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Properties

Label 3675.2.a.b.1.1
Level $3675$
Weight $2$
Character 3675.1
Self dual yes
Analytic conductor $29.345$
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] = [3675,2,Mod(1,3675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3675, 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("3675.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.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: \(29.3450227428\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3675.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} +2.00000 q^{6} +1.00000 q^{9} +2.00000 q^{11} -2.00000 q^{12} +1.00000 q^{13} -4.00000 q^{16} +2.00000 q^{17} -2.00000 q^{18} +5.00000 q^{19} -4.00000 q^{22} -6.00000 q^{23} -2.00000 q^{26} -1.00000 q^{27} +10.0000 q^{29} +3.00000 q^{31} +8.00000 q^{32} -2.00000 q^{33} -4.00000 q^{34} +2.00000 q^{36} -2.00000 q^{37} -10.0000 q^{38} -1.00000 q^{39} +8.00000 q^{41} -1.00000 q^{43} +4.00000 q^{44} +12.0000 q^{46} +2.00000 q^{47} +4.00000 q^{48} -2.00000 q^{51} +2.00000 q^{52} +4.00000 q^{53} +2.00000 q^{54} -5.00000 q^{57} -20.0000 q^{58} +10.0000 q^{59} -7.00000 q^{61} -6.00000 q^{62} -8.00000 q^{64} +4.00000 q^{66} +3.00000 q^{67} +4.00000 q^{68} +6.00000 q^{69} -8.00000 q^{71} -14.0000 q^{73} +4.00000 q^{74} +10.0000 q^{76} +2.00000 q^{78} +1.00000 q^{81} -16.0000 q^{82} +6.00000 q^{83} +2.00000 q^{86} -10.0000 q^{87} -12.0000 q^{92} -3.00000 q^{93} -4.00000 q^{94} -8.00000 q^{96} +17.0000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −2.00000 −0.577350
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −2.00000 −0.471405
\(19\) 5.00000 1.14708 0.573539 − 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 − 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 − 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 8.00000 1.41421
\(33\) −2.00000 −0.348155
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −10.0000 −1.62221
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 − 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 − 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 12.0000 1.76930
\(47\) 2.00000 0.291730 0.145865 − 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 4.00000 0.577350
\(49\) 0 0
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 2.00000 0.277350
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 2.00000 0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) −5.00000 −0.662266
\(58\) −20.0000 −2.62613
\(59\) 10.0000 1.30189 0.650945 − 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 − 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) −6.00000 −0.762001
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 3.00000 0.366508 0.183254 − 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) 4.00000 0.485071
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 10.0000 1.14708
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −16.0000 −1.76690
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) −10.0000 −1.07211
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −12.0000 −1.25109
\(93\) −3.00000 −0.311086
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) −8.00000 −0.816497
\(97\) 17.0000 1.72609 0.863044 − 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 − 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 4.00000 0.396059
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −8.00000 −0.777029
\(107\) −12.0000 −1.16008 −0.580042 − 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −2.00000 −0.192450
\(109\) 5.00000 0.478913 0.239457 − 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 4.00000 0.376288 0.188144 − 0.982141i \(-0.439753\pi\)
0.188144 + 0.982141i \(0.439753\pi\)
\(114\) 10.0000 0.936586
\(115\) 0 0
\(116\) 20.0000 1.85695
\(117\) 1.00000 0.0924500
\(118\) −20.0000 −1.84115
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 14.0000 1.26750
\(123\) −8.00000 −0.721336
\(124\) 6.00000 0.538816
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) −6.00000 −0.518321
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 − 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) −12.0000 −1.02151
\(139\) −20.0000 −1.69638 −0.848189 − 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) 16.0000 1.34269
\(143\) 2.00000 0.167248
\(144\) −4.00000 −0.333333
\(145\) 0 0
\(146\) 28.0000 2.31730
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) 10.0000 0.819232 0.409616 − 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 7.00000 0.569652 0.284826 − 0.958579i \(-0.408064\pi\)
0.284826 + 0.958579i \(0.408064\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) −13.0000 −1.03751 −0.518756 − 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 0 0
\(159\) −4.00000 −0.317221
\(160\) 0 0
\(161\) 0 0
\(162\) −2.00000 −0.157135
\(163\) −11.0000 −0.861586 −0.430793 − 0.902451i \(-0.641766\pi\)
−0.430793 + 0.902451i \(0.641766\pi\)
\(164\) 16.0000 1.24939
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 5.00000 0.382360
\(172\) −2.00000 −0.152499
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 20.0000 1.51620
\(175\) 0 0
\(176\) −8.00000 −0.603023
\(177\) −10.0000 −0.751646
\(178\) 0 0
\(179\) −10.0000 −0.747435 −0.373718 − 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) −17.0000 −1.26360 −0.631800 − 0.775131i \(-0.717684\pi\)
−0.631800 + 0.775131i \(0.717684\pi\)
\(182\) 0 0
\(183\) 7.00000 0.517455
\(184\) 0 0
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) 4.00000 0.292509
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) 0 0
\(191\) 22.0000 1.59186 0.795932 − 0.605386i \(-0.206981\pi\)
0.795932 + 0.605386i \(0.206981\pi\)
\(192\) 8.00000 0.577350
\(193\) −11.0000 −0.791797 −0.395899 − 0.918294i \(-0.629567\pi\)
−0.395899 + 0.918294i \(0.629567\pi\)
\(194\) −34.0000 −2.44106
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) −4.00000 −0.284268
\(199\) 5.00000 0.354441 0.177220 − 0.984171i \(-0.443289\pi\)
0.177220 + 0.984171i \(0.443289\pi\)
\(200\) 0 0
\(201\) −3.00000 −0.211604
\(202\) 24.0000 1.68863
\(203\) 0 0
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) −6.00000 −0.417029
\(208\) −4.00000 −0.277350
\(209\) 10.0000 0.691714
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 − 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 8.00000 0.549442
\(213\) 8.00000 0.548151
\(214\) 24.0000 1.64061
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) 2.00000 0.134535
\(222\) −4.00000 −0.268462
\(223\) −19.0000 −1.27233 −0.636167 − 0.771551i \(-0.719481\pi\)
−0.636167 + 0.771551i \(0.719481\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −8.00000 −0.532152
\(227\) −8.00000 −0.530979 −0.265489 − 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) −10.0000 −0.662266
\(229\) 15.0000 0.991228 0.495614 − 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 24.0000 1.57229 0.786146 − 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 20.0000 1.30189
\(237\) 0 0
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 − 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) 23.0000 1.48156 0.740780 − 0.671748i \(-0.234456\pi\)
0.740780 + 0.671748i \(0.234456\pi\)
\(242\) 14.0000 0.899954
\(243\) −1.00000 −0.0641500
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) 16.0000 1.02012
\(247\) 5.00000 0.318142
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 12.0000 0.748539 0.374270 − 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) −2.00000 −0.124515
\(259\) 0 0
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) 24.0000 1.48272
\(263\) −16.0000 −0.986602 −0.493301 − 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 6.00000 0.366508
\(269\) 10.0000 0.609711 0.304855 − 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −8.00000 −0.485071
\(273\) 0 0
\(274\) −36.0000 −2.17484
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) 3.00000 0.180253 0.0901263 − 0.995930i \(-0.471273\pi\)
0.0901263 + 0.995930i \(0.471273\pi\)
\(278\) 40.0000 2.39904
\(279\) 3.00000 0.179605
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 − 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 4.00000 0.238197
\(283\) −9.00000 −0.534994 −0.267497 − 0.963559i \(-0.586197\pi\)
−0.267497 + 0.963559i \(0.586197\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 8.00000 0.471405
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −17.0000 −0.996558
\(292\) −28.0000 −1.63858
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −2.00000 −0.116052
\(298\) −20.0000 −1.15857
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) −14.0000 −0.805609
\(303\) 12.0000 0.689382
\(304\) −20.0000 −1.14708
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) 7.00000 0.399511 0.199756 − 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 − 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 11.0000 0.621757 0.310878 − 0.950450i \(-0.399377\pi\)
0.310878 + 0.950450i \(0.399377\pi\)
\(314\) 26.0000 1.46726
\(315\) 0 0
\(316\) 0 0
\(317\) 8.00000 0.449325 0.224662 − 0.974437i \(-0.427872\pi\)
0.224662 + 0.974437i \(0.427872\pi\)
\(318\) 8.00000 0.448618
\(319\) 20.0000 1.11979
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 10.0000 0.556415
\(324\) 2.00000 0.111111
\(325\) 0 0
\(326\) 22.0000 1.21847
\(327\) −5.00000 −0.276501
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 − 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 12.0000 0.658586
\(333\) −2.00000 −0.109599
\(334\) −24.0000 −1.31322
\(335\) 0 0
\(336\) 0 0
\(337\) 23.0000 1.25289 0.626445 − 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 24.0000 1.30543
\(339\) −4.00000 −0.217250
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) −10.0000 −0.540738
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) −2.00000 −0.107366 −0.0536828 − 0.998558i \(-0.517096\pi\)
−0.0536828 + 0.998558i \(0.517096\pi\)
\(348\) −20.0000 −1.07211
\(349\) −10.0000 −0.535288 −0.267644 − 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 16.0000 0.852803
\(353\) 6.00000 0.319348 0.159674 − 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 20.0000 1.06299
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 20.0000 1.05703
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 34.0000 1.78700
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) −14.0000 −0.731792
\(367\) 27.0000 1.40939 0.704694 − 0.709511i \(-0.251084\pi\)
0.704694 + 0.709511i \(0.251084\pi\)
\(368\) 24.0000 1.25109
\(369\) 8.00000 0.416463
\(370\) 0 0
\(371\) 0 0
\(372\) −6.00000 −0.311086
\(373\) 29.0000 1.50156 0.750782 − 0.660551i \(-0.229677\pi\)
0.750782 + 0.660551i \(0.229677\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) 10.0000 0.515026
\(378\) 0 0
\(379\) 25.0000 1.28416 0.642082 − 0.766636i \(-0.278071\pi\)
0.642082 + 0.766636i \(0.278071\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) −44.0000 −2.25124
\(383\) 36.0000 1.83951 0.919757 − 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) −1.00000 −0.0508329
\(388\) 34.0000 1.72609
\(389\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) −36.0000 −1.81365
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) 7.00000 0.351320 0.175660 − 0.984451i \(-0.443794\pi\)
0.175660 + 0.984451i \(0.443794\pi\)
\(398\) −10.0000 −0.501255
\(399\) 0 0
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 − 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 6.00000 0.299253
\(403\) 3.00000 0.149441
\(404\) −24.0000 −1.19404
\(405\) 0 0
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) −5.00000 −0.247234 −0.123617 − 0.992330i \(-0.539449\pi\)
−0.123617 + 0.992330i \(0.539449\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 12.0000 0.589768
\(415\) 0 0
\(416\) 8.00000 0.392232
\(417\) 20.0000 0.979404
\(418\) −20.0000 −0.978232
\(419\) 20.0000 0.977064 0.488532 − 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 − 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 26.0000 1.26566
\(423\) 2.00000 0.0972433
\(424\) 0 0
\(425\) 0 0
\(426\) −16.0000 −0.775203
\(427\) 0 0
\(428\) −24.0000 −1.16008
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 − 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 4.00000 0.192450
\(433\) −29.0000 −1.39365 −0.696826 − 0.717241i \(-0.745405\pi\)
−0.696826 + 0.717241i \(0.745405\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) −30.0000 −1.43509
\(438\) −28.0000 −1.33789
\(439\) 35.0000 1.67046 0.835229 − 0.549902i \(-0.185335\pi\)
0.835229 + 0.549902i \(0.185335\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 4.00000 0.189832
\(445\) 0 0
\(446\) 38.0000 1.79935
\(447\) −10.0000 −0.472984
\(448\) 0 0
\(449\) 20.0000 0.943858 0.471929 − 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) 16.0000 0.753411
\(452\) 8.00000 0.376288
\(453\) −7.00000 −0.328889
\(454\) 16.0000 0.750917
\(455\) 0 0
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −30.0000 −1.40181
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 − 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 − 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) −40.0000 −1.85695
\(465\) 0 0
\(466\) −48.0000 −2.22356
\(467\) −38.0000 −1.75843 −0.879215 − 0.476425i \(-0.841932\pi\)
−0.879215 + 0.476425i \(0.841932\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) 13.0000 0.599008
\(472\) 0 0
\(473\) −2.00000 −0.0919601
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 4.00000 0.183147
\(478\) −40.0000 −1.82956
\(479\) −30.0000 −1.37073 −0.685367 − 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) −46.0000 −2.09524
\(483\) 0 0
\(484\) −14.0000 −0.636364
\(485\) 0 0
\(486\) 2.00000 0.0907218
\(487\) 13.0000 0.589086 0.294543 − 0.955638i \(-0.404833\pi\)
0.294543 + 0.955638i \(0.404833\pi\)
\(488\) 0 0
\(489\) 11.0000 0.497437
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 − 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) −16.0000 −0.721336
\(493\) 20.0000 0.900755
\(494\) −10.0000 −0.449921
\(495\) 0 0
\(496\) −12.0000 −0.538816
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) −5.00000 −0.223831 −0.111915 − 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 24.0000 1.07117
\(503\) 16.0000 0.713405 0.356702 − 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 24.0000 1.06693
\(507\) 12.0000 0.532939
\(508\) 16.0000 0.709885
\(509\) 10.0000 0.443242 0.221621 − 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −32.0000 −1.41421
\(513\) −5.00000 −0.220755
\(514\) −24.0000 −1.05859
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) 4.00000 0.175920
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 − 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) −20.0000 −0.875376
\(523\) 31.0000 1.35554 0.677768 − 0.735276i \(-0.262948\pi\)
0.677768 + 0.735276i \(0.262948\pi\)
\(524\) −24.0000 −1.04844
\(525\) 0 0
\(526\) 32.0000 1.39527
\(527\) 6.00000 0.261364
\(528\) 8.00000 0.348155
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 10.0000 0.433963
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 10.0000 0.431532
\(538\) −20.0000 −0.862261
\(539\) 0 0
\(540\) 0 0
\(541\) −3.00000 −0.128980 −0.0644900 − 0.997918i \(-0.520542\pi\)
−0.0644900 + 0.997918i \(0.520542\pi\)
\(542\) −16.0000 −0.687259
\(543\) 17.0000 0.729540
\(544\) 16.0000 0.685994
\(545\) 0 0
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 − 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 36.0000 1.53784
\(549\) −7.00000 −0.298753
\(550\) 0 0
\(551\) 50.0000 2.13007
\(552\) 0 0
\(553\) 0 0
\(554\) −6.00000 −0.254916
\(555\) 0 0
\(556\) −40.0000 −1.69638
\(557\) −42.0000 −1.77960 −0.889799 − 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) −6.00000 −0.254000
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 36.0000 1.51857
\(563\) 6.00000 0.252870 0.126435 − 0.991975i \(-0.459647\pi\)
0.126435 + 0.991975i \(0.459647\pi\)
\(564\) −4.00000 −0.168430
\(565\) 0 0
\(566\) 18.0000 0.756596
\(567\) 0 0
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 − 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −13.0000 −0.544033 −0.272017 − 0.962293i \(-0.587691\pi\)
−0.272017 + 0.962293i \(0.587691\pi\)
\(572\) 4.00000 0.167248
\(573\) −22.0000 −0.919063
\(574\) 0 0
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) −13.0000 −0.541197 −0.270599 − 0.962692i \(-0.587222\pi\)
−0.270599 + 0.962692i \(0.587222\pi\)
\(578\) 26.0000 1.08146
\(579\) 11.0000 0.457144
\(580\) 0 0
\(581\) 0 0
\(582\) 34.0000 1.40935
\(583\) 8.00000 0.331326
\(584\) 0 0
\(585\) 0 0
\(586\) −12.0000 −0.495715
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 15.0000 0.618064
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 8.00000 0.328798
\(593\) 16.0000 0.657041 0.328521 − 0.944497i \(-0.393450\pi\)
0.328521 + 0.944497i \(0.393450\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 20.0000 0.819232
\(597\) −5.00000 −0.204636
\(598\) 12.0000 0.490716
\(599\) −20.0000 −0.817178 −0.408589 − 0.912719i \(-0.633979\pi\)
−0.408589 + 0.912719i \(0.633979\pi\)
\(600\) 0 0
\(601\) 13.0000 0.530281 0.265141 − 0.964210i \(-0.414582\pi\)
0.265141 + 0.964210i \(0.414582\pi\)
\(602\) 0 0
\(603\) 3.00000 0.122169
\(604\) 14.0000 0.569652
\(605\) 0 0
\(606\) −24.0000 −0.974933
\(607\) −8.00000 −0.324710 −0.162355 − 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 40.0000 1.62221
\(609\) 0 0
\(610\) 0 0
\(611\) 2.00000 0.0809113
\(612\) 4.00000 0.161690
\(613\) 14.0000 0.565455 0.282727 − 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −14.0000 −0.564994
\(615\) 0 0
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 − 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) −8.00000 −0.321807
\(619\) 25.0000 1.00483 0.502417 − 0.864625i \(-0.332444\pi\)
0.502417 + 0.864625i \(0.332444\pi\)
\(620\) 0 0
\(621\) 6.00000 0.240772
\(622\) −36.0000 −1.44347
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −22.0000 −0.879297
\(627\) −10.0000 −0.399362
\(628\) −26.0000 −1.03751
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) −23.0000 −0.915616 −0.457808 − 0.889051i \(-0.651365\pi\)
−0.457808 + 0.889051i \(0.651365\pi\)
\(632\) 0 0
\(633\) 13.0000 0.516704
\(634\) −16.0000 −0.635441
\(635\) 0 0
\(636\) −8.00000 −0.317221
\(637\) 0 0
\(638\) −40.0000 −1.58362
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 12.0000 0.473972 0.236986 − 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) −24.0000 −0.947204
\(643\) 36.0000 1.41970 0.709851 − 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −20.0000 −0.786889
\(647\) −28.0000 −1.10079 −0.550397 − 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 0 0
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) −22.0000 −0.861586
\(653\) 14.0000 0.547862 0.273931 − 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) −32.0000 −1.24939
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) −40.0000 −1.55818 −0.779089 − 0.626913i \(-0.784318\pi\)
−0.779089 + 0.626913i \(0.784318\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 − 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) −24.0000 −0.932786
\(663\) −2.00000 −0.0776736
\(664\) 0 0
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) −60.0000 −2.32321
\(668\) 24.0000 0.928588
\(669\) 19.0000 0.734582
\(670\) 0 0
\(671\) −14.0000 −0.540464
\(672\) 0 0
\(673\) −6.00000 −0.231283 −0.115642 − 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) −46.0000 −1.77185
\(675\) 0 0
\(676\) −24.0000 −0.923077
\(677\) 42.0000 1.61419 0.807096 − 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 8.00000 0.307238
\(679\) 0 0
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) −12.0000 −0.459504
\(683\) 24.0000 0.918334 0.459167 − 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 10.0000 0.382360
\(685\) 0 0
\(686\) 0 0
\(687\) −15.0000 −0.572286
\(688\) 4.00000 0.152499
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 − 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 12.0000 0.456172
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) 20.0000 0.757011
\(699\) −24.0000 −0.907763
\(700\) 0 0
\(701\) 22.0000 0.830929 0.415464 − 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 2.00000 0.0754851
\(703\) −10.0000 −0.377157
\(704\) −16.0000 −0.603023
\(705\) 0 0
\(706\) −12.0000 −0.451626
\(707\) 0 0
\(708\) −20.0000 −0.751646
\(709\) 25.0000 0.938895 0.469447 − 0.882960i \(-0.344453\pi\)
0.469447 + 0.882960i \(0.344453\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −18.0000 −0.674105
\(714\) 0 0
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) −20.0000 −0.746914
\(718\) 0 0
\(719\) −30.0000 −1.11881 −0.559406 − 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −12.0000 −0.446594
\(723\) −23.0000 −0.855379
\(724\) −34.0000 −1.26360
\(725\) 0 0
\(726\) −14.0000 −0.519589
\(727\) −43.0000 −1.59478 −0.797391 − 0.603463i \(-0.793787\pi\)
−0.797391 + 0.603463i \(0.793787\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −2.00000 −0.0739727
\(732\) 14.0000 0.517455
\(733\) −34.0000 −1.25582 −0.627909 − 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) −54.0000 −1.99318
\(735\) 0 0
\(736\) −48.0000 −1.76930
\(737\) 6.00000 0.221013
\(738\) −16.0000 −0.588968
\(739\) −20.0000 −0.735712 −0.367856 − 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −5.00000 −0.183680
\(742\) 0 0
\(743\) 4.00000 0.146746 0.0733729 − 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −58.0000 −2.12353
\(747\) 6.00000 0.219529
\(748\) 8.00000 0.292509
\(749\) 0 0
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 − 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) −8.00000 −0.291730
\(753\) 12.0000 0.437304
\(754\) −20.0000 −0.728357
\(755\) 0 0
\(756\) 0 0
\(757\) 23.0000 0.835949 0.417975 − 0.908459i \(-0.362740\pi\)
0.417975 + 0.908459i \(0.362740\pi\)
\(758\) −50.0000 −1.81608
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 − 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 16.0000 0.579619
\(763\) 0 0
\(764\) 44.0000 1.59186
\(765\) 0 0
\(766\) −72.0000 −2.60147
\(767\) 10.0000 0.361079
\(768\) −16.0000 −0.577350
\(769\) −35.0000 −1.26213 −0.631066 − 0.775729i \(-0.717382\pi\)
−0.631066 + 0.775729i \(0.717382\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) −22.0000 −0.791797
\(773\) −24.0000 −0.863220 −0.431610 − 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 2.00000 0.0718885
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 40.0000 1.43315
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 24.0000 0.858238
\(783\) −10.0000 −0.357371
\(784\) 0 0
\(785\) 0 0
\(786\) −24.0000 −0.856052
\(787\) 7.00000 0.249523 0.124762 − 0.992187i \(-0.460183\pi\)
0.124762 + 0.992187i \(0.460183\pi\)
\(788\) 36.0000 1.28245
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −7.00000 −0.248577
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 10.0000 0.354441
\(797\) 52.0000 1.84193 0.920967 − 0.389640i \(-0.127401\pi\)
0.920967 + 0.389640i \(0.127401\pi\)
\(798\) 0 0
\(799\) 4.00000 0.141510
\(800\) 0 0
\(801\) 0 0
\(802\) −24.0000 −0.847469
\(803\) −28.0000 −0.988099
\(804\) −6.00000 −0.211604
\(805\) 0 0
\(806\) −6.00000 −0.211341
\(807\) −10.0000 −0.352017
\(808\) 0 0
\(809\) −20.0000 −0.703163 −0.351581 − 0.936157i \(-0.614356\pi\)
−0.351581 + 0.936157i \(0.614356\pi\)
\(810\) 0 0
\(811\) −27.0000 −0.948098 −0.474049 − 0.880498i \(-0.657208\pi\)
−0.474049 + 0.880498i \(0.657208\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) 8.00000 0.280056
\(817\) −5.00000 −0.174928
\(818\) 10.0000 0.349642
\(819\) 0 0
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 − 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 36.0000 1.25564
\(823\) −41.0000 −1.42917 −0.714585 − 0.699549i \(-0.753384\pi\)
−0.714585 + 0.699549i \(0.753384\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 − 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) −12.0000 −0.417029
\(829\) 30.0000 1.04194 0.520972 − 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 0 0
\(831\) −3.00000 −0.104069
\(832\) −8.00000 −0.277350
\(833\) 0 0
\(834\) −40.0000 −1.38509
\(835\) 0 0
\(836\) 20.0000 0.691714
\(837\) −3.00000 −0.103695
\(838\) −40.0000 −1.38178
\(839\) −10.0000 −0.345238 −0.172619 − 0.984989i \(-0.555223\pi\)
−0.172619 + 0.984989i \(0.555223\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) −44.0000 −1.51634
\(843\) 18.0000 0.619953
\(844\) −26.0000 −0.894957
\(845\) 0 0
\(846\) −4.00000 −0.137523
\(847\) 0 0
\(848\) −16.0000 −0.549442
\(849\) 9.00000 0.308879
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) 16.0000 0.548151
\(853\) 51.0000 1.74621 0.873103 − 0.487535i \(-0.162104\pi\)
0.873103 + 0.487535i \(0.162104\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −28.0000 −0.956462 −0.478231 − 0.878234i \(-0.658722\pi\)
−0.478231 + 0.878234i \(0.658722\pi\)
\(858\) 4.00000 0.136558
\(859\) −40.0000 −1.36478 −0.682391 − 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 36.0000 1.22616
\(863\) −16.0000 −0.544646 −0.272323 − 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) −8.00000 −0.272166
\(865\) 0 0
\(866\) 58.0000 1.97092
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 3.00000 0.101651
\(872\) 0 0
\(873\) 17.0000 0.575363
\(874\) 60.0000 2.02953
\(875\) 0 0
\(876\) 28.0000 0.946032
\(877\) −27.0000 −0.911725 −0.455863 − 0.890050i \(-0.650669\pi\)
−0.455863 + 0.890050i \(0.650669\pi\)
\(878\) −70.0000 −2.36239
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −32.0000 −1.07811 −0.539054 − 0.842271i \(-0.681218\pi\)
−0.539054 + 0.842271i \(0.681218\pi\)
\(882\) 0 0
\(883\) −41.0000 −1.37976 −0.689880 − 0.723924i \(-0.742337\pi\)
−0.689880 + 0.723924i \(0.742337\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −48.0000 −1.61259
\(887\) −18.0000 −0.604381 −0.302190 − 0.953248i \(-0.597718\pi\)
−0.302190 + 0.953248i \(0.597718\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) −38.0000 −1.27233
\(893\) 10.0000 0.334637
\(894\) 20.0000 0.668900
\(895\) 0 0
\(896\) 0 0
\(897\) 6.00000 0.200334
\(898\) −40.0000 −1.33482
\(899\) 30.0000 1.00056
\(900\) 0 0
\(901\) 8.00000 0.266519
\(902\) −32.0000 −1.06548
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 14.0000 0.465119
\(907\) −12.0000 −0.398453 −0.199227 − 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) −16.0000 −0.530979
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −58.0000 −1.92163 −0.960813 − 0.277198i \(-0.910594\pi\)
−0.960813 + 0.277198i \(0.910594\pi\)
\(912\) 20.0000 0.662266
\(913\) 12.0000 0.397142
\(914\) 44.0000 1.45539
\(915\) 0 0
\(916\) 30.0000 0.991228
\(917\) 0 0
\(918\) 4.00000 0.132020
\(919\) 55.0000 1.81428 0.907141 − 0.420826i \(-0.138260\pi\)
0.907141 + 0.420826i \(0.138260\pi\)
\(920\) 0 0
\(921\) −7.00000 −0.230658
\(922\) 24.0000 0.790398
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) 0 0
\(926\) −48.0000 −1.57738
\(927\) −4.00000 −0.131377
\(928\) 80.0000 2.62613
\(929\) 50.0000 1.64045 0.820223 − 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 48.0000 1.57229
\(933\) −18.0000 −0.589294
\(934\) 76.0000 2.48680
\(935\) 0 0
\(936\) 0 0
\(937\) −33.0000 −1.07806 −0.539032 − 0.842286i \(-0.681210\pi\)
−0.539032 + 0.842286i \(0.681210\pi\)
\(938\) 0 0
\(939\) −11.0000 −0.358971
\(940\) 0 0
\(941\) −22.0000 −0.717180 −0.358590 − 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) −26.0000 −0.847126
\(943\) −48.0000 −1.56310
\(944\) −40.0000 −1.30189
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 18.0000 0.584921 0.292461 − 0.956278i \(-0.405526\pi\)
0.292461 + 0.956278i \(0.405526\pi\)
\(948\) 0 0
\(949\) −14.0000 −0.454459
\(950\) 0 0
\(951\) −8.00000 −0.259418
\(952\) 0 0
\(953\) −56.0000 −1.81402 −0.907009 − 0.421111i \(-0.861640\pi\)
−0.907009 + 0.421111i \(0.861640\pi\)
\(954\) −8.00000 −0.259010
\(955\) 0 0
\(956\) 40.0000 1.29369
\(957\) −20.0000 −0.646508
\(958\) 60.0000 1.93851
\(959\) 0 0
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 4.00000 0.128965
\(963\) −12.0000 −0.386695
\(964\) 46.0000 1.48156
\(965\) 0 0
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 − 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) −10.0000 −0.321246
\(970\) 0 0
\(971\) −42.0000 −1.34784 −0.673922 − 0.738802i \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) −2.00000 −0.0641500
\(973\) 0 0
\(974\) −26.0000 −0.833094
\(975\) 0 0
\(976\) 28.0000 0.896258
\(977\) −2.00000 −0.0639857 −0.0319928 − 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) −22.0000 −0.703482
\(979\) 0 0
\(980\) 0 0
\(981\) 5.00000 0.159638
\(982\) 16.0000 0.510581
\(983\) 36.0000 1.14822 0.574111 − 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −40.0000 −1.27386
\(987\) 0 0
\(988\) 10.0000 0.318142
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) 17.0000 0.540023 0.270011 − 0.962857i \(-0.412973\pi\)
0.270011 + 0.962857i \(0.412973\pi\)
\(992\) 24.0000 0.762001
\(993\) −12.0000 −0.380808
\(994\) 0 0
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) 42.0000 1.33015 0.665077 − 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) 10.0000 0.316544
\(999\) 2.00000 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 3675.2.a.b.1.1 1
5.4 even 2 3675.2.a.q.1.1 1
7.6 odd 2 75.2.a.a.1.1 ✓ 1
21.20 even 2 225.2.a.e.1.1 1
28.27 even 2 1200.2.a.c.1.1 1
35.13 even 4 75.2.b.a.49.2 2
35.27 even 4 75.2.b.a.49.1 2
35.34 odd 2 75.2.a.c.1.1 yes 1
56.13 odd 2 4800.2.a.bb.1.1 1
56.27 even 2 4800.2.a.br.1.1 1
77.76 even 2 9075.2.a.s.1.1 1
84.83 odd 2 3600.2.a.j.1.1 1
105.62 odd 4 225.2.b.a.199.2 2
105.83 odd 4 225.2.b.a.199.1 2
105.104 even 2 225.2.a.a.1.1 1
140.27 odd 4 1200.2.f.d.49.2 2
140.83 odd 4 1200.2.f.d.49.1 2
140.139 even 2 1200.2.a.p.1.1 1
280.13 even 4 4800.2.f.l.3649.1 2
280.27 odd 4 4800.2.f.y.3649.1 2
280.69 odd 2 4800.2.a.bq.1.1 1
280.83 odd 4 4800.2.f.y.3649.2 2
280.139 even 2 4800.2.a.be.1.1 1
280.237 even 4 4800.2.f.l.3649.2 2
385.384 even 2 9075.2.a.a.1.1 1
420.83 even 4 3600.2.f.p.2449.2 2
420.167 even 4 3600.2.f.p.2449.1 2
420.419 odd 2 3600.2.a.bk.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.a.a.1.1 ✓ 1 7.6 odd 2
75.2.a.c.1.1 yes 1 35.34 odd 2
75.2.b.a.49.1 2 35.27 even 4
75.2.b.a.49.2 2 35.13 even 4
225.2.a.a.1.1 1 105.104 even 2
225.2.a.e.1.1 1 21.20 even 2
225.2.b.a.199.1 2 105.83 odd 4
225.2.b.a.199.2 2 105.62 odd 4
1200.2.a.c.1.1 1 28.27 even 2
1200.2.a.p.1.1 1 140.139 even 2
1200.2.f.d.49.1 2 140.83 odd 4
1200.2.f.d.49.2 2 140.27 odd 4
3600.2.a.j.1.1 1 84.83 odd 2
3600.2.a.bk.1.1 1 420.419 odd 2
3600.2.f.p.2449.1 2 420.167 even 4
3600.2.f.p.2449.2 2 420.83 even 4
3675.2.a.b.1.1 1 1.1 even 1 trivial
3675.2.a.q.1.1 1 5.4 even 2
4800.2.a.bb.1.1 1 56.13 odd 2
4800.2.a.be.1.1 1 280.139 even 2
4800.2.a.bq.1.1 1 280.69 odd 2
4800.2.a.br.1.1 1 56.27 even 2
4800.2.f.l.3649.1 2 280.13 even 4
4800.2.f.l.3649.2 2 280.237 even 4
4800.2.f.y.3649.1 2 280.27 odd 4
4800.2.f.y.3649.2 2 280.83 odd 4
9075.2.a.a.1.1 1 385.384 even 2
9075.2.a.s.1.1 1 77.76 even 2