Properties

Label 3381.2.a.b.1.1
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3381,2,Mod(1,3381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3381, 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("3381.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.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: \(26.9974209234\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3381.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} -3.00000 q^{15} -1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -3.00000 q^{20} +2.00000 q^{22} +1.00000 q^{23} -3.00000 q^{24} +4.00000 q^{25} -1.00000 q^{26} -1.00000 q^{27} +3.00000 q^{30} -10.0000 q^{31} -5.00000 q^{32} +2.00000 q^{33} +1.00000 q^{34} -1.00000 q^{36} -2.00000 q^{37} -1.00000 q^{39} +9.00000 q^{40} -4.00000 q^{41} +4.00000 q^{43} +2.00000 q^{44} +3.00000 q^{45} -1.00000 q^{46} -7.00000 q^{47} +1.00000 q^{48} -4.00000 q^{50} +1.00000 q^{51} -1.00000 q^{52} -9.00000 q^{53} +1.00000 q^{54} -6.00000 q^{55} +4.00000 q^{59} +3.00000 q^{60} +2.00000 q^{61} +10.0000 q^{62} +7.00000 q^{64} +3.00000 q^{65} -2.00000 q^{66} -3.00000 q^{67} +1.00000 q^{68} -1.00000 q^{69} -15.0000 q^{71} +3.00000 q^{72} +13.0000 q^{73} +2.00000 q^{74} -4.00000 q^{75} +1.00000 q^{78} +4.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} +4.00000 q^{82} -6.00000 q^{83} -3.00000 q^{85} -4.00000 q^{86} -6.00000 q^{88} +6.00000 q^{89} -3.00000 q^{90} -1.00000 q^{92} +10.0000 q^{93} +7.00000 q^{94} +5.00000 q^{96} -16.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) −3.00000 −0.948683
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) −1.00000 −0.250000
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 1.00000 0.208514
\(24\) −3.00000 −0.612372
\(25\) 4.00000 0.800000
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 3.00000 0.547723
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) −5.00000 −0.883883
\(33\) 2.00000 0.348155
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 9.00000 1.42302
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 2.00000 0.301511
\(45\) 3.00000 0.447214
\(46\) −1.00000 −0.147442
\(47\) −7.00000 −1.02105 −0.510527 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −4.00000 −0.565685
\(51\) 1.00000 0.140028
\(52\) −1.00000 −0.138675
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 1.00000 0.136083
\(55\) −6.00000 −0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 3.00000 0.387298
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 10.0000 1.27000
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 3.00000 0.372104
\(66\) −2.00000 −0.246183
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 1.00000 0.121268
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −15.0000 −1.78017 −0.890086 0.455792i \(-0.849356\pi\)
−0.890086 + 0.455792i \(0.849356\pi\)
\(72\) 3.00000 0.353553
\(73\) 13.0000 1.52153 0.760767 0.649025i \(-0.224823\pi\)
0.760767 + 0.649025i \(0.224823\pi\)
\(74\) 2.00000 0.232495
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 1.00000 0.113228
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −3.00000 −0.316228
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 10.0000 1.03695
\(94\) 7.00000 0.721995
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) −16.0000 −1.62455 −0.812277 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) −4.00000 −0.400000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −11.0000 −1.08386 −0.541931 0.840423i \(-0.682307\pi\)
−0.541931 + 0.840423i \(0.682307\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) 9.00000 0.874157
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 1.00000 0.0962250
\(109\) 8.00000 0.766261 0.383131 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) 6.00000 0.572078
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 7.00000 0.658505 0.329252 0.944242i \(-0.393203\pi\)
0.329252 + 0.944242i \(0.393203\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) −9.00000 −0.821584
\(121\) −7.00000 −0.636364
\(122\) −2.00000 −0.181071
\(123\) 4.00000 0.360668
\(124\) 10.0000 0.898027
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 3.00000 0.265165
\(129\) −4.00000 −0.352180
\(130\) −3.00000 −0.263117
\(131\) 3.00000 0.262111 0.131056 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) 3.00000 0.259161
\(135\) −3.00000 −0.258199
\(136\) −3.00000 −0.257248
\(137\) −17.0000 −1.45241 −0.726204 0.687479i \(-0.758717\pi\)
−0.726204 + 0.687479i \(0.758717\pi\)
\(138\) 1.00000 0.0851257
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 0 0
\(141\) 7.00000 0.589506
\(142\) 15.0000 1.25877
\(143\) −2.00000 −0.167248
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −13.0000 −1.07589
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 5.00000 0.409616 0.204808 0.978802i \(-0.434343\pi\)
0.204808 + 0.978802i \(0.434343\pi\)
\(150\) 4.00000 0.326599
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) −30.0000 −2.40966
\(156\) 1.00000 0.0800641
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −4.00000 −0.318223
\(159\) 9.00000 0.713746
\(160\) −15.0000 −1.18585
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 4.00000 0.312348
\(165\) 6.00000 0.467099
\(166\) 6.00000 0.465690
\(167\) 19.0000 1.47026 0.735132 0.677924i \(-0.237120\pi\)
0.735132 + 0.677924i \(0.237120\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 3.00000 0.230089
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 16.0000 1.21646 0.608229 0.793762i \(-0.291880\pi\)
0.608229 + 0.793762i \(0.291880\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) −4.00000 −0.300658
\(178\) −6.00000 −0.449719
\(179\) −11.0000 −0.822179 −0.411089 0.911595i \(-0.634852\pi\)
−0.411089 + 0.911595i \(0.634852\pi\)
\(180\) −3.00000 −0.223607
\(181\) 26.0000 1.93256 0.966282 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 3.00000 0.221163
\(185\) −6.00000 −0.441129
\(186\) −10.0000 −0.733236
\(187\) 2.00000 0.146254
\(188\) 7.00000 0.510527
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −7.00000 −0.505181
\(193\) −11.0000 −0.791797 −0.395899 0.918294i \(-0.629567\pi\)
−0.395899 + 0.918294i \(0.629567\pi\)
\(194\) 16.0000 1.14873
\(195\) −3.00000 −0.214834
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 2.00000 0.142134
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 12.0000 0.848528
\(201\) 3.00000 0.211604
\(202\) −18.0000 −1.26648
\(203\) 0 0
\(204\) −1.00000 −0.0700140
\(205\) −12.0000 −0.838116
\(206\) 11.0000 0.766406
\(207\) 1.00000 0.0695048
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) 9.00000 0.618123
\(213\) 15.0000 1.02778
\(214\) −4.00000 −0.273434
\(215\) 12.0000 0.818393
\(216\) −3.00000 −0.204124
\(217\) 0 0
\(218\) −8.00000 −0.541828
\(219\) −13.0000 −0.878459
\(220\) 6.00000 0.404520
\(221\) −1.00000 −0.0672673
\(222\) −2.00000 −0.134231
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) −7.00000 −0.465633
\(227\) 14.0000 0.929213 0.464606 0.885517i \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) 0 0
\(229\) −28.0000 −1.85029 −0.925146 0.379611i \(-0.876058\pi\)
−0.925146 + 0.379611i \(0.876058\pi\)
\(230\) −3.00000 −0.197814
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −21.0000 −1.36989
\(236\) −4.00000 −0.260378
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 3.00000 0.193649
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −4.00000 −0.255031
\(247\) 0 0
\(248\) −30.0000 −1.90500
\(249\) 6.00000 0.380235
\(250\) 3.00000 0.189737
\(251\) −10.0000 −0.631194 −0.315597 0.948893i \(-0.602205\pi\)
−0.315597 + 0.948893i \(0.602205\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) −2.00000 −0.125491
\(255\) 3.00000 0.187867
\(256\) −17.0000 −1.06250
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) −3.00000 −0.186052
\(261\) 0 0
\(262\) −3.00000 −0.185341
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 6.00000 0.369274
\(265\) −27.0000 −1.65860
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 3.00000 0.183254
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 3.00000 0.182574
\(271\) −14.0000 −0.850439 −0.425220 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) 17.0000 1.02701
\(275\) −8.00000 −0.482418
\(276\) 1.00000 0.0601929
\(277\) 13.0000 0.781094 0.390547 0.920583i \(-0.372286\pi\)
0.390547 + 0.920583i \(0.372286\pi\)
\(278\) −10.0000 −0.599760
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) 27.0000 1.61068 0.805342 0.592810i \(-0.201981\pi\)
0.805342 + 0.592810i \(0.201981\pi\)
\(282\) −7.00000 −0.416844
\(283\) −31.0000 −1.84276 −0.921379 0.388664i \(-0.872937\pi\)
−0.921379 + 0.388664i \(0.872937\pi\)
\(284\) 15.0000 0.890086
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) −5.00000 −0.294628
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 16.0000 0.937937
\(292\) −13.0000 −0.760767
\(293\) 13.0000 0.759468 0.379734 0.925096i \(-0.376015\pi\)
0.379734 + 0.925096i \(0.376015\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) −6.00000 −0.348743
\(297\) 2.00000 0.116052
\(298\) −5.00000 −0.289642
\(299\) 1.00000 0.0578315
\(300\) 4.00000 0.230940
\(301\) 0 0
\(302\) −4.00000 −0.230174
\(303\) −18.0000 −1.03407
\(304\) 0 0
\(305\) 6.00000 0.343559
\(306\) 1.00000 0.0571662
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 11.0000 0.625768
\(310\) 30.0000 1.70389
\(311\) −21.0000 −1.19080 −0.595400 0.803429i \(-0.703007\pi\)
−0.595400 + 0.803429i \(0.703007\pi\)
\(312\) −3.00000 −0.169842
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) −9.00000 −0.504695
\(319\) 0 0
\(320\) 21.0000 1.17394
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 4.00000 0.221880
\(326\) 20.0000 1.10770
\(327\) −8.00000 −0.442401
\(328\) −12.0000 −0.662589
\(329\) 0 0
\(330\) −6.00000 −0.330289
\(331\) −34.0000 −1.86881 −0.934405 0.356214i \(-0.884068\pi\)
−0.934405 + 0.356214i \(0.884068\pi\)
\(332\) 6.00000 0.329293
\(333\) −2.00000 −0.109599
\(334\) −19.0000 −1.03963
\(335\) −9.00000 −0.491723
\(336\) 0 0
\(337\) −12.0000 −0.653682 −0.326841 0.945079i \(-0.605984\pi\)
−0.326841 + 0.945079i \(0.605984\pi\)
\(338\) 12.0000 0.652714
\(339\) −7.00000 −0.380188
\(340\) 3.00000 0.162698
\(341\) 20.0000 1.08306
\(342\) 0 0
\(343\) 0 0
\(344\) 12.0000 0.646997
\(345\) −3.00000 −0.161515
\(346\) −16.0000 −0.860165
\(347\) 3.00000 0.161048 0.0805242 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) 0 0
\(349\) 5.00000 0.267644 0.133822 0.991005i \(-0.457275\pi\)
0.133822 + 0.991005i \(0.457275\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 10.0000 0.533002
\(353\) 4.00000 0.212899 0.106449 0.994318i \(-0.466052\pi\)
0.106449 + 0.994318i \(0.466052\pi\)
\(354\) 4.00000 0.212598
\(355\) −45.0000 −2.38835
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 11.0000 0.581368
\(359\) 22.0000 1.16112 0.580558 0.814219i \(-0.302835\pi\)
0.580558 + 0.814219i \(0.302835\pi\)
\(360\) 9.00000 0.474342
\(361\) −19.0000 −1.00000
\(362\) −26.0000 −1.36653
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 39.0000 2.04135
\(366\) 2.00000 0.104542
\(367\) 19.0000 0.991792 0.495896 0.868382i \(-0.334840\pi\)
0.495896 + 0.868382i \(0.334840\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −4.00000 −0.208232
\(370\) 6.00000 0.311925
\(371\) 0 0
\(372\) −10.0000 −0.518476
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −2.00000 −0.103418
\(375\) 3.00000 0.154919
\(376\) −21.0000 −1.08299
\(377\) 0 0
\(378\) 0 0
\(379\) −19.0000 −0.975964 −0.487982 0.872854i \(-0.662267\pi\)
−0.487982 + 0.872854i \(0.662267\pi\)
\(380\) 0 0
\(381\) −2.00000 −0.102463
\(382\) 0 0
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 11.0000 0.559885
\(387\) 4.00000 0.203331
\(388\) 16.0000 0.812277
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 3.00000 0.151911
\(391\) −1.00000 −0.0505722
\(392\) 0 0
\(393\) −3.00000 −0.151330
\(394\) 18.0000 0.906827
\(395\) 12.0000 0.603786
\(396\) 2.00000 0.100504
\(397\) −21.0000 −1.05396 −0.526980 0.849878i \(-0.676676\pi\)
−0.526980 + 0.849878i \(0.676676\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −11.0000 −0.549314 −0.274657 0.961542i \(-0.588564\pi\)
−0.274657 + 0.961542i \(0.588564\pi\)
\(402\) −3.00000 −0.149626
\(403\) −10.0000 −0.498135
\(404\) −18.0000 −0.895533
\(405\) 3.00000 0.149071
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 3.00000 0.148522
\(409\) −31.0000 −1.53285 −0.766426 0.642333i \(-0.777967\pi\)
−0.766426 + 0.642333i \(0.777967\pi\)
\(410\) 12.0000 0.592638
\(411\) 17.0000 0.838548
\(412\) 11.0000 0.541931
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) −18.0000 −0.883585
\(416\) −5.00000 −0.245145
\(417\) −10.0000 −0.489702
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 22.0000 1.07094
\(423\) −7.00000 −0.340352
\(424\) −27.0000 −1.31124
\(425\) −4.00000 −0.194029
\(426\) −15.0000 −0.726752
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) 2.00000 0.0965609
\(430\) −12.0000 −0.578691
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 1.00000 0.0481125
\(433\) 22.0000 1.05725 0.528626 0.848855i \(-0.322707\pi\)
0.528626 + 0.848855i \(0.322707\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −8.00000 −0.383131
\(437\) 0 0
\(438\) 13.0000 0.621164
\(439\) −22.0000 −1.05000 −0.525001 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) −18.0000 −0.858116
\(441\) 0 0
\(442\) 1.00000 0.0475651
\(443\) −11.0000 −0.522626 −0.261313 0.965254i \(-0.584155\pi\)
−0.261313 + 0.965254i \(0.584155\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 18.0000 0.853282
\(446\) 4.00000 0.189405
\(447\) −5.00000 −0.236492
\(448\) 0 0
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) −4.00000 −0.188562
\(451\) 8.00000 0.376705
\(452\) −7.00000 −0.329252
\(453\) −4.00000 −0.187936
\(454\) −14.0000 −0.657053
\(455\) 0 0
\(456\) 0 0
\(457\) 16.0000 0.748448 0.374224 0.927338i \(-0.377909\pi\)
0.374224 + 0.927338i \(0.377909\pi\)
\(458\) 28.0000 1.30835
\(459\) 1.00000 0.0466760
\(460\) −3.00000 −0.139876
\(461\) 42.0000 1.95614 0.978068 0.208288i \(-0.0667892\pi\)
0.978068 + 0.208288i \(0.0667892\pi\)
\(462\) 0 0
\(463\) 40.0000 1.85896 0.929479 0.368875i \(-0.120257\pi\)
0.929479 + 0.368875i \(0.120257\pi\)
\(464\) 0 0
\(465\) 30.0000 1.39122
\(466\) 24.0000 1.11178
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) 21.0000 0.968658
\(471\) 10.0000 0.460776
\(472\) 12.0000 0.552345
\(473\) −8.00000 −0.367840
\(474\) 4.00000 0.183726
\(475\) 0 0
\(476\) 0 0
\(477\) −9.00000 −0.412082
\(478\) 16.0000 0.731823
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 15.0000 0.684653
\(481\) −2.00000 −0.0911922
\(482\) 6.00000 0.273293
\(483\) 0 0
\(484\) 7.00000 0.318182
\(485\) −48.0000 −2.17957
\(486\) 1.00000 0.0453609
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 6.00000 0.271607
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) 9.00000 0.406164 0.203082 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) −4.00000 −0.180334
\(493\) 0 0
\(494\) 0 0
\(495\) −6.00000 −0.269680
\(496\) 10.0000 0.449013
\(497\) 0 0
\(498\) −6.00000 −0.268866
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 3.00000 0.134164
\(501\) −19.0000 −0.848857
\(502\) 10.0000 0.446322
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 54.0000 2.40297
\(506\) 2.00000 0.0889108
\(507\) 12.0000 0.532939
\(508\) −2.00000 −0.0887357
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) −3.00000 −0.132842
\(511\) 0 0
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) −33.0000 −1.45415
\(516\) 4.00000 0.176090
\(517\) 14.0000 0.615719
\(518\) 0 0
\(519\) −16.0000 −0.702322
\(520\) 9.00000 0.394676
\(521\) 29.0000 1.27051 0.635257 0.772301i \(-0.280894\pi\)
0.635257 + 0.772301i \(0.280894\pi\)
\(522\) 0 0
\(523\) −17.0000 −0.743358 −0.371679 0.928361i \(-0.621218\pi\)
−0.371679 + 0.928361i \(0.621218\pi\)
\(524\) −3.00000 −0.131056
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 10.0000 0.435607
\(528\) −2.00000 −0.0870388
\(529\) 1.00000 0.0434783
\(530\) 27.0000 1.17281
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) 6.00000 0.259645
\(535\) 12.0000 0.518805
\(536\) −9.00000 −0.388741
\(537\) 11.0000 0.474685
\(538\) 4.00000 0.172452
\(539\) 0 0
\(540\) 3.00000 0.129099
\(541\) 5.00000 0.214967 0.107483 0.994207i \(-0.465721\pi\)
0.107483 + 0.994207i \(0.465721\pi\)
\(542\) 14.0000 0.601351
\(543\) −26.0000 −1.11577
\(544\) 5.00000 0.214373
\(545\) 24.0000 1.02805
\(546\) 0 0
\(547\) −26.0000 −1.11168 −0.555840 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) 17.0000 0.726204
\(549\) 2.00000 0.0853579
\(550\) 8.00000 0.341121
\(551\) 0 0
\(552\) −3.00000 −0.127688
\(553\) 0 0
\(554\) −13.0000 −0.552317
\(555\) 6.00000 0.254686
\(556\) −10.0000 −0.424094
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 10.0000 0.423334
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −2.00000 −0.0844401
\(562\) −27.0000 −1.13893
\(563\) −40.0000 −1.68580 −0.842900 0.538071i \(-0.819153\pi\)
−0.842900 + 0.538071i \(0.819153\pi\)
\(564\) −7.00000 −0.294753
\(565\) 21.0000 0.883477
\(566\) 31.0000 1.30303
\(567\) 0 0
\(568\) −45.0000 −1.88816
\(569\) 33.0000 1.38343 0.691716 0.722170i \(-0.256855\pi\)
0.691716 + 0.722170i \(0.256855\pi\)
\(570\) 0 0
\(571\) 7.00000 0.292941 0.146470 0.989215i \(-0.453209\pi\)
0.146470 + 0.989215i \(0.453209\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 7.00000 0.291667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 16.0000 0.665512
\(579\) 11.0000 0.457144
\(580\) 0 0
\(581\) 0 0
\(582\) −16.0000 −0.663221
\(583\) 18.0000 0.745484
\(584\) 39.0000 1.61383
\(585\) 3.00000 0.124035
\(586\) −13.0000 −0.537025
\(587\) 37.0000 1.52715 0.763577 0.645717i \(-0.223441\pi\)
0.763577 + 0.645717i \(0.223441\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −12.0000 −0.494032
\(591\) 18.0000 0.740421
\(592\) 2.00000 0.0821995
\(593\) −26.0000 −1.06769 −0.533846 0.845582i \(-0.679254\pi\)
−0.533846 + 0.845582i \(0.679254\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −5.00000 −0.204808
\(597\) 16.0000 0.654836
\(598\) −1.00000 −0.0408930
\(599\) 39.0000 1.59350 0.796748 0.604311i \(-0.206552\pi\)
0.796748 + 0.604311i \(0.206552\pi\)
\(600\) −12.0000 −0.489898
\(601\) −18.0000 −0.734235 −0.367118 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(602\) 0 0
\(603\) −3.00000 −0.122169
\(604\) −4.00000 −0.162758
\(605\) −21.0000 −0.853771
\(606\) 18.0000 0.731200
\(607\) −6.00000 −0.243532 −0.121766 0.992559i \(-0.538856\pi\)
−0.121766 + 0.992559i \(0.538856\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −6.00000 −0.242933
\(611\) −7.00000 −0.283190
\(612\) 1.00000 0.0404226
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −20.0000 −0.807134
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) −29.0000 −1.16750 −0.583748 0.811935i \(-0.698414\pi\)
−0.583748 + 0.811935i \(0.698414\pi\)
\(618\) −11.0000 −0.442485
\(619\) 27.0000 1.08522 0.542611 0.839984i \(-0.317436\pi\)
0.542611 + 0.839984i \(0.317436\pi\)
\(620\) 30.0000 1.20483
\(621\) −1.00000 −0.0401286
\(622\) 21.0000 0.842023
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) −29.0000 −1.16000
\(626\) −8.00000 −0.319744
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) 2.00000 0.0797452
\(630\) 0 0
\(631\) −5.00000 −0.199047 −0.0995234 0.995035i \(-0.531732\pi\)
−0.0995234 + 0.995035i \(0.531732\pi\)
\(632\) 12.0000 0.477334
\(633\) 22.0000 0.874421
\(634\) −10.0000 −0.397151
\(635\) 6.00000 0.238103
\(636\) −9.00000 −0.356873
\(637\) 0 0
\(638\) 0 0
\(639\) −15.0000 −0.593391
\(640\) 9.00000 0.355756
\(641\) 33.0000 1.30342 0.651711 0.758468i \(-0.274052\pi\)
0.651711 + 0.758468i \(0.274052\pi\)
\(642\) 4.00000 0.157867
\(643\) −12.0000 −0.473234 −0.236617 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 0 0
\(645\) −12.0000 −0.472500
\(646\) 0 0
\(647\) −8.00000 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 3.00000 0.117851
\(649\) −8.00000 −0.314027
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 8.00000 0.312825
\(655\) 9.00000 0.351659
\(656\) 4.00000 0.156174
\(657\) 13.0000 0.507178
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) −6.00000 −0.233550
\(661\) 8.00000 0.311164 0.155582 0.987823i \(-0.450275\pi\)
0.155582 + 0.987823i \(0.450275\pi\)
\(662\) 34.0000 1.32145
\(663\) 1.00000 0.0388368
\(664\) −18.0000 −0.698535
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 0 0
\(668\) −19.0000 −0.735132
\(669\) 4.00000 0.154649
\(670\) 9.00000 0.347700
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) 50.0000 1.92736 0.963679 0.267063i \(-0.0860531\pi\)
0.963679 + 0.267063i \(0.0860531\pi\)
\(674\) 12.0000 0.462223
\(675\) −4.00000 −0.153960
\(676\) 12.0000 0.461538
\(677\) 11.0000 0.422764 0.211382 0.977403i \(-0.432204\pi\)
0.211382 + 0.977403i \(0.432204\pi\)
\(678\) 7.00000 0.268833
\(679\) 0 0
\(680\) −9.00000 −0.345134
\(681\) −14.0000 −0.536481
\(682\) −20.0000 −0.765840
\(683\) 41.0000 1.56882 0.784411 0.620242i \(-0.212966\pi\)
0.784411 + 0.620242i \(0.212966\pi\)
\(684\) 0 0
\(685\) −51.0000 −1.94861
\(686\) 0 0
\(687\) 28.0000 1.06827
\(688\) −4.00000 −0.152499
\(689\) −9.00000 −0.342873
\(690\) 3.00000 0.114208
\(691\) 22.0000 0.836919 0.418460 0.908235i \(-0.362570\pi\)
0.418460 + 0.908235i \(0.362570\pi\)
\(692\) −16.0000 −0.608229
\(693\) 0 0
\(694\) −3.00000 −0.113878
\(695\) 30.0000 1.13796
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) −5.00000 −0.189253
\(699\) 24.0000 0.907763
\(700\) 0 0
\(701\) −39.0000 −1.47301 −0.736505 0.676432i \(-0.763525\pi\)
−0.736505 + 0.676432i \(0.763525\pi\)
\(702\) 1.00000 0.0377426
\(703\) 0 0
\(704\) −14.0000 −0.527645
\(705\) 21.0000 0.790906
\(706\) −4.00000 −0.150542
\(707\) 0 0
\(708\) 4.00000 0.150329
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 45.0000 1.68882
\(711\) 4.00000 0.150012
\(712\) 18.0000 0.674579
\(713\) −10.0000 −0.374503
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) 11.0000 0.411089
\(717\) 16.0000 0.597531
\(718\) −22.0000 −0.821033
\(719\) −15.0000 −0.559406 −0.279703 0.960087i \(-0.590236\pi\)
−0.279703 + 0.960087i \(0.590236\pi\)
\(720\) −3.00000 −0.111803
\(721\) 0 0
\(722\) 19.0000 0.707107
\(723\) 6.00000 0.223142
\(724\) −26.0000 −0.966282
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −39.0000 −1.44345
\(731\) −4.00000 −0.147945
\(732\) 2.00000 0.0739221
\(733\) 24.0000 0.886460 0.443230 0.896408i \(-0.353832\pi\)
0.443230 + 0.896408i \(0.353832\pi\)
\(734\) −19.0000 −0.701303
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) 6.00000 0.221013
\(738\) 4.00000 0.147242
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 6.00000 0.220564
\(741\) 0 0
\(742\) 0 0
\(743\) −18.0000 −0.660356 −0.330178 0.943919i \(-0.607109\pi\)
−0.330178 + 0.943919i \(0.607109\pi\)
\(744\) 30.0000 1.09985
\(745\) 15.0000 0.549557
\(746\) 26.0000 0.951928
\(747\) −6.00000 −0.219529
\(748\) −2.00000 −0.0731272
\(749\) 0 0
\(750\) −3.00000 −0.109545
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 7.00000 0.255264
\(753\) 10.0000 0.364420
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) 12.0000 0.436147 0.218074 0.975932i \(-0.430023\pi\)
0.218074 + 0.975932i \(0.430023\pi\)
\(758\) 19.0000 0.690111
\(759\) 2.00000 0.0725954
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 2.00000 0.0724524
\(763\) 0 0
\(764\) 0 0
\(765\) −3.00000 −0.108465
\(766\) −6.00000 −0.216789
\(767\) 4.00000 0.144432
\(768\) 17.0000 0.613435
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 11.0000 0.395899
\(773\) −9.00000 −0.323708 −0.161854 0.986815i \(-0.551747\pi\)
−0.161854 + 0.986815i \(0.551747\pi\)
\(774\) −4.00000 −0.143777
\(775\) −40.0000 −1.43684
\(776\) −48.0000 −1.72310
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 0 0
\(780\) 3.00000 0.107417
\(781\) 30.0000 1.07348
\(782\) 1.00000 0.0357599
\(783\) 0 0
\(784\) 0 0
\(785\) −30.0000 −1.07075
\(786\) 3.00000 0.107006
\(787\) −51.0000 −1.81795 −0.908977 0.416847i \(-0.863135\pi\)
−0.908977 + 0.416847i \(0.863135\pi\)
\(788\) 18.0000 0.641223
\(789\) 16.0000 0.569615
\(790\) −12.0000 −0.426941
\(791\) 0 0
\(792\) −6.00000 −0.213201
\(793\) 2.00000 0.0710221
\(794\) 21.0000 0.745262
\(795\) 27.0000 0.957591
\(796\) 16.0000 0.567105
\(797\) −19.0000 −0.673015 −0.336507 0.941681i \(-0.609246\pi\)
−0.336507 + 0.941681i \(0.609246\pi\)
\(798\) 0 0
\(799\) 7.00000 0.247642
\(800\) −20.0000 −0.707107
\(801\) 6.00000 0.212000
\(802\) 11.0000 0.388424
\(803\) −26.0000 −0.917520
\(804\) −3.00000 −0.105802
\(805\) 0 0
\(806\) 10.0000 0.352235
\(807\) 4.00000 0.140807
\(808\) 54.0000 1.89971
\(809\) −54.0000 −1.89854 −0.949269 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) −3.00000 −0.105409
\(811\) 30.0000 1.05344 0.526721 0.850038i \(-0.323421\pi\)
0.526721 + 0.850038i \(0.323421\pi\)
\(812\) 0 0
\(813\) 14.0000 0.491001
\(814\) −4.00000 −0.140200
\(815\) −60.0000 −2.10171
\(816\) −1.00000 −0.0350070
\(817\) 0 0
\(818\) 31.0000 1.08389
\(819\) 0 0
\(820\) 12.0000 0.419058
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) −17.0000 −0.592943
\(823\) −26.0000 −0.906303 −0.453152 0.891434i \(-0.649700\pi\)
−0.453152 + 0.891434i \(0.649700\pi\)
\(824\) −33.0000 −1.14961
\(825\) 8.00000 0.278524
\(826\) 0 0
\(827\) 48.0000 1.66912 0.834562 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 3.00000 0.104194 0.0520972 0.998642i \(-0.483409\pi\)
0.0520972 + 0.998642i \(0.483409\pi\)
\(830\) 18.0000 0.624789
\(831\) −13.0000 −0.450965
\(832\) 7.00000 0.242681
\(833\) 0 0
\(834\) 10.0000 0.346272
\(835\) 57.0000 1.97257
\(836\) 0 0
\(837\) 10.0000 0.345651
\(838\) 12.0000 0.414533
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −8.00000 −0.275698
\(843\) −27.0000 −0.929929
\(844\) 22.0000 0.757271
\(845\) −36.0000 −1.23844
\(846\) 7.00000 0.240665
\(847\) 0 0
\(848\) 9.00000 0.309061
\(849\) 31.0000 1.06392
\(850\) 4.00000 0.137199
\(851\) −2.00000 −0.0685591
\(852\) −15.0000 −0.513892
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 46.0000 1.57133 0.785665 0.618652i \(-0.212321\pi\)
0.785665 + 0.618652i \(0.212321\pi\)
\(858\) −2.00000 −0.0682789
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) −12.0000 −0.409197
\(861\) 0 0
\(862\) 32.0000 1.08992
\(863\) −21.0000 −0.714848 −0.357424 0.933942i \(-0.616345\pi\)
−0.357424 + 0.933942i \(0.616345\pi\)
\(864\) 5.00000 0.170103
\(865\) 48.0000 1.63205
\(866\) −22.0000 −0.747590
\(867\) 16.0000 0.543388
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) −3.00000 −0.101651
\(872\) 24.0000 0.812743
\(873\) −16.0000 −0.541518
\(874\) 0 0
\(875\) 0 0
\(876\) 13.0000 0.439229
\(877\) 19.0000 0.641584 0.320792 0.947150i \(-0.396051\pi\)
0.320792 + 0.947150i \(0.396051\pi\)
\(878\) 22.0000 0.742464
\(879\) −13.0000 −0.438479
\(880\) 6.00000 0.202260
\(881\) 9.00000 0.303218 0.151609 0.988441i \(-0.451555\pi\)
0.151609 + 0.988441i \(0.451555\pi\)
\(882\) 0 0
\(883\) 24.0000 0.807664 0.403832 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(884\) 1.00000 0.0336336
\(885\) −12.0000 −0.403376
\(886\) 11.0000 0.369552
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 6.00000 0.201347
\(889\) 0 0
\(890\) −18.0000 −0.603361
\(891\) −2.00000 −0.0670025
\(892\) 4.00000 0.133930
\(893\) 0 0
\(894\) 5.00000 0.167225
\(895\) −33.0000 −1.10307
\(896\) 0 0
\(897\) −1.00000 −0.0333890
\(898\) −14.0000 −0.467186
\(899\) 0 0
\(900\) −4.00000 −0.133333
\(901\) 9.00000 0.299833
\(902\) −8.00000 −0.266371
\(903\) 0 0
\(904\) 21.0000 0.698450
\(905\) 78.0000 2.59281
\(906\) 4.00000 0.132891
\(907\) −21.0000 −0.697294 −0.348647 0.937254i \(-0.613359\pi\)
−0.348647 + 0.937254i \(0.613359\pi\)
\(908\) −14.0000 −0.464606
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) −16.0000 −0.529233
\(915\) −6.00000 −0.198354
\(916\) 28.0000 0.925146
\(917\) 0 0
\(918\) −1.00000 −0.0330049
\(919\) −49.0000 −1.61636 −0.808180 0.588935i \(-0.799547\pi\)
−0.808180 + 0.588935i \(0.799547\pi\)
\(920\) 9.00000 0.296721
\(921\) −20.0000 −0.659022
\(922\) −42.0000 −1.38320
\(923\) −15.0000 −0.493731
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) −40.0000 −1.31448
\(927\) −11.0000 −0.361287
\(928\) 0 0
\(929\) −24.0000 −0.787414 −0.393707 0.919236i \(-0.628808\pi\)
−0.393707 + 0.919236i \(0.628808\pi\)
\(930\) −30.0000 −0.983739
\(931\) 0 0
\(932\) 24.0000 0.786146
\(933\) 21.0000 0.687509
\(934\) −28.0000 −0.916188
\(935\) 6.00000 0.196221
\(936\) 3.00000 0.0980581
\(937\) 8.00000 0.261349 0.130674 0.991425i \(-0.458286\pi\)
0.130674 + 0.991425i \(0.458286\pi\)
\(938\) 0 0
\(939\) −8.00000 −0.261070
\(940\) 21.0000 0.684944
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) −10.0000 −0.325818
\(943\) −4.00000 −0.130258
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) 47.0000 1.52729 0.763647 0.645634i \(-0.223407\pi\)
0.763647 + 0.645634i \(0.223407\pi\)
\(948\) 4.00000 0.129914
\(949\) 13.0000 0.421998
\(950\) 0 0
\(951\) −10.0000 −0.324272
\(952\) 0 0
\(953\) −14.0000 −0.453504 −0.226752 0.973952i \(-0.572811\pi\)
−0.226752 + 0.973952i \(0.572811\pi\)
\(954\) 9.00000 0.291386
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) 0 0
\(958\) 16.0000 0.516937
\(959\) 0 0
\(960\) −21.0000 −0.677772
\(961\) 69.0000 2.22581
\(962\) 2.00000 0.0644826
\(963\) 4.00000 0.128898
\(964\) 6.00000 0.193247
\(965\) −33.0000 −1.06231
\(966\) 0 0
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) −21.0000 −0.674966
\(969\) 0 0
\(970\) 48.0000 1.54119
\(971\) 4.00000 0.128366 0.0641831 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) −4.00000 −0.128103
\(976\) −2.00000 −0.0640184
\(977\) 39.0000 1.24772 0.623860 0.781536i \(-0.285563\pi\)
0.623860 + 0.781536i \(0.285563\pi\)
\(978\) −20.0000 −0.639529
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) 8.00000 0.255420
\(982\) −9.00000 −0.287202
\(983\) 14.0000 0.446531 0.223265 0.974758i \(-0.428328\pi\)
0.223265 + 0.974758i \(0.428328\pi\)
\(984\) 12.0000 0.382546
\(985\) −54.0000 −1.72058
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 6.00000 0.190693
\(991\) −50.0000 −1.58830 −0.794151 0.607720i \(-0.792084\pi\)
−0.794151 + 0.607720i \(0.792084\pi\)
\(992\) 50.0000 1.58750
\(993\) 34.0000 1.07896
\(994\) 0 0
\(995\) −48.0000 −1.52170
\(996\) −6.00000 −0.190117
\(997\) −42.0000 −1.33015 −0.665077 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) −36.0000 −1.13956
\(999\) 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 3381.2.a.b.1.1 1
7.3 odd 6 483.2.i.d.415.1 yes 2
7.5 odd 6 483.2.i.d.277.1 2
7.6 odd 2 3381.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.d.277.1 2 7.5 odd 6
483.2.i.d.415.1 yes 2 7.3 odd 6
3381.2.a.b.1.1 1 1.1 even 1 trivial
3381.2.a.c.1.1 1 7.6 odd 2