Properties

Label 3525.2.a.m.1.1
Level $3525$
Weight $2$
Character 3525.1
Self dual yes
Analytic conductor $28.147$
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] = [3525,2,Mod(1,3525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3525, 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("3525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.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: \(28.1472667125\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 141)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} +3.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} -2.00000 q^{12} -2.00000 q^{13} +6.00000 q^{14} -4.00000 q^{16} +6.00000 q^{17} +2.00000 q^{18} -6.00000 q^{19} -3.00000 q^{21} -10.0000 q^{22} -9.00000 q^{23} -4.00000 q^{26} -1.00000 q^{27} +6.00000 q^{28} +1.00000 q^{29} -2.00000 q^{31} -8.00000 q^{32} +5.00000 q^{33} +12.0000 q^{34} +2.00000 q^{36} -1.00000 q^{37} -12.0000 q^{38} +2.00000 q^{39} +6.00000 q^{41} -6.00000 q^{42} -2.00000 q^{43} -10.0000 q^{44} -18.0000 q^{46} -1.00000 q^{47} +4.00000 q^{48} +2.00000 q^{49} -6.00000 q^{51} -4.00000 q^{52} -2.00000 q^{54} +6.00000 q^{57} +2.00000 q^{58} -12.0000 q^{59} -2.00000 q^{61} -4.00000 q^{62} +3.00000 q^{63} -8.00000 q^{64} +10.0000 q^{66} -2.00000 q^{67} +12.0000 q^{68} +9.00000 q^{69} -2.00000 q^{71} +2.00000 q^{73} -2.00000 q^{74} -12.0000 q^{76} -15.0000 q^{77} +4.00000 q^{78} -15.0000 q^{79} +1.00000 q^{81} +12.0000 q^{82} +4.00000 q^{83} -6.00000 q^{84} -4.00000 q^{86} -1.00000 q^{87} +10.0000 q^{89} -6.00000 q^{91} -18.0000 q^{92} +2.00000 q^{93} -2.00000 q^{94} +8.00000 q^{96} -1.00000 q^{97} +4.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) −2.00000 −0.577350
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 6.00000 1.60357
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 2.00000 0.471405
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) −10.0000 −2.13201
\(23\) −9.00000 −1.87663 −0.938315 0.345782i \(-0.887614\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) 6.00000 1.13389
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −8.00000 −1.41421
\(33\) 5.00000 0.870388
\(34\) 12.0000 2.05798
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) −12.0000 −1.94666
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −6.00000 −0.925820
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) −10.0000 −1.50756
\(45\) 0 0
\(46\) −18.0000 −2.65396
\(47\) −1.00000 −0.145865
\(48\) 4.00000 0.577350
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) −4.00000 −0.554700
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −2.00000 −0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 2.00000 0.262613
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −4.00000 −0.508001
\(63\) 3.00000 0.377964
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 10.0000 1.23091
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 12.0000 1.45521
\(69\) 9.00000 1.08347
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −12.0000 −1.37649
\(77\) −15.0000 −1.70941
\(78\) 4.00000 0.452911
\(79\) −15.0000 −1.68763 −0.843816 0.536633i \(-0.819696\pi\)
−0.843816 + 0.536633i \(0.819696\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 12.0000 1.32518
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −6.00000 −0.654654
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) −18.0000 −1.87663
\(93\) 2.00000 0.207390
\(94\) −2.00000 −0.206284
\(95\) 0 0
\(96\) 8.00000 0.816497
\(97\) −1.00000 −0.101535 −0.0507673 0.998711i \(-0.516167\pi\)
−0.0507673 + 0.998711i \(0.516167\pi\)
\(98\) 4.00000 0.404061
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) −4.00000 −0.398015 −0.199007 0.979998i \(-0.563772\pi\)
−0.199007 + 0.979998i \(0.563772\pi\)
\(102\) −12.0000 −1.18818
\(103\) 13.0000 1.28093 0.640464 0.767988i \(-0.278742\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 17.0000 1.64345 0.821726 0.569883i \(-0.193011\pi\)
0.821726 + 0.569883i \(0.193011\pi\)
\(108\) −2.00000 −0.192450
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) −12.0000 −1.13389
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 12.0000 1.12390
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) −2.00000 −0.184900
\(118\) −24.0000 −2.20938
\(119\) 18.0000 1.65006
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −4.00000 −0.362143
\(123\) −6.00000 −0.541002
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 6.00000 0.534522
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −22.0000 −1.92215 −0.961074 0.276289i \(-0.910895\pi\)
−0.961074 + 0.276289i \(0.910895\pi\)
\(132\) 10.0000 0.870388
\(133\) −18.0000 −1.56080
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 18.0000 1.53226
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) 1.00000 0.0842152
\(142\) −4.00000 −0.335673
\(143\) 10.0000 0.836242
\(144\) −4.00000 −0.333333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) −2.00000 −0.164957
\(148\) −2.00000 −0.164399
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) −30.0000 −2.41747
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) −30.0000 −2.38667
\(159\) 0 0
\(160\) 0 0
\(161\) −27.0000 −2.12790
\(162\) 2.00000 0.157135
\(163\) −18.0000 −1.40987 −0.704934 0.709273i \(-0.749024\pi\)
−0.704934 + 0.709273i \(0.749024\pi\)
\(164\) 12.0000 0.937043
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 1.00000 0.0773823 0.0386912 0.999251i \(-0.487681\pi\)
0.0386912 + 0.999251i \(0.487681\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) −4.00000 −0.304997
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) 20.0000 1.50756
\(177\) 12.0000 0.901975
\(178\) 20.0000 1.49906
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) −12.0000 −0.889499
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) −30.0000 −2.19382
\(188\) −2.00000 −0.145865
\(189\) −3.00000 −0.218218
\(190\) 0 0
\(191\) 26.0000 1.88129 0.940647 0.339387i \(-0.110219\pi\)
0.940647 + 0.339387i \(0.110219\pi\)
\(192\) 8.00000 0.577350
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 4.00000 0.285714
\(197\) 24.0000 1.70993 0.854965 0.518686i \(-0.173579\pi\)
0.854965 + 0.518686i \(0.173579\pi\)
\(198\) −10.0000 −0.710669
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 2.00000 0.141069
\(202\) −8.00000 −0.562878
\(203\) 3.00000 0.210559
\(204\) −12.0000 −0.840168
\(205\) 0 0
\(206\) 26.0000 1.81151
\(207\) −9.00000 −0.625543
\(208\) 8.00000 0.554700
\(209\) 30.0000 2.07514
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 0 0
\(213\) 2.00000 0.137038
\(214\) 34.0000 2.32419
\(215\) 0 0
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) 12.0000 0.812743
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 2.00000 0.134231
\(223\) 12.0000 0.803579 0.401790 0.915732i \(-0.368388\pi\)
0.401790 + 0.915732i \(0.368388\pi\)
\(224\) −24.0000 −1.60357
\(225\) 0 0
\(226\) 28.0000 1.86253
\(227\) 3.00000 0.199117 0.0995585 0.995032i \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) 12.0000 0.794719
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 15.0000 0.986928
\(232\) 0 0
\(233\) −21.0000 −1.37576 −0.687878 0.725826i \(-0.741458\pi\)
−0.687878 + 0.725826i \(0.741458\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) −24.0000 −1.56227
\(237\) 15.0000 0.974355
\(238\) 36.0000 2.33353
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 0 0
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) 28.0000 1.79991
\(243\) −1.00000 −0.0641500
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) −12.0000 −0.765092
\(247\) 12.0000 0.763542
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 14.0000 0.883672 0.441836 0.897096i \(-0.354327\pi\)
0.441836 + 0.897096i \(0.354327\pi\)
\(252\) 6.00000 0.377964
\(253\) 45.0000 2.82913
\(254\) −40.0000 −2.50982
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −11.0000 −0.686161 −0.343081 0.939306i \(-0.611470\pi\)
−0.343081 + 0.939306i \(0.611470\pi\)
\(258\) 4.00000 0.249029
\(259\) −3.00000 −0.186411
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) −44.0000 −2.71833
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −36.0000 −2.20730
\(267\) −10.0000 −0.611990
\(268\) −4.00000 −0.244339
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 0 0
\(271\) 9.00000 0.546711 0.273356 0.961913i \(-0.411866\pi\)
0.273356 + 0.961913i \(0.411866\pi\)
\(272\) −24.0000 −1.45521
\(273\) 6.00000 0.363137
\(274\) −12.0000 −0.724947
\(275\) 0 0
\(276\) 18.0000 1.08347
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −20.0000 −1.19952
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) −15.0000 −0.894825 −0.447412 0.894328i \(-0.647654\pi\)
−0.447412 + 0.894328i \(0.647654\pi\)
\(282\) 2.00000 0.119098
\(283\) 5.00000 0.297219 0.148610 0.988896i \(-0.452520\pi\)
0.148610 + 0.988896i \(0.452520\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) 20.0000 1.18262
\(287\) 18.0000 1.06251
\(288\) −8.00000 −0.471405
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 1.00000 0.0586210
\(292\) 4.00000 0.234082
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) −4.00000 −0.233285
\(295\) 0 0
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) 18.0000 1.04097
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) −20.0000 −1.15087
\(303\) 4.00000 0.229794
\(304\) 24.0000 1.37649
\(305\) 0 0
\(306\) 12.0000 0.685994
\(307\) 17.0000 0.970241 0.485121 0.874447i \(-0.338776\pi\)
0.485121 + 0.874447i \(0.338776\pi\)
\(308\) −30.0000 −1.70941
\(309\) −13.0000 −0.739544
\(310\) 0 0
\(311\) 3.00000 0.170114 0.0850572 0.996376i \(-0.472893\pi\)
0.0850572 + 0.996376i \(0.472893\pi\)
\(312\) 0 0
\(313\) 30.0000 1.69570 0.847850 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) −26.0000 −1.46726
\(315\) 0 0
\(316\) −30.0000 −1.68763
\(317\) 17.0000 0.954815 0.477408 0.878682i \(-0.341577\pi\)
0.477408 + 0.878682i \(0.341577\pi\)
\(318\) 0 0
\(319\) −5.00000 −0.279946
\(320\) 0 0
\(321\) −17.0000 −0.948847
\(322\) −54.0000 −3.00930
\(323\) −36.0000 −2.00309
\(324\) 2.00000 0.111111
\(325\) 0 0
\(326\) −36.0000 −1.99386
\(327\) −6.00000 −0.331801
\(328\) 0 0
\(329\) −3.00000 −0.165395
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 8.00000 0.439057
\(333\) −1.00000 −0.0547997
\(334\) 2.00000 0.109435
\(335\) 0 0
\(336\) 12.0000 0.654654
\(337\) 13.0000 0.708155 0.354078 0.935216i \(-0.384795\pi\)
0.354078 + 0.935216i \(0.384795\pi\)
\(338\) −18.0000 −0.979071
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) 10.0000 0.541530
\(342\) −12.0000 −0.648886
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) −2.00000 −0.107211
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 40.0000 2.13201
\(353\) −36.0000 −1.91609 −0.958043 0.286623i \(-0.907467\pi\)
−0.958043 + 0.286623i \(0.907467\pi\)
\(354\) 24.0000 1.27559
\(355\) 0 0
\(356\) 20.0000 1.06000
\(357\) −18.0000 −0.952661
\(358\) −18.0000 −0.951330
\(359\) 3.00000 0.158334 0.0791670 0.996861i \(-0.474774\pi\)
0.0791670 + 0.996861i \(0.474774\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 24.0000 1.26141
\(363\) −14.0000 −0.734809
\(364\) −12.0000 −0.628971
\(365\) 0 0
\(366\) 4.00000 0.209083
\(367\) 24.0000 1.25279 0.626395 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) 36.0000 1.87663
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) 4.00000 0.207390
\(373\) −20.0000 −1.03556 −0.517780 0.855514i \(-0.673242\pi\)
−0.517780 + 0.855514i \(0.673242\pi\)
\(374\) −60.0000 −3.10253
\(375\) 0 0
\(376\) 0 0
\(377\) −2.00000 −0.103005
\(378\) −6.00000 −0.308607
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) 20.0000 1.02463
\(382\) 52.0000 2.66055
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −32.0000 −1.62876
\(387\) −2.00000 −0.101666
\(388\) −2.00000 −0.101535
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) −54.0000 −2.73090
\(392\) 0 0
\(393\) 22.0000 1.10975
\(394\) 48.0000 2.41821
\(395\) 0 0
\(396\) −10.0000 −0.502519
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 20.0000 1.00251
\(399\) 18.0000 0.901127
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 4.00000 0.199502
\(403\) 4.00000 0.199254
\(404\) −8.00000 −0.398015
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) 5.00000 0.247841
\(408\) 0 0
\(409\) 16.0000 0.791149 0.395575 0.918434i \(-0.370545\pi\)
0.395575 + 0.918434i \(0.370545\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) 26.0000 1.28093
\(413\) −36.0000 −1.77144
\(414\) −18.0000 −0.884652
\(415\) 0 0
\(416\) 16.0000 0.784465
\(417\) 10.0000 0.489702
\(418\) 60.0000 2.93470
\(419\) 3.00000 0.146560 0.0732798 0.997311i \(-0.476653\pi\)
0.0732798 + 0.997311i \(0.476653\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) −32.0000 −1.55774
\(423\) −1.00000 −0.0486217
\(424\) 0 0
\(425\) 0 0
\(426\) 4.00000 0.193801
\(427\) −6.00000 −0.290360
\(428\) 34.0000 1.64345
\(429\) −10.0000 −0.482805
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 4.00000 0.192450
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) −12.0000 −0.576018
\(435\) 0 0
\(436\) 12.0000 0.574696
\(437\) 54.0000 2.58317
\(438\) −4.00000 −0.191127
\(439\) −15.0000 −0.715911 −0.357955 0.933739i \(-0.616526\pi\)
−0.357955 + 0.933739i \(0.616526\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) −24.0000 −1.14156
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 24.0000 1.13643
\(447\) 0 0
\(448\) −24.0000 −1.13389
\(449\) 5.00000 0.235965 0.117982 0.993016i \(-0.462357\pi\)
0.117982 + 0.993016i \(0.462357\pi\)
\(450\) 0 0
\(451\) −30.0000 −1.41264
\(452\) 28.0000 1.31701
\(453\) 10.0000 0.469841
\(454\) 6.00000 0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) −11.0000 −0.514558 −0.257279 0.966337i \(-0.582826\pi\)
−0.257279 + 0.966337i \(0.582826\pi\)
\(458\) 0 0
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −27.0000 −1.25752 −0.628758 0.777601i \(-0.716436\pi\)
−0.628758 + 0.777601i \(0.716436\pi\)
\(462\) 30.0000 1.39573
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) −42.0000 −1.94561
\(467\) 23.0000 1.06431 0.532157 0.846646i \(-0.321382\pi\)
0.532157 + 0.846646i \(0.321382\pi\)
\(468\) −4.00000 −0.184900
\(469\) −6.00000 −0.277054
\(470\) 0 0
\(471\) 13.0000 0.599008
\(472\) 0 0
\(473\) 10.0000 0.459800
\(474\) 30.0000 1.37795
\(475\) 0 0
\(476\) 36.0000 1.65006
\(477\) 0 0
\(478\) −8.00000 −0.365911
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) −10.0000 −0.455488
\(483\) 27.0000 1.22854
\(484\) 28.0000 1.27273
\(485\) 0 0
\(486\) −2.00000 −0.0907218
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) 18.0000 0.813988
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) −12.0000 −0.541002
\(493\) 6.00000 0.270226
\(494\) 24.0000 1.07981
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −6.00000 −0.269137
\(498\) −8.00000 −0.358489
\(499\) 2.00000 0.0895323 0.0447661 0.998997i \(-0.485746\pi\)
0.0447661 + 0.998997i \(0.485746\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 28.0000 1.24970
\(503\) −9.00000 −0.401290 −0.200645 0.979664i \(-0.564304\pi\)
−0.200645 + 0.979664i \(0.564304\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 90.0000 4.00099
\(507\) 9.00000 0.399704
\(508\) −40.0000 −1.77471
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 32.0000 1.41421
\(513\) 6.00000 0.264906
\(514\) −22.0000 −0.970378
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 5.00000 0.219900
\(518\) −6.00000 −0.263625
\(519\) 0 0
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 2.00000 0.0875376
\(523\) −11.0000 −0.480996 −0.240498 0.970650i \(-0.577311\pi\)
−0.240498 + 0.970650i \(0.577311\pi\)
\(524\) −44.0000 −1.92215
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) −12.0000 −0.522728
\(528\) −20.0000 −0.870388
\(529\) 58.0000 2.52174
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) −36.0000 −1.56080
\(533\) −12.0000 −0.519778
\(534\) −20.0000 −0.865485
\(535\) 0 0
\(536\) 0 0
\(537\) 9.00000 0.388379
\(538\) −8.00000 −0.344904
\(539\) −10.0000 −0.430730
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 18.0000 0.773166
\(543\) −12.0000 −0.514969
\(544\) −48.0000 −2.05798
\(545\) 0 0
\(546\) 12.0000 0.513553
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) −12.0000 −0.512615
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) −45.0000 −1.91359
\(554\) −20.0000 −0.849719
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) −1.00000 −0.0423714 −0.0211857 0.999776i \(-0.506744\pi\)
−0.0211857 + 0.999776i \(0.506744\pi\)
\(558\) −4.00000 −0.169334
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 30.0000 1.26660
\(562\) −30.0000 −1.26547
\(563\) −8.00000 −0.337160 −0.168580 0.985688i \(-0.553918\pi\)
−0.168580 + 0.985688i \(0.553918\pi\)
\(564\) 2.00000 0.0842152
\(565\) 0 0
\(566\) 10.0000 0.420331
\(567\) 3.00000 0.125988
\(568\) 0 0
\(569\) 21.0000 0.880366 0.440183 0.897908i \(-0.354914\pi\)
0.440183 + 0.897908i \(0.354914\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 20.0000 0.836242
\(573\) −26.0000 −1.08617
\(574\) 36.0000 1.50261
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 38.0000 1.58059
\(579\) 16.0000 0.664937
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 2.00000 0.0829027
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) −20.0000 −0.825488 −0.412744 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) −4.00000 −0.164957
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) −24.0000 −0.987228
\(592\) 4.00000 0.164399
\(593\) 22.0000 0.903432 0.451716 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) 10.0000 0.410305
\(595\) 0 0
\(596\) 0 0
\(597\) −10.0000 −0.409273
\(598\) 36.0000 1.47215
\(599\) 19.0000 0.776319 0.388159 0.921592i \(-0.373111\pi\)
0.388159 + 0.921592i \(0.373111\pi\)
\(600\) 0 0
\(601\) −25.0000 −1.01977 −0.509886 0.860242i \(-0.670312\pi\)
−0.509886 + 0.860242i \(0.670312\pi\)
\(602\) −12.0000 −0.489083
\(603\) −2.00000 −0.0814463
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) 8.00000 0.324978
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 48.0000 1.94666
\(609\) −3.00000 −0.121566
\(610\) 0 0
\(611\) 2.00000 0.0809113
\(612\) 12.0000 0.485071
\(613\) −3.00000 −0.121169 −0.0605844 0.998163i \(-0.519296\pi\)
−0.0605844 + 0.998163i \(0.519296\pi\)
\(614\) 34.0000 1.37213
\(615\) 0 0
\(616\) 0 0
\(617\) 16.0000 0.644136 0.322068 0.946717i \(-0.395622\pi\)
0.322068 + 0.946717i \(0.395622\pi\)
\(618\) −26.0000 −1.04587
\(619\) −35.0000 −1.40677 −0.703384 0.710810i \(-0.748329\pi\)
−0.703384 + 0.710810i \(0.748329\pi\)
\(620\) 0 0
\(621\) 9.00000 0.361158
\(622\) 6.00000 0.240578
\(623\) 30.0000 1.20192
\(624\) −8.00000 −0.320256
\(625\) 0 0
\(626\) 60.0000 2.39808
\(627\) −30.0000 −1.19808
\(628\) −26.0000 −1.03751
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 34.0000 1.35352 0.676759 0.736204i \(-0.263384\pi\)
0.676759 + 0.736204i \(0.263384\pi\)
\(632\) 0 0
\(633\) 16.0000 0.635943
\(634\) 34.0000 1.35031
\(635\) 0 0
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) −10.0000 −0.395904
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) −45.0000 −1.77739 −0.888697 0.458496i \(-0.848388\pi\)
−0.888697 + 0.458496i \(0.848388\pi\)
\(642\) −34.0000 −1.34187
\(643\) 44.0000 1.73519 0.867595 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) −54.0000 −2.12790
\(645\) 0 0
\(646\) −72.0000 −2.83280
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 0 0
\(649\) 60.0000 2.35521
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) −36.0000 −1.40987
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −12.0000 −0.469237
\(655\) 0 0
\(656\) −24.0000 −0.937043
\(657\) 2.00000 0.0780274
\(658\) −6.00000 −0.233904
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) 40.0000 1.55464
\(663\) 12.0000 0.466041
\(664\) 0 0
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −9.00000 −0.348481
\(668\) 2.00000 0.0773823
\(669\) −12.0000 −0.463947
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) 24.0000 0.925820
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) −18.0000 −0.692308
\(677\) −46.0000 −1.76792 −0.883962 0.467559i \(-0.845134\pi\)
−0.883962 + 0.467559i \(0.845134\pi\)
\(678\) −28.0000 −1.07533
\(679\) −3.00000 −0.115129
\(680\) 0 0
\(681\) −3.00000 −0.114960
\(682\) 20.0000 0.765840
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) −12.0000 −0.458831
\(685\) 0 0
\(686\) −30.0000 −1.14541
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) 0 0
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 0 0
\(693\) −15.0000 −0.569803
\(694\) 56.0000 2.12573
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) −56.0000 −2.11963
\(699\) 21.0000 0.794293
\(700\) 0 0
\(701\) −43.0000 −1.62409 −0.812044 0.583597i \(-0.801645\pi\)
−0.812044 + 0.583597i \(0.801645\pi\)
\(702\) 4.00000 0.150970
\(703\) 6.00000 0.226294
\(704\) 40.0000 1.50756
\(705\) 0 0
\(706\) −72.0000 −2.70976
\(707\) −12.0000 −0.451306
\(708\) 24.0000 0.901975
\(709\) −43.0000 −1.61490 −0.807449 0.589937i \(-0.799153\pi\)
−0.807449 + 0.589937i \(0.799153\pi\)
\(710\) 0 0
\(711\) −15.0000 −0.562544
\(712\) 0 0
\(713\) 18.0000 0.674105
\(714\) −36.0000 −1.34727
\(715\) 0 0
\(716\) −18.0000 −0.672692
\(717\) 4.00000 0.149383
\(718\) 6.00000 0.223918
\(719\) −26.0000 −0.969636 −0.484818 0.874615i \(-0.661114\pi\)
−0.484818 + 0.874615i \(0.661114\pi\)
\(720\) 0 0
\(721\) 39.0000 1.45244
\(722\) 34.0000 1.26535
\(723\) 5.00000 0.185952
\(724\) 24.0000 0.891953
\(725\) 0 0
\(726\) −28.0000 −1.03918
\(727\) −46.0000 −1.70605 −0.853023 0.521874i \(-0.825233\pi\)
−0.853023 + 0.521874i \(0.825233\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 4.00000 0.147844
\(733\) 7.00000 0.258551 0.129275 0.991609i \(-0.458735\pi\)
0.129275 + 0.991609i \(0.458735\pi\)
\(734\) 48.0000 1.77171
\(735\) 0 0
\(736\) 72.0000 2.65396
\(737\) 10.0000 0.368355
\(738\) 12.0000 0.441726
\(739\) 41.0000 1.50821 0.754105 0.656754i \(-0.228071\pi\)
0.754105 + 0.656754i \(0.228071\pi\)
\(740\) 0 0
\(741\) −12.0000 −0.440831
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −40.0000 −1.46450
\(747\) 4.00000 0.146352
\(748\) −60.0000 −2.19382
\(749\) 51.0000 1.86350
\(750\) 0 0
\(751\) −2.00000 −0.0729810 −0.0364905 0.999334i \(-0.511618\pi\)
−0.0364905 + 0.999334i \(0.511618\pi\)
\(752\) 4.00000 0.145865
\(753\) −14.0000 −0.510188
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) −6.00000 −0.218218
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −10.0000 −0.363216
\(759\) −45.0000 −1.63340
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 40.0000 1.44905
\(763\) 18.0000 0.651644
\(764\) 52.0000 1.88129
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 24.0000 0.866590
\(768\) −16.0000 −0.577350
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) 11.0000 0.396155
\(772\) −32.0000 −1.15171
\(773\) −12.0000 −0.431610 −0.215805 0.976436i \(-0.569238\pi\)
−0.215805 + 0.976436i \(0.569238\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 0 0
\(777\) 3.00000 0.107624
\(778\) 4.00000 0.143407
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) 10.0000 0.357828
\(782\) −108.000 −3.86207
\(783\) −1.00000 −0.0357371
\(784\) −8.00000 −0.285714
\(785\) 0 0
\(786\) 44.0000 1.56943
\(787\) −8.00000 −0.285169 −0.142585 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) 48.0000 1.70993
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 42.0000 1.49335
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 36.0000 1.27759
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 33.0000 1.16892 0.584460 0.811423i \(-0.301306\pi\)
0.584460 + 0.811423i \(0.301306\pi\)
\(798\) 36.0000 1.27439
\(799\) −6.00000 −0.212265
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 24.0000 0.847469
\(803\) −10.0000 −0.352892
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 4.00000 0.140807
\(808\) 0 0
\(809\) 31.0000 1.08990 0.544951 0.838468i \(-0.316548\pi\)
0.544951 + 0.838468i \(0.316548\pi\)
\(810\) 0 0
\(811\) −37.0000 −1.29925 −0.649623 0.760257i \(-0.725073\pi\)
−0.649623 + 0.760257i \(0.725073\pi\)
\(812\) 6.00000 0.210559
\(813\) −9.00000 −0.315644
\(814\) 10.0000 0.350500
\(815\) 0 0
\(816\) 24.0000 0.840168
\(817\) 12.0000 0.419827
\(818\) 32.0000 1.11885
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 12.0000 0.418548
\(823\) 39.0000 1.35945 0.679727 0.733465i \(-0.262098\pi\)
0.679727 + 0.733465i \(0.262098\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −72.0000 −2.50520
\(827\) 44.0000 1.53003 0.765015 0.644013i \(-0.222732\pi\)
0.765015 + 0.644013i \(0.222732\pi\)
\(828\) −18.0000 −0.625543
\(829\) −30.0000 −1.04194 −0.520972 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 16.0000 0.554700
\(833\) 12.0000 0.415775
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) 60.0000 2.07514
\(837\) 2.00000 0.0691301
\(838\) 6.00000 0.207267
\(839\) −21.0000 −0.725001 −0.362500 0.931984i \(-0.618077\pi\)
−0.362500 + 0.931984i \(0.618077\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 40.0000 1.37849
\(843\) 15.0000 0.516627
\(844\) −32.0000 −1.10149
\(845\) 0 0
\(846\) −2.00000 −0.0687614
\(847\) 42.0000 1.44314
\(848\) 0 0
\(849\) −5.00000 −0.171600
\(850\) 0 0
\(851\) 9.00000 0.308516
\(852\) 4.00000 0.137038
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) −12.0000 −0.410632
\(855\) 0 0
\(856\) 0 0
\(857\) −41.0000 −1.40053 −0.700267 0.713881i \(-0.746936\pi\)
−0.700267 + 0.713881i \(0.746936\pi\)
\(858\) −20.0000 −0.682789
\(859\) −14.0000 −0.477674 −0.238837 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) 0 0
\(861\) −18.0000 −0.613438
\(862\) 36.0000 1.22616
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 8.00000 0.272166
\(865\) 0 0
\(866\) 0 0
\(867\) −19.0000 −0.645274
\(868\) −12.0000 −0.407307
\(869\) 75.0000 2.54420
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 0 0
\(873\) −1.00000 −0.0338449
\(874\) 108.000 3.65315
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) −34.0000 −1.14810 −0.574049 0.818821i \(-0.694628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) −30.0000 −1.01245
\(879\) 9.00000 0.303562
\(880\) 0 0
\(881\) 33.0000 1.11180 0.555899 0.831250i \(-0.312374\pi\)
0.555899 + 0.831250i \(0.312374\pi\)
\(882\) 4.00000 0.134687
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) −24.0000 −0.807207
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 0 0
\(889\) −60.0000 −2.01234
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 24.0000 0.803579
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −18.0000 −0.601003
\(898\) 10.0000 0.333704
\(899\) −2.00000 −0.0667037
\(900\) 0 0
\(901\) 0 0
\(902\) −60.0000 −1.99778
\(903\) 6.00000 0.199667
\(904\) 0 0
\(905\) 0 0
\(906\) 20.0000 0.664455
\(907\) −13.0000 −0.431658 −0.215829 0.976431i \(-0.569245\pi\)
−0.215829 + 0.976431i \(0.569245\pi\)
\(908\) 6.00000 0.199117
\(909\) −4.00000 −0.132672
\(910\) 0 0
\(911\) 22.0000 0.728893 0.364446 0.931224i \(-0.381258\pi\)
0.364446 + 0.931224i \(0.381258\pi\)
\(912\) −24.0000 −0.794719
\(913\) −20.0000 −0.661903
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) 0 0
\(917\) −66.0000 −2.17951
\(918\) −12.0000 −0.396059
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) −17.0000 −0.560169
\(922\) −54.0000 −1.77840
\(923\) 4.00000 0.131662
\(924\) 30.0000 0.986928
\(925\) 0 0
\(926\) 16.0000 0.525793
\(927\) 13.0000 0.426976
\(928\) −8.00000 −0.262613
\(929\) −50.0000 −1.64045 −0.820223 0.572043i \(-0.806151\pi\)
−0.820223 + 0.572043i \(0.806151\pi\)
\(930\) 0 0
\(931\) −12.0000 −0.393284
\(932\) −42.0000 −1.37576
\(933\) −3.00000 −0.0982156
\(934\) 46.0000 1.50517
\(935\) 0 0
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) −12.0000 −0.391814
\(939\) −30.0000 −0.979013
\(940\) 0 0
\(941\) −42.0000 −1.36916 −0.684580 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 26.0000 0.847126
\(943\) −54.0000 −1.75848
\(944\) 48.0000 1.56227
\(945\) 0 0
\(946\) 20.0000 0.650256
\(947\) −58.0000 −1.88475 −0.942373 0.334563i \(-0.891411\pi\)
−0.942373 + 0.334563i \(0.891411\pi\)
\(948\) 30.0000 0.974355
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) −17.0000 −0.551263
\(952\) 0 0
\(953\) 11.0000 0.356325 0.178162 0.984001i \(-0.442985\pi\)
0.178162 + 0.984001i \(0.442985\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −8.00000 −0.258738
\(957\) 5.00000 0.161627
\(958\) 48.0000 1.55081
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 4.00000 0.128965
\(963\) 17.0000 0.547817
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 54.0000 1.73742
\(967\) −43.0000 −1.38279 −0.691393 0.722478i \(-0.743003\pi\)
−0.691393 + 0.722478i \(0.743003\pi\)
\(968\) 0 0
\(969\) 36.0000 1.15649
\(970\) 0 0
\(971\) −61.0000 −1.95758 −0.978792 0.204859i \(-0.934327\pi\)
−0.978792 + 0.204859i \(0.934327\pi\)
\(972\) −2.00000 −0.0641500
\(973\) −30.0000 −0.961756
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 36.0000 1.15115
\(979\) −50.0000 −1.59801
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 56.0000 1.78703
\(983\) −12.0000 −0.382741 −0.191370 0.981518i \(-0.561293\pi\)
−0.191370 + 0.981518i \(0.561293\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 3.00000 0.0954911
\(988\) 24.0000 0.763542
\(989\) 18.0000 0.572367
\(990\) 0 0
\(991\) −33.0000 −1.04828 −0.524140 0.851632i \(-0.675613\pi\)
−0.524140 + 0.851632i \(0.675613\pi\)
\(992\) 16.0000 0.508001
\(993\) −20.0000 −0.634681
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) −20.0000 −0.633406 −0.316703 0.948525i \(-0.602576\pi\)
−0.316703 + 0.948525i \(0.602576\pi\)
\(998\) 4.00000 0.126618
\(999\) 1.00000 0.0316386
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 3525.2.a.m.1.1 1
5.4 even 2 141.2.a.a.1.1 1
15.14 odd 2 423.2.a.f.1.1 1
20.19 odd 2 2256.2.a.c.1.1 1
35.34 odd 2 6909.2.a.a.1.1 1
40.19 odd 2 9024.2.a.bv.1.1 1
40.29 even 2 9024.2.a.t.1.1 1
60.59 even 2 6768.2.a.t.1.1 1
235.234 odd 2 6627.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
141.2.a.a.1.1 1 5.4 even 2
423.2.a.f.1.1 1 15.14 odd 2
2256.2.a.c.1.1 1 20.19 odd 2
3525.2.a.m.1.1 1 1.1 even 1 trivial
6627.2.a.a.1.1 1 235.234 odd 2
6768.2.a.t.1.1 1 60.59 even 2
6909.2.a.a.1.1 1 35.34 odd 2
9024.2.a.t.1.1 1 40.29 even 2
9024.2.a.bv.1.1 1 40.19 odd 2