00000 0.218797
\(753\) −27.0000 −0.983935
\(754\) 36.0000 1.31104
\(755\) 0 0
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 8.00000 0.290573
\(759\) 0 0
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 5.00000 0.181131
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −18.0000 −0.650366
\(767\) 12.0000 0.433295
\(768\) −1.00000 −0.0360844
\(769\) −19.0000 −0.685158 −0.342579 0.939489i \(-0.611300\pi\)
−0.342579 + 0.939489i \(0.611300\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 19.0000 0.683825
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 10.0000 0.359443
\(775\) 0 0
\(776\) 1.00000 0.0358979
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) 0 0
\(780\) 0 0
\(781\) −18.0000 −0.644091
\(782\) 0 0
\(783\) −9.00000 −0.321634
\(784\) 0 0
\(785\) 0 0
\(786\) 9.00000 0.321019
\(787\) −50.0000 −1.78231 −0.891154 0.453701i \(-0.850103\pi\)
−0.891154 + 0.453701i \(0.850103\pi\)
\(788\) −6.00000 −0.213741
\(789\) −6.00000 −0.213606
\(790\) 0 0
\(791\) 0 0
\(792\) 3.00000 0.106600
\(793\) −40.0000 −1.42044
\(794\) 4.00000 0.141955
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 33.0000 1.16892 0.584460 0.811423i \(-0.301306\pi\)
0.584460 + 0.811423i \(0.301306\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 24.0000 0.847469
\(803\) −6.00000 −0.211735
\(804\) −10.0000 −0.352673
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) −21.0000 −0.739235
\(808\) −18.0000 −0.633238
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 0 0
\(813\) −11.0000 −0.385787
\(814\) −24.0000 −0.841200
\(815\) 0 0
\(816\) 0 0
\(817\) −40.0000 −1.39942
\(818\) −25.0000 −0.874105
\(819\) 0 0
\(820\) 0 0
\(821\) −3.00000 −0.104701 −0.0523504 0.998629i \(-0.516671\pi\)
−0.0523504 + 0.998629i \(0.516671\pi\)
\(822\) 18.0000 0.627822
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) −15.0000 −0.521601 −0.260801 0.965393i \(-0.583986\pi\)
−0.260801 + 0.965393i \(0.583986\pi\)
\(828\) 0 0
\(829\) −4.00000 −0.138926 −0.0694629 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) 4.00000 0.138675
\(833\) 0 0
\(834\) −2.00000 −0.0692543
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) 1.00000 0.0345651
\(838\) 0 0
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) −22.0000 −0.758170
\(843\) −6.00000 −0.206651
\(844\) 14.0000 0.481900
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) 0 0
\(848\) 3.00000 0.103020
\(849\) 14.0000 0.480479
\(850\) 0 0
\(851\) 0 0
\(852\) 6.00000 0.205557
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) −12.0000 −0.409673
\(859\) 50.0000 1.70598 0.852989 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 34.0000 1.15537
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) −3.00000 −0.101768
\(870\) 0 0
\(871\) 40.0000 1.35535
\(872\) 14.0000 0.474100
\(873\) 1.00000 0.0338449
\(874\) 0 0
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) −32.0000 −1.08056 −0.540282 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) 35.0000 1.18119
\(879\) 33.0000 1.11306
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) −32.0000 −1.07689 −0.538443 0.842662i \(-0.680987\pi\)
−0.538443 + 0.842662i \(0.680987\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 33.0000 1.10866
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 8.00000 0.268462
\(889\) 0 0
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 19.0000 0.636167
\(893\) −24.0000 −0.803129
\(894\) −18.0000 −0.602010
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 12.0000 0.400445
\(899\) −9.00000 −0.300167
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 1.00000 0.0332228
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) 27.0000 0.896026
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) 6.00000 0.198789 0.0993944 0.995048i \(-0.468309\pi\)
0.0993944 + 0.995048i \(0.468309\pi\)
\(912\) 4.00000 0.132453
\(913\) 27.0000 0.893570
\(914\) 1.00000 0.0330771
\(915\) 0 0
\(916\) −4.00000 −0.132164
\(917\) 0 0
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 8.00000 0.263609
\(922\) 30.0000 0.987997
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) −8.00000 −0.262754
\(928\) 9.00000 0.295439
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 24.0000 0.786146
\(933\) −24.0000 −0.785725
\(934\) −36.0000 −1.17796
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) −35.0000 −1.14340 −0.571700 0.820463i \(-0.693716\pi\)
−0.571700 + 0.820463i \(0.693716\pi\)
\(938\) 0 0
\(939\) −31.0000 −1.01165
\(940\) 0 0
\(941\) −9.00000 −0.293392 −0.146696 0.989182i \(-0.546864\pi\)
−0.146696 + 0.989182i \(0.546864\pi\)
\(942\) −4.00000 −0.130327
\(943\) 0 0
\(944\) 3.00000 0.0976417
\(945\) 0 0
\(946\) 30.0000 0.975384
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 1.00000 0.0324785
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) 9.00000 0.291845
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 3.00000 0.0971286
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) −27.0000 −0.872786
\(958\) −18.0000 −0.581554
\(959\) 0 0
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) −32.0000 −1.03172
\(963\) 3.00000 0.0966736
\(964\) −1.00000 −0.0322078
\(965\) 0 0
\(966\) 0 0
\(967\) 1.00000 0.0321578 0.0160789 0.999871i \(-0.494882\pi\)
0.0160789 + 0.999871i \(0.494882\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) 0 0
\(971\) −39.0000 −1.25157 −0.625785 0.779996i \(-0.715221\pi\)
−0.625785 + 0.779996i \(0.715221\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −41.0000 −1.31372
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −16.0000 −0.511624
\(979\) 18.0000 0.575282
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) −33.0000 −1.05307
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −16.0000 −0.509028
\(989\) 0 0
\(990\) 0 0
\(991\) −13.0000 −0.412959 −0.206479 0.978451i \(-0.566201\pi\)
−0.206479 + 0.978451i \(0.566201\pi\)
\(992\) −1.00000 −0.0317500
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) 0 0
\(996\) −9.00000 −0.285176
\(997\) −14.0000 −0.443384 −0.221692 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) 2.00000 0.0633089
\(999\) 8.00000 0.253109
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 7350.2.a.ce.1.1 1
5.4 even 2 294.2.a.d.1.1 1
7.2 even 3 1050.2.i.e.151.1 2
7.4 even 3 1050.2.i.e.751.1 2
7.6 odd 2 7350.2.a.cw.1.1 1
15.14 odd 2 882.2.a.g.1.1 1
20.19 odd 2 2352.2.a.m.1.1 1
35.2 odd 12 1050.2.o.b.949.2 4
35.4 even 6 42.2.e.b.37.1 yes 2
35.9 even 6 42.2.e.b.25.1 2
35.18 odd 12 1050.2.o.b.499.2 4
35.19 odd 6 294.2.e.f.67.1 2
35.23 odd 12 1050.2.o.b.949.1 4
35.24 odd 6 294.2.e.f.79.1 2
35.32 odd 12 1050.2.o.b.499.1 4
35.34 odd 2 294.2.a.a.1.1 1
40.19 odd 2 9408.2.a.bu.1.1 1
40.29 even 2 9408.2.a.d.1.1 1
60.59 even 2 7056.2.a.g.1.1 1
105.44 odd 6 126.2.g.b.109.1 2
105.59 even 6 882.2.g.b.667.1 2
105.74 odd 6 126.2.g.b.37.1 2
105.89 even 6 882.2.g.b.361.1 2
105.104 even 2 882.2.a.k.1.1 1
140.19 even 6 2352.2.q.m.1537.1 2
140.39 odd 6 336.2.q.d.289.1 2
140.59 even 6 2352.2.q.m.961.1 2
140.79 odd 6 336.2.q.d.193.1 2
140.139 even 2 2352.2.a.n.1.1 1
280.69 odd 2 9408.2.a.db.1.1 1
280.109 even 6 1344.2.q.v.961.1 2
280.139 even 2 9408.2.a.bm.1.1 1
280.149 even 6 1344.2.q.v.193.1 2
280.179 odd 6 1344.2.q.j.961.1 2
280.219 odd 6 1344.2.q.j.193.1 2
315.4 even 6 1134.2.h.p.541.1 2
315.74 odd 6 1134.2.e.p.919.1 2
315.79 even 6 1134.2.h.p.109.1 2
315.149 odd 6 1134.2.e.p.865.1 2
315.184 even 6 1134.2.e.a.865.1 2
315.214 even 6 1134.2.e.a.919.1 2
315.254 odd 6 1134.2.h.a.109.1 2
315.284 odd 6 1134.2.h.a.541.1 2
420.179 even 6 1008.2.s.n.289.1 2
420.359 even 6 1008.2.s.n.865.1 2
420.419 odd 2 7056.2.a.bz.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.e.b.25.1 2 35.9 even 6
42.2.e.b.37.1 yes 2 35.4 even 6
126.2.g.b.37.1 2 105.74 odd 6
126.2.g.b.109.1 2 105.44 odd 6
294.2.a.a.1.1 1 35.34 odd 2
294.2.a.d.1.1 1 5.4 even 2
294.2.e.f.67.1 2 35.19 odd 6
294.2.e.f.79.1 2 35.24 odd 6
336.2.q.d.193.1 2 140.79 odd 6
336.2.q.d.289.1 2 140.39 odd 6
882.2.a.g.1.1 1 15.14 odd 2
882.2.a.k.1.1 1 105.104 even 2
882.2.g.b.361.1 2 105.89 even 6
882.2.g.b.667.1 2 105.59 even 6
1008.2.s.n.289.1 2 420.179 even 6
1008.2.s.n.865.1 2 420.359 even 6
1050.2.i.e.151.1 2 7.2 even 3
1050.2.i.e.751.1 2 7.4 even 3
1050.2.o.b.499.1 4 35.32 odd 12
1050.2.o.b.499.2 4 35.18 odd 12
1050.2.o.b.949.1 4 35.23 odd 12
1050.2.o.b.949.2 4 35.2 odd 12
1134.2.e.a.865.1 2 315.184 even 6
1134.2.e.a.919.1 2 315.214 even 6
1134.2.e.p.865.1 2 315.149 odd 6
1134.2.e.p.919.1 2 315.74 odd 6
1134.2.h.a.109.1 2 315.254 odd 6
1134.2.h.a.541.1 2 315.284 odd 6
1134.2.h.p.109.1 2 315.79 even 6
1134.2.h.p.541.1 2 315.4 even 6
1344.2.q.j.193.1 2 280.219 odd 6
1344.2.q.j.961.1 2 280.179 odd 6
1344.2.q.v.193.1 2 280.149 even 6
1344.2.q.v.961.1 2 280.109 even 6
2352.2.a.m.1.1 1 20.19 odd 2
2352.2.a.n.1.1 1 140.139 even 2
2352.2.q.m.961.1 2 140.59 even 6
2352.2.q.m.1537.1 2 140.19 even 6
7056.2.a.g.1.1 1 60.59 even 2
7056.2.a.bz.1.1 1 420.419 odd 2
7350.2.a.ce.1.1 1 1.1 even 1 trivial
7350.2.a.cw.1.1 1 7.6 odd 2
9408.2.a.d.1.1 1 40.29 even 2
9408.2.a.bm.1.1 1 280.139 even 2
9408.2.a.bu.1.1 1 40.19 odd 2
9408.2.a.db.1.1 1 280.69 odd 2