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Properties

Label 1925.2.a.k.1.1
Level $1925$
Weight $2$
Character 1925.1
Self dual yes
Analytic conductor $15.371$
Analytic rank $1$
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] = [1925,2,Mod(1,1925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1925, 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("1925.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1925.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: \(15.3712023891\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 385)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1925.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} -2.00000 q^{12} -6.00000 q^{13} -1.00000 q^{14} -1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} -2.00000 q^{21} +1.00000 q^{22} +6.00000 q^{23} -6.00000 q^{24} -6.00000 q^{26} -4.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} +5.00000 q^{32} +2.00000 q^{33} -6.00000 q^{34} -1.00000 q^{36} +4.00000 q^{37} -12.0000 q^{39} -6.00000 q^{41} -2.00000 q^{42} -8.00000 q^{43} -1.00000 q^{44} +6.00000 q^{46} -10.0000 q^{47} -2.00000 q^{48} +1.00000 q^{49} -12.0000 q^{51} +6.00000 q^{52} +4.00000 q^{53} -4.00000 q^{54} +3.00000 q^{56} -6.00000 q^{58} +12.0000 q^{59} -10.0000 q^{61} -1.00000 q^{63} +7.00000 q^{64} +2.00000 q^{66} -2.00000 q^{67} +6.00000 q^{68} +12.0000 q^{69} +8.00000 q^{71} -3.00000 q^{72} +2.00000 q^{73} +4.00000 q^{74} -1.00000 q^{77} -12.0000 q^{78} +8.00000 q^{79} -11.0000 q^{81} -6.00000 q^{82} +2.00000 q^{84} -8.00000 q^{86} -12.0000 q^{87} -3.00000 q^{88} -6.00000 q^{89} +6.00000 q^{91} -6.00000 q^{92} -10.0000 q^{94} +10.0000 q^{96} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 − 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 2.00000 1.15470 0.577350 − 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) −1.00000 −0.377964
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −2.00000 −0.577350
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 1.00000 0.213201
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −6.00000 −1.22474
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) −4.00000 −0.769800
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.00000 0.883883
\(33\) 2.00000 0.348155
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) −12.0000 −1.92154
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −2.00000 −0.308607
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −10.0000 −1.45865 −0.729325 − 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) −2.00000 −0.288675
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −12.0000 −1.68034
\(52\) 6.00000 0.832050
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 6.00000 0.727607
\(69\) 12.0000 1.44463
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −3.00000 −0.353553
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) −12.0000 −1.35873
\(79\) 8.00000 0.900070 0.450035 − 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) −6.00000 −0.662589
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) −12.0000 −1.28654
\(88\) −3.00000 −0.319801
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) −10.0000 −1.03142
\(95\) 0 0
\(96\) 10.0000 1.02062
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 1.00000 0.101015
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 − 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) −12.0000 −1.18818
\(103\) 14.0000 1.37946 0.689730 − 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 18.0000 1.76505
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) 12.0000 1.16008 0.580042 − 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 4.00000 0.384900
\(109\) 6.00000 0.574696 0.287348 − 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 1.00000 0.0944911
\(113\) −12.0000 −1.12887 −0.564433 − 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) −6.00000 −0.554700
\(118\) 12.0000 1.10469
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −10.0000 −0.905357
\(123\) −12.0000 −1.08200
\(124\) 0 0
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) −12.0000 −1.06483 −0.532414 − 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −3.00000 −0.265165
\(129\) −16.0000 −1.40872
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) 18.0000 1.54349
\(137\) −20.0000 −1.70872 −0.854358 − 0.519685i \(-0.826049\pi\)
−0.854358 + 0.519685i \(0.826049\pi\)
\(138\) 12.0000 1.02151
\(139\) 12.0000 1.01783 0.508913 − 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −20.0000 −1.68430
\(142\) 8.00000 0.671345
\(143\) −6.00000 −0.501745
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 2.00000 0.164957
\(148\) −4.00000 −0.328798
\(149\) −10.0000 −0.819232 −0.409616 − 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 − 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) 12.0000 0.960769
\(157\) 4.00000 0.319235 0.159617 − 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) 8.00000 0.636446
\(159\) 8.00000 0.634441
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) −11.0000 −0.864242
\(163\) −18.0000 −1.40987 −0.704934 − 0.709273i \(-0.749024\pi\)
−0.704934 + 0.709273i \(0.749024\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) 20.0000 1.54765 0.773823 − 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) 6.00000 0.462910
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −18.0000 −1.36851 −0.684257 − 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −12.0000 −0.909718
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 24.0000 1.80395
\(178\) −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
\(181\) 10.0000 0.743294 0.371647 − 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 6.00000 0.444750
\(183\) −20.0000 −1.47844
\(184\) −18.0000 −1.32698
\(185\) 0 0
\(186\) 0 0
\(187\) −6.00000 −0.438763
\(188\) 10.0000 0.729325
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 − 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 14.0000 1.01036
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 1.00000 0.0710669
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) −14.0000 −0.985037
\(203\) 6.00000 0.421117
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) 14.0000 0.975426
\(207\) 6.00000 0.417029
\(208\) 6.00000 0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 − 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −4.00000 −0.274721
\(213\) 16.0000 1.09630
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 12.0000 0.816497
\(217\) 0 0
\(218\) 6.00000 0.406371
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 36.0000 2.42162
\(222\) 8.00000 0.536925
\(223\) 2.00000 0.133930 0.0669650 − 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) −5.00000 −0.334077
\(225\) 0 0
\(226\) −12.0000 −0.798228
\(227\) −8.00000 −0.530979 −0.265489 − 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 18.0000 1.18176
\(233\) 22.0000 1.44127 0.720634 − 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 16.0000 1.03931
\(238\) 6.00000 0.388922
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 1.00000 0.0642824
\(243\) −10.0000 −0.641500
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) −12.0000 −0.765092
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 1.00000 0.0629941
\(253\) 6.00000 0.377217
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −4.00000 −0.249513 −0.124757 − 0.992187i \(-0.539815\pi\)
−0.124757 + 0.992187i \(0.539815\pi\)
\(258\) −16.0000 −0.996116
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 12.0000 0.741362
\(263\) 28.0000 1.72655 0.863277 − 0.504730i \(-0.168408\pi\)
0.863277 + 0.504730i \(0.168408\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) 2.00000 0.122169
\(269\) −30.0000 −1.82913 −0.914566 − 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 − 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 6.00000 0.363803
\(273\) 12.0000 0.726273
\(274\) −20.0000 −1.20824
\(275\) 0 0
\(276\) −12.0000 −0.722315
\(277\) 2.00000 0.120168 0.0600842 − 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 12.0000 0.719712
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 − 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −20.0000 −1.19098
\(283\) −16.0000 −0.951101 −0.475551 − 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 6.00000 0.354169
\(288\) 5.00000 0.294628
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) 26.0000 1.51894 0.759468 − 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) −12.0000 −0.697486
\(297\) −4.00000 −0.232104
\(298\) −10.0000 −0.579284
\(299\) −36.0000 −2.08193
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 4.00000 0.230174
\(303\) −28.0000 −1.60856
\(304\) 0 0
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) −20.0000 −1.14146 −0.570730 − 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 1.00000 0.0569803
\(309\) 28.0000 1.59286
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 − 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 36.0000 2.03810
\(313\) −28.0000 −1.58265 −0.791327 − 0.611393i \(-0.790609\pi\)
−0.791327 + 0.611393i \(0.790609\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 8.00000 0.449325 0.224662 − 0.974437i \(-0.427872\pi\)
0.224662 + 0.974437i \(0.427872\pi\)
\(318\) 8.00000 0.448618
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) 24.0000 1.33955
\(322\) −6.00000 −0.334367
\(323\) 0 0
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) −18.0000 −0.996928
\(327\) 12.0000 0.663602
\(328\) 18.0000 0.993884
\(329\) 10.0000 0.551318
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 − 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 4.00000 0.219199
\(334\) 20.0000 1.09435
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) −6.00000 −0.326841 −0.163420 − 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 23.0000 1.25104
\(339\) −24.0000 −1.30350
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 24.0000 1.29399
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 4.00000 0.214731 0.107366 − 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 12.0000 0.643268
\(349\) −10.0000 −0.535288 −0.267644 − 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 24.0000 1.28103
\(352\) 5.00000 0.266501
\(353\) −20.0000 −1.06449 −0.532246 − 0.846590i \(-0.678652\pi\)
−0.532246 + 0.846590i \(0.678652\pi\)
\(354\) 24.0000 1.27559
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 12.0000 0.635107
\(358\) 12.0000 0.634220
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 10.0000 0.525588
\(363\) 2.00000 0.104973
\(364\) −6.00000 −0.314485
\(365\) 0 0
\(366\) −20.0000 −1.04542
\(367\) 22.0000 1.14839 0.574195 − 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) −6.00000 −0.312772
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) 14.0000 0.724893 0.362446 − 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) 30.0000 1.54713
\(377\) 36.0000 1.85409
\(378\) 4.00000 0.205738
\(379\) 12.0000 0.616399 0.308199 − 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) −24.0000 −1.22956
\(382\) 8.00000 0.409316
\(383\) −18.0000 −0.919757 −0.459879 − 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) −6.00000 −0.306186
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −8.00000 −0.406663
\(388\) 0 0
\(389\) 26.0000 1.31825 0.659126 − 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) −3.00000 −0.151523
\(393\) 24.0000 1.21064
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) −28.0000 −1.40528 −0.702640 − 0.711546i \(-0.747995\pi\)
−0.702640 + 0.711546i \(0.747995\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) 0 0
\(401\) −2.00000 −0.0998752 −0.0499376 − 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) −4.00000 −0.199502
\(403\) 0 0
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) 4.00000 0.198273
\(408\) 36.0000 1.78227
\(409\) 14.0000 0.692255 0.346128 − 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) −40.0000 −1.97305
\(412\) −14.0000 −0.689730
\(413\) −12.0000 −0.590481
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) −30.0000 −1.47087
\(417\) 24.0000 1.17529
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 − 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 − 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) 8.00000 0.389434
\(423\) −10.0000 −0.486217
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) 10.0000 0.483934
\(428\) −12.0000 −0.580042
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 4.00000 0.192450
\(433\) −16.0000 −0.768911 −0.384455 − 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) 0 0
\(438\) 4.00000 0.191127
\(439\) 40.0000 1.90910 0.954548 − 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 36.0000 1.71235
\(443\) −26.0000 −1.23530 −0.617649 − 0.786454i \(-0.711915\pi\)
−0.617649 + 0.786454i \(0.711915\pi\)
\(444\) −8.00000 −0.379663
\(445\) 0 0
\(446\) 2.00000 0.0947027
\(447\) −20.0000 −0.945968
\(448\) −7.00000 −0.330719
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 12.0000 0.564433
\(453\) 8.00000 0.375873
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) 0 0
\(457\) −2.00000 −0.0935561 −0.0467780 − 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) −22.0000 −1.02799
\(459\) 24.0000 1.12022
\(460\) 0 0
\(461\) 10.0000 0.465746 0.232873 − 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) −2.00000 −0.0930484
\(463\) −26.0000 −1.20832 −0.604161 − 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 22.0000 1.01913
\(467\) −30.0000 −1.38823 −0.694117 − 0.719862i \(-0.744205\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) 6.00000 0.277350
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) 8.00000 0.368621
\(472\) −36.0000 −1.65703
\(473\) −8.00000 −0.367840
\(474\) 16.0000 0.734904
\(475\) 0 0
\(476\) −6.00000 −0.275010
\(477\) 4.00000 0.183147
\(478\) 0 0
\(479\) −4.00000 −0.182765 −0.0913823 − 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) 10.0000 0.455488
\(483\) −12.0000 −0.546019
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) 2.00000 0.0906287 0.0453143 − 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 30.0000 1.35804
\(489\) −36.0000 −1.62798
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 12.0000 0.541002
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) −36.0000 −1.61158 −0.805791 − 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) 40.0000 1.78707
\(502\) −12.0000 −0.535586
\(503\) 24.0000 1.07011 0.535054 − 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) 6.00000 0.266733
\(507\) 46.0000 2.04293
\(508\) 12.0000 0.532414
\(509\) 18.0000 0.797836 0.398918 − 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) −4.00000 −0.176432
\(515\) 0 0
\(516\) 16.0000 0.704361
\(517\) −10.0000 −0.439799
\(518\) −4.00000 −0.175750
\(519\) −36.0000 −1.58022
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 − 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) −6.00000 −0.262613
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 28.0000 1.22086
\(527\) 0 0
\(528\) −2.00000 −0.0870388
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 36.0000 1.55933
\(534\) −12.0000 −0.519291
\(535\) 0 0
\(536\) 6.00000 0.259161
\(537\) 24.0000 1.03568
\(538\) −30.0000 −1.29339
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −26.0000 −1.11783 −0.558914 − 0.829226i \(-0.688782\pi\)
−0.558914 + 0.829226i \(0.688782\pi\)
\(542\) 16.0000 0.687259
\(543\) 20.0000 0.858282
\(544\) −30.0000 −1.28624
\(545\) 0 0
\(546\) 12.0000 0.513553
\(547\) −8.00000 −0.342055 −0.171028 − 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 20.0000 0.854358
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 0 0
\(552\) −36.0000 −1.53226
\(553\) −8.00000 −0.340195
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) −18.0000 −0.762684 −0.381342 − 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 48.0000 2.03018
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) 6.00000 0.253095
\(563\) −12.0000 −0.505740 −0.252870 − 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 20.0000 0.842152
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) 11.0000 0.461957
\(568\) −24.0000 −1.00702
\(569\) 10.0000 0.419222 0.209611 − 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 − 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 6.00000 0.250873
\(573\) 16.0000 0.668410
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −8.00000 −0.333044 −0.166522 − 0.986038i \(-0.553254\pi\)
−0.166522 + 0.986038i \(0.553254\pi\)
\(578\) 19.0000 0.790296
\(579\) −28.0000 −1.16364
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 4.00000 0.165663
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 26.0000 1.07405
\(587\) 18.0000 0.742940 0.371470 − 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 0 0
\(590\) 0 0
\(591\) −36.0000 −1.48084
\(592\) −4.00000 −0.164399
\(593\) −38.0000 −1.56047 −0.780236 − 0.625485i \(-0.784901\pi\)
−0.780236 + 0.625485i \(0.784901\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) −16.0000 −0.654836
\(598\) −36.0000 −1.47215
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 − 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 8.00000 0.326056
\(603\) −2.00000 −0.0814463
\(604\) −4.00000 −0.162758
\(605\) 0 0
\(606\) −28.0000 −1.13742
\(607\) 20.0000 0.811775 0.405887 − 0.913923i \(-0.366962\pi\)
0.405887 + 0.913923i \(0.366962\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 60.0000 2.42734
\(612\) 6.00000 0.242536
\(613\) −34.0000 −1.37325 −0.686624 − 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) 36.0000 1.44931 0.724653 − 0.689114i \(-0.242000\pi\)
0.724653 + 0.689114i \(0.242000\pi\)
\(618\) 28.0000 1.12633
\(619\) −36.0000 −1.44696 −0.723481 − 0.690344i \(-0.757459\pi\)
−0.723481 + 0.690344i \(0.757459\pi\)
\(620\) 0 0
\(621\) −24.0000 −0.963087
\(622\) −8.00000 −0.320771
\(623\) 6.00000 0.240385
\(624\) 12.0000 0.480384
\(625\) 0 0
\(626\) −28.0000 −1.11911
\(627\) 0 0
\(628\) −4.00000 −0.159617
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 − 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −24.0000 −0.954669
\(633\) 16.0000 0.635943
\(634\) 8.00000 0.317721
\(635\) 0 0
\(636\) −8.00000 −0.317221
\(637\) −6.00000 −0.237729
\(638\) −6.00000 −0.237542
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 − 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 24.0000 0.947204
\(643\) −6.00000 −0.236617 −0.118308 − 0.992977i \(-0.537747\pi\)
−0.118308 + 0.992977i \(0.537747\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 − 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 33.0000 1.29636
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 18.0000 0.704934
\(653\) −36.0000 −1.40879 −0.704394 − 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) 12.0000 0.469237
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 2.00000 0.0780274
\(658\) 10.0000 0.389841
\(659\) 24.0000 0.934907 0.467454 − 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 − 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 20.0000 0.777322
\(663\) 72.0000 2.79625
\(664\) 0 0
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) −36.0000 −1.39393
\(668\) −20.0000 −0.773823
\(669\) 4.00000 0.154649
\(670\) 0 0
\(671\) −10.0000 −0.386046
\(672\) −10.0000 −0.385758
\(673\) 26.0000 1.00223 0.501113 − 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −6.00000 −0.231111
\(675\) 0 0
\(676\) −23.0000 −0.884615
\(677\) 18.0000 0.691796 0.345898 − 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −24.0000 −0.921714
\(679\) 0 0
\(680\) 0 0
\(681\) −16.0000 −0.613121
\(682\) 0 0
\(683\) 26.0000 0.994862 0.497431 − 0.867503i \(-0.334277\pi\)
0.497431 + 0.867503i \(0.334277\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) −44.0000 −1.67870
\(688\) 8.00000 0.304997
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 4.00000 0.152167 0.0760836 − 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) 18.0000 0.684257
\(693\) −1.00000 −0.0379869
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) 36.0000 1.36458
\(697\) 36.0000 1.36360
\(698\) −10.0000 −0.378506
\(699\) 44.0000 1.66423
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 − 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 24.0000 0.905822
\(703\) 0 0
\(704\) 7.00000 0.263822
\(705\) 0 0
\(706\) −20.0000 −0.752710
\(707\) 14.0000 0.526524
\(708\) −24.0000 −0.901975
\(709\) 46.0000 1.72757 0.863783 − 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 18.0000 0.674579
\(713\) 0 0
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) 24.0000 0.895049 0.447524 − 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) −19.0000 −0.707107
\(723\) 20.0000 0.743808
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) −22.0000 −0.815935 −0.407967 − 0.912996i \(-0.633762\pi\)
−0.407967 + 0.912996i \(0.633762\pi\)
\(728\) −18.0000 −0.667124
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) 20.0000 0.739221
\(733\) −50.0000 −1.84679 −0.923396 − 0.383849i \(-0.874598\pi\)
−0.923396 + 0.383849i \(0.874598\pi\)
\(734\) 22.0000 0.812035
\(735\) 0 0
\(736\) 30.0000 1.10581
\(737\) −2.00000 −0.0736709
\(738\) −6.00000 −0.220863
\(739\) 16.0000 0.588570 0.294285 − 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −4.00000 −0.146845
\(743\) 12.0000 0.440237 0.220119 − 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) 6.00000 0.219382
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 − 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 10.0000 0.364662
\(753\) −24.0000 −0.874609
\(754\) 36.0000 1.31104
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 12.0000 0.435860
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 − 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) −24.0000 −0.869428
\(763\) −6.00000 −0.217215
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −18.0000 −0.650366
\(767\) −72.0000 −2.59977
\(768\) −34.0000 −1.22687
\(769\) 2.00000 0.0721218 0.0360609 − 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) −8.00000 −0.288113
\(772\) 14.0000 0.503871
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) 0 0
\(777\) −8.00000 −0.286998
\(778\) 26.0000 0.932145
\(779\) 0 0
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) −36.0000 −1.28736
\(783\) 24.0000 0.857690
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 24.0000 0.856052
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 18.0000 0.641223
\(789\) 56.0000 1.99365
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) −3.00000 −0.106600
\(793\) 60.0000 2.13066
\(794\) −28.0000 −0.993683
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 44.0000 1.55856 0.779280 − 0.626676i \(-0.215585\pi\)
0.779280 + 0.626676i \(0.215585\pi\)
\(798\) 0 0
\(799\) 60.0000 2.12265
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) −2.00000 −0.0706225
\(803\) 2.00000 0.0705785
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 0 0
\(807\) −60.0000 −2.11210
\(808\) 42.0000 1.47755
\(809\) −50.0000 −1.75791 −0.878953 − 0.476908i \(-0.841757\pi\)
−0.878953 + 0.476908i \(0.841757\pi\)
\(810\) 0 0
\(811\) −32.0000 −1.12367 −0.561836 − 0.827249i \(-0.689905\pi\)
−0.561836 + 0.827249i \(0.689905\pi\)
\(812\) −6.00000 −0.210559
\(813\) 32.0000 1.12229
\(814\) 4.00000 0.140200
\(815\) 0 0
\(816\) 12.0000 0.420084
\(817\) 0 0
\(818\) 14.0000 0.489499
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) 26.0000 0.907406 0.453703 − 0.891153i \(-0.350103\pi\)
0.453703 + 0.891153i \(0.350103\pi\)
\(822\) −40.0000 −1.39516
\(823\) −46.0000 −1.60346 −0.801730 − 0.597687i \(-0.796087\pi\)
−0.801730 + 0.597687i \(0.796087\pi\)
\(824\) −42.0000 −1.46314
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −6.00000 −0.208514
\(829\) −42.0000 −1.45872 −0.729360 − 0.684130i \(-0.760182\pi\)
−0.729360 + 0.684130i \(0.760182\pi\)
\(830\) 0 0
\(831\) 4.00000 0.138758
\(832\) −42.0000 −1.45609
\(833\) −6.00000 −0.207888
\(834\) 24.0000 0.831052
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) −16.0000 −0.552381 −0.276191 − 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 26.0000 0.896019
\(843\) 12.0000 0.413302
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) −10.0000 −0.343807
\(847\) −1.00000 −0.0343604
\(848\) −4.00000 −0.137361
\(849\) −32.0000 −1.09824
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) −16.0000 −0.548151
\(853\) −26.0000 −0.890223 −0.445112 − 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) −36.0000 −1.23045
\(857\) 26.0000 0.888143 0.444072 − 0.895991i \(-0.353534\pi\)
0.444072 + 0.895991i \(0.353534\pi\)
\(858\) −12.0000 −0.409673
\(859\) −44.0000 −1.50126 −0.750630 − 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) 0 0
\(863\) −6.00000 −0.204242 −0.102121 − 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) −20.0000 −0.680414
\(865\) 0 0
\(866\) −16.0000 −0.543702
\(867\) 38.0000 1.29055
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) −18.0000 −0.609557
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) 22.0000 0.742887 0.371444 − 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 40.0000 1.34993
\(879\) 52.0000 1.75392
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 − 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 1.00000 0.0336718
\(883\) −2.00000 −0.0673054 −0.0336527 − 0.999434i \(-0.510714\pi\)
−0.0336527 + 0.999434i \(0.510714\pi\)
\(884\) −36.0000 −1.21081
\(885\) 0 0
\(886\) −26.0000 −0.873487
\(887\) −28.0000 −0.940148 −0.470074 − 0.882627i \(-0.655773\pi\)
−0.470074 + 0.882627i \(0.655773\pi\)
\(888\) −24.0000 −0.805387
\(889\) 12.0000 0.402467
\(890\) 0 0
\(891\) −11.0000 −0.368514
\(892\) −2.00000 −0.0669650
\(893\) 0 0
\(894\) −20.0000 −0.668900
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) −72.0000 −2.40401
\(898\) −30.0000 −1.00111
\(899\) 0 0
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) −6.00000 −0.199778
\(903\) 16.0000 0.532447
\(904\) 36.0000 1.19734
\(905\) 0 0
\(906\) 8.00000 0.265782
\(907\) −50.0000 −1.66022 −0.830111 − 0.557598i \(-0.811723\pi\)
−0.830111 + 0.557598i \(0.811723\pi\)
\(908\) 8.00000 0.265489
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 − 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −2.00000 −0.0661541
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) −12.0000 −0.396275
\(918\) 24.0000 0.792118
\(919\) 20.0000 0.659739 0.329870 − 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) −40.0000 −1.31804
\(922\) 10.0000 0.329332
\(923\) −48.0000 −1.57994
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) −26.0000 −0.854413
\(927\) 14.0000 0.459820
\(928\) −30.0000 −0.984798
\(929\) −34.0000 −1.11550 −0.557752 − 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −22.0000 −0.720634
\(933\) −16.0000 −0.523816
\(934\) −30.0000 −0.981630
\(935\) 0 0
\(936\) 18.0000 0.588348
\(937\) −26.0000 −0.849383 −0.424691 − 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 2.00000 0.0653023
\(939\) −56.0000 −1.82749
\(940\) 0 0
\(941\) 26.0000 0.847576 0.423788 − 0.905761i \(-0.360700\pi\)
0.423788 + 0.905761i \(0.360700\pi\)
\(942\) 8.00000 0.260654
\(943\) −36.0000 −1.17232
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) −26.0000 −0.844886 −0.422443 − 0.906389i \(-0.638827\pi\)
−0.422443 + 0.906389i \(0.638827\pi\)
\(948\) −16.0000 −0.519656
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 16.0000 0.518836
\(952\) −18.0000 −0.583383
\(953\) −26.0000 −0.842223 −0.421111 − 0.907009i \(-0.638360\pi\)
−0.421111 + 0.907009i \(0.638360\pi\)
\(954\) 4.00000 0.129505
\(955\) 0 0
\(956\) 0 0
\(957\) −12.0000 −0.387905
\(958\) −4.00000 −0.129234
\(959\) 20.0000 0.645834
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −24.0000 −0.773791
\(963\) 12.0000 0.386695
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) −12.0000 −0.386094
\(967\) −20.0000 −0.643157 −0.321578 − 0.946883i \(-0.604213\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(968\) −3.00000 −0.0964237
\(969\) 0 0
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 − 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 10.0000 0.320750
\(973\) −12.0000 −0.384702
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 52.0000 1.66363 0.831814 − 0.555055i \(-0.187303\pi\)
0.831814 + 0.555055i \(0.187303\pi\)
\(978\) −36.0000 −1.15115
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) −12.0000 −0.382935
\(983\) 54.0000 1.72233 0.861166 − 0.508323i \(-0.169735\pi\)
0.861166 + 0.508323i \(0.169735\pi\)
\(984\) 36.0000 1.14764
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) 20.0000 0.636607
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) −48.0000 −1.52477 −0.762385 − 0.647124i \(-0.775972\pi\)
−0.762385 + 0.647124i \(0.775972\pi\)
\(992\) 0 0
\(993\) 40.0000 1.26936
\(994\) −8.00000 −0.253745
\(995\) 0 0
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 − 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −36.0000 −1.13956
\(999\) −16.0000 −0.506218
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 1925.2.a.k.1.1 1
5.2 odd 4 385.2.b.a.309.2 yes 2
5.3 odd 4 385.2.b.a.309.1 ✓ 2
5.4 even 2 1925.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
385.2.b.a.309.1 ✓ 2 5.3 odd 4
385.2.b.a.309.2 yes 2 5.2 odd 4
1925.2.a.d.1.1 1 5.4 even 2
1925.2.a.k.1.1 1 1.1 even 1 trivial