Properties

Label 3381.2.a.t.1.2
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+2.23607 q^{2} +1.00000 q^{3} +3.00000 q^{4} +3.23607 q^{5} +2.23607 q^{6} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.23607 q^{2} +1.00000 q^{3} +3.00000 q^{4} +3.23607 q^{5} +2.23607 q^{6} +2.23607 q^{8} +1.00000 q^{9} +7.23607 q^{10} +4.00000 q^{11} +3.00000 q^{12} -4.47214 q^{13} +3.23607 q^{15} -1.00000 q^{16} +7.23607 q^{17} +2.23607 q^{18} -2.76393 q^{19} +9.70820 q^{20} +8.94427 q^{22} +1.00000 q^{23} +2.23607 q^{24} +5.47214 q^{25} -10.0000 q^{26} +1.00000 q^{27} -4.47214 q^{29} +7.23607 q^{30} -2.47214 q^{31} -6.70820 q^{32} +4.00000 q^{33} +16.1803 q^{34} +3.00000 q^{36} -4.47214 q^{37} -6.18034 q^{38} -4.47214 q^{39} +7.23607 q^{40} -6.94427 q^{41} +7.70820 q^{43} +12.0000 q^{44} +3.23607 q^{45} +2.23607 q^{46} +4.00000 q^{47} -1.00000 q^{48} +12.2361 q^{50} +7.23607 q^{51} -13.4164 q^{52} -0.763932 q^{53} +2.23607 q^{54} +12.9443 q^{55} -2.76393 q^{57} -10.0000 q^{58} -12.9443 q^{59} +9.70820 q^{60} +4.47214 q^{61} -5.52786 q^{62} -13.0000 q^{64} -14.4721 q^{65} +8.94427 q^{66} +5.23607 q^{67} +21.7082 q^{68} +1.00000 q^{69} -8.00000 q^{71} +2.23607 q^{72} +10.9443 q^{73} -10.0000 q^{74} +5.47214 q^{75} -8.29180 q^{76} -10.0000 q^{78} -3.70820 q^{79} -3.23607 q^{80} +1.00000 q^{81} -15.5279 q^{82} -4.00000 q^{83} +23.4164 q^{85} +17.2361 q^{86} -4.47214 q^{87} +8.94427 q^{88} -3.23607 q^{89} +7.23607 q^{90} +3.00000 q^{92} -2.47214 q^{93} +8.94427 q^{94} -8.94427 q^{95} -6.70820 q^{96} +0.472136 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{9} + 10 q^{10} + 8 q^{11} + 6 q^{12} + 2 q^{15} - 2 q^{16} + 10 q^{17} - 10 q^{19} + 6 q^{20} + 2 q^{23} + 2 q^{25} - 20 q^{26} + 2 q^{27} + 10 q^{30} + 4 q^{31} + 8 q^{33} + 10 q^{34} + 6 q^{36} + 10 q^{38} + 10 q^{40} + 4 q^{41} + 2 q^{43} + 24 q^{44} + 2 q^{45} + 8 q^{47} - 2 q^{48} + 20 q^{50} + 10 q^{51} - 6 q^{53} + 8 q^{55} - 10 q^{57} - 20 q^{58} - 8 q^{59} + 6 q^{60} - 20 q^{62} - 26 q^{64} - 20 q^{65} + 6 q^{67} + 30 q^{68} + 2 q^{69} - 16 q^{71} + 4 q^{73} - 20 q^{74} + 2 q^{75} - 30 q^{76} - 20 q^{78} + 6 q^{79} - 2 q^{80} + 2 q^{81} - 40 q^{82} - 8 q^{83} + 20 q^{85} + 30 q^{86} - 2 q^{89} + 10 q^{90} + 6 q^{92} + 4 q^{93} - 8 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23607 1.58114 0.790569 0.612372i \(-0.209785\pi\)
0.790569 + 0.612372i \(0.209785\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.00000 1.50000
\(5\) 3.23607 1.44721 0.723607 0.690212i \(-0.242483\pi\)
0.723607 + 0.690212i \(0.242483\pi\)
\(6\) 2.23607 0.912871
\(7\) 0 0
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) 7.23607 2.28825
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 3.00000 0.866025
\(13\) −4.47214 −1.24035 −0.620174 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 0 0
\(15\) 3.23607 0.835549
\(16\) −1.00000 −0.250000
\(17\) 7.23607 1.75500 0.877502 0.479573i \(-0.159208\pi\)
0.877502 + 0.479573i \(0.159208\pi\)
\(18\) 2.23607 0.527046
\(19\) −2.76393 −0.634089 −0.317045 0.948411i \(-0.602691\pi\)
−0.317045 + 0.948411i \(0.602691\pi\)
\(20\) 9.70820 2.17082
\(21\) 0 0
\(22\) 8.94427 1.90693
\(23\) 1.00000 0.208514
\(24\) 2.23607 0.456435
\(25\) 5.47214 1.09443
\(26\) −10.0000 −1.96116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 7.23607 1.32112
\(31\) −2.47214 −0.444009 −0.222004 0.975046i \(-0.571260\pi\)
−0.222004 + 0.975046i \(0.571260\pi\)
\(32\) −6.70820 −1.18585
\(33\) 4.00000 0.696311
\(34\) 16.1803 2.77491
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) −4.47214 −0.735215 −0.367607 0.929981i \(-0.619823\pi\)
−0.367607 + 0.929981i \(0.619823\pi\)
\(38\) −6.18034 −1.00258
\(39\) −4.47214 −0.716115
\(40\) 7.23607 1.14412
\(41\) −6.94427 −1.08451 −0.542257 0.840213i \(-0.682430\pi\)
−0.542257 + 0.840213i \(0.682430\pi\)
\(42\) 0 0
\(43\) 7.70820 1.17549 0.587745 0.809046i \(-0.300016\pi\)
0.587745 + 0.809046i \(0.300016\pi\)
\(44\) 12.0000 1.80907
\(45\) 3.23607 0.482405
\(46\) 2.23607 0.329690
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 12.2361 1.73044
\(51\) 7.23607 1.01325
\(52\) −13.4164 −1.86052
\(53\) −0.763932 −0.104934 −0.0524671 0.998623i \(-0.516708\pi\)
−0.0524671 + 0.998623i \(0.516708\pi\)
\(54\) 2.23607 0.304290
\(55\) 12.9443 1.74541
\(56\) 0 0
\(57\) −2.76393 −0.366092
\(58\) −10.0000 −1.31306
\(59\) −12.9443 −1.68520 −0.842600 0.538539i \(-0.818976\pi\)
−0.842600 + 0.538539i \(0.818976\pi\)
\(60\) 9.70820 1.25332
\(61\) 4.47214 0.572598 0.286299 0.958140i \(-0.407575\pi\)
0.286299 + 0.958140i \(0.407575\pi\)
\(62\) −5.52786 −0.702039
\(63\) 0 0
\(64\) −13.0000 −1.62500
\(65\) −14.4721 −1.79505
\(66\) 8.94427 1.10096
\(67\) 5.23607 0.639688 0.319844 0.947470i \(-0.396370\pi\)
0.319844 + 0.947470i \(0.396370\pi\)
\(68\) 21.7082 2.63251
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 2.23607 0.263523
\(73\) 10.9443 1.28093 0.640465 0.767987i \(-0.278742\pi\)
0.640465 + 0.767987i \(0.278742\pi\)
\(74\) −10.0000 −1.16248
\(75\) 5.47214 0.631868
\(76\) −8.29180 −0.951134
\(77\) 0 0
\(78\) −10.0000 −1.13228
\(79\) −3.70820 −0.417206 −0.208603 0.978000i \(-0.566892\pi\)
−0.208603 + 0.978000i \(0.566892\pi\)
\(80\) −3.23607 −0.361803
\(81\) 1.00000 0.111111
\(82\) −15.5279 −1.71477
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 23.4164 2.53987
\(86\) 17.2361 1.85861
\(87\) −4.47214 −0.479463
\(88\) 8.94427 0.953463
\(89\) −3.23607 −0.343023 −0.171511 0.985182i \(-0.554865\pi\)
−0.171511 + 0.985182i \(0.554865\pi\)
\(90\) 7.23607 0.762749
\(91\) 0 0
\(92\) 3.00000 0.312772
\(93\) −2.47214 −0.256349
\(94\) 8.94427 0.922531
\(95\) −8.94427 −0.917663
\(96\) −6.70820 −0.684653
\(97\) 0.472136 0.0479381 0.0239691 0.999713i \(-0.492370\pi\)
0.0239691 + 0.999713i \(0.492370\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 16.4164 1.64164
\(101\) −10.9443 −1.08900 −0.544498 0.838762i \(-0.683280\pi\)
−0.544498 + 0.838762i \(0.683280\pi\)
\(102\) 16.1803 1.60209
\(103\) 6.76393 0.666470 0.333235 0.942844i \(-0.391860\pi\)
0.333235 + 0.942844i \(0.391860\pi\)
\(104\) −10.0000 −0.980581
\(105\) 0 0
\(106\) −1.70820 −0.165915
\(107\) −0.944272 −0.0912862 −0.0456431 0.998958i \(-0.514534\pi\)
−0.0456431 + 0.998958i \(0.514534\pi\)
\(108\) 3.00000 0.288675
\(109\) 2.94427 0.282010 0.141005 0.990009i \(-0.454967\pi\)
0.141005 + 0.990009i \(0.454967\pi\)
\(110\) 28.9443 2.75973
\(111\) −4.47214 −0.424476
\(112\) 0 0
\(113\) 16.1803 1.52212 0.761059 0.648682i \(-0.224680\pi\)
0.761059 + 0.648682i \(0.224680\pi\)
\(114\) −6.18034 −0.578842
\(115\) 3.23607 0.301765
\(116\) −13.4164 −1.24568
\(117\) −4.47214 −0.413449
\(118\) −28.9443 −2.66454
\(119\) 0 0
\(120\) 7.23607 0.660560
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) −6.94427 −0.626144
\(124\) −7.41641 −0.666013
\(125\) 1.52786 0.136656
\(126\) 0 0
\(127\) 10.4721 0.929252 0.464626 0.885507i \(-0.346189\pi\)
0.464626 + 0.885507i \(0.346189\pi\)
\(128\) −15.6525 −1.38350
\(129\) 7.70820 0.678670
\(130\) −32.3607 −2.83822
\(131\) 0.944272 0.0825014 0.0412507 0.999149i \(-0.486866\pi\)
0.0412507 + 0.999149i \(0.486866\pi\)
\(132\) 12.0000 1.04447
\(133\) 0 0
\(134\) 11.7082 1.01143
\(135\) 3.23607 0.278516
\(136\) 16.1803 1.38745
\(137\) 3.23607 0.276476 0.138238 0.990399i \(-0.455856\pi\)
0.138238 + 0.990399i \(0.455856\pi\)
\(138\) 2.23607 0.190347
\(139\) −8.94427 −0.758643 −0.379322 0.925265i \(-0.623843\pi\)
−0.379322 + 0.925265i \(0.623843\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) −17.8885 −1.50117
\(143\) −17.8885 −1.49592
\(144\) −1.00000 −0.0833333
\(145\) −14.4721 −1.20185
\(146\) 24.4721 2.02533
\(147\) 0 0
\(148\) −13.4164 −1.10282
\(149\) 1.70820 0.139942 0.0699708 0.997549i \(-0.477709\pi\)
0.0699708 + 0.997549i \(0.477709\pi\)
\(150\) 12.2361 0.999071
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −6.18034 −0.501292
\(153\) 7.23607 0.585001
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) −13.4164 −1.07417
\(157\) 12.4721 0.995385 0.497692 0.867354i \(-0.334181\pi\)
0.497692 + 0.867354i \(0.334181\pi\)
\(158\) −8.29180 −0.659660
\(159\) −0.763932 −0.0605838
\(160\) −21.7082 −1.71618
\(161\) 0 0
\(162\) 2.23607 0.175682
\(163\) −19.4164 −1.52081 −0.760405 0.649449i \(-0.775000\pi\)
−0.760405 + 0.649449i \(0.775000\pi\)
\(164\) −20.8328 −1.62677
\(165\) 12.9443 1.00771
\(166\) −8.94427 −0.694210
\(167\) 4.94427 0.382599 0.191300 0.981532i \(-0.438730\pi\)
0.191300 + 0.981532i \(0.438730\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 52.3607 4.01588
\(171\) −2.76393 −0.211363
\(172\) 23.1246 1.76324
\(173\) 4.47214 0.340010 0.170005 0.985443i \(-0.445622\pi\)
0.170005 + 0.985443i \(0.445622\pi\)
\(174\) −10.0000 −0.758098
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) −12.9443 −0.972951
\(178\) −7.23607 −0.542366
\(179\) 3.05573 0.228396 0.114198 0.993458i \(-0.463570\pi\)
0.114198 + 0.993458i \(0.463570\pi\)
\(180\) 9.70820 0.723607
\(181\) −23.8885 −1.77562 −0.887811 0.460209i \(-0.847775\pi\)
−0.887811 + 0.460209i \(0.847775\pi\)
\(182\) 0 0
\(183\) 4.47214 0.330590
\(184\) 2.23607 0.164845
\(185\) −14.4721 −1.06401
\(186\) −5.52786 −0.405323
\(187\) 28.9443 2.11661
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) −20.0000 −1.45095
\(191\) 10.4721 0.757737 0.378869 0.925450i \(-0.376313\pi\)
0.378869 + 0.925450i \(0.376313\pi\)
\(192\) −13.0000 −0.938194
\(193\) −9.41641 −0.677808 −0.338904 0.940821i \(-0.610056\pi\)
−0.338904 + 0.940821i \(0.610056\pi\)
\(194\) 1.05573 0.0757969
\(195\) −14.4721 −1.03637
\(196\) 0 0
\(197\) −9.41641 −0.670891 −0.335446 0.942060i \(-0.608887\pi\)
−0.335446 + 0.942060i \(0.608887\pi\)
\(198\) 8.94427 0.635642
\(199\) 3.70820 0.262868 0.131434 0.991325i \(-0.458042\pi\)
0.131434 + 0.991325i \(0.458042\pi\)
\(200\) 12.2361 0.865221
\(201\) 5.23607 0.369324
\(202\) −24.4721 −1.72185
\(203\) 0 0
\(204\) 21.7082 1.51988
\(205\) −22.4721 −1.56952
\(206\) 15.1246 1.05378
\(207\) 1.00000 0.0695048
\(208\) 4.47214 0.310087
\(209\) −11.0557 −0.764741
\(210\) 0 0
\(211\) 22.4721 1.54705 0.773523 0.633768i \(-0.218493\pi\)
0.773523 + 0.633768i \(0.218493\pi\)
\(212\) −2.29180 −0.157401
\(213\) −8.00000 −0.548151
\(214\) −2.11146 −0.144336
\(215\) 24.9443 1.70119
\(216\) 2.23607 0.152145
\(217\) 0 0
\(218\) 6.58359 0.445897
\(219\) 10.9443 0.739545
\(220\) 38.8328 2.61811
\(221\) −32.3607 −2.17681
\(222\) −10.0000 −0.671156
\(223\) −9.88854 −0.662186 −0.331093 0.943598i \(-0.607417\pi\)
−0.331093 + 0.943598i \(0.607417\pi\)
\(224\) 0 0
\(225\) 5.47214 0.364809
\(226\) 36.1803 2.40668
\(227\) 22.4721 1.49153 0.745764 0.666210i \(-0.232085\pi\)
0.745764 + 0.666210i \(0.232085\pi\)
\(228\) −8.29180 −0.549138
\(229\) 14.9443 0.987545 0.493773 0.869591i \(-0.335618\pi\)
0.493773 + 0.869591i \(0.335618\pi\)
\(230\) 7.23607 0.477132
\(231\) 0 0
\(232\) −10.0000 −0.656532
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) −10.0000 −0.653720
\(235\) 12.9443 0.844391
\(236\) −38.8328 −2.52780
\(237\) −3.70820 −0.240874
\(238\) 0 0
\(239\) −12.9443 −0.837295 −0.418648 0.908149i \(-0.637496\pi\)
−0.418648 + 0.908149i \(0.637496\pi\)
\(240\) −3.23607 −0.208887
\(241\) −28.4721 −1.83405 −0.917026 0.398828i \(-0.869417\pi\)
−0.917026 + 0.398828i \(0.869417\pi\)
\(242\) 11.1803 0.718699
\(243\) 1.00000 0.0641500
\(244\) 13.4164 0.858898
\(245\) 0 0
\(246\) −15.5279 −0.990020
\(247\) 12.3607 0.786491
\(248\) −5.52786 −0.351020
\(249\) −4.00000 −0.253490
\(250\) 3.41641 0.216073
\(251\) 1.52786 0.0964379 0.0482190 0.998837i \(-0.484645\pi\)
0.0482190 + 0.998837i \(0.484645\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 23.4164 1.46928
\(255\) 23.4164 1.46639
\(256\) −9.00000 −0.562500
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 17.2361 1.07307
\(259\) 0 0
\(260\) −43.4164 −2.69257
\(261\) −4.47214 −0.276818
\(262\) 2.11146 0.130446
\(263\) −2.47214 −0.152438 −0.0762192 0.997091i \(-0.524285\pi\)
−0.0762192 + 0.997091i \(0.524285\pi\)
\(264\) 8.94427 0.550482
\(265\) −2.47214 −0.151862
\(266\) 0 0
\(267\) −3.23607 −0.198044
\(268\) 15.7082 0.959531
\(269\) −8.47214 −0.516555 −0.258278 0.966071i \(-0.583155\pi\)
−0.258278 + 0.966071i \(0.583155\pi\)
\(270\) 7.23607 0.440373
\(271\) 26.4721 1.60807 0.804034 0.594583i \(-0.202683\pi\)
0.804034 + 0.594583i \(0.202683\pi\)
\(272\) −7.23607 −0.438751
\(273\) 0 0
\(274\) 7.23607 0.437147
\(275\) 21.8885 1.31993
\(276\) 3.00000 0.180579
\(277\) −19.8885 −1.19499 −0.597493 0.801874i \(-0.703837\pi\)
−0.597493 + 0.801874i \(0.703837\pi\)
\(278\) −20.0000 −1.19952
\(279\) −2.47214 −0.148003
\(280\) 0 0
\(281\) −6.65248 −0.396853 −0.198427 0.980116i \(-0.563583\pi\)
−0.198427 + 0.980116i \(0.563583\pi\)
\(282\) 8.94427 0.532624
\(283\) 7.70820 0.458205 0.229103 0.973402i \(-0.426421\pi\)
0.229103 + 0.973402i \(0.426421\pi\)
\(284\) −24.0000 −1.42414
\(285\) −8.94427 −0.529813
\(286\) −40.0000 −2.36525
\(287\) 0 0
\(288\) −6.70820 −0.395285
\(289\) 35.3607 2.08004
\(290\) −32.3607 −1.90028
\(291\) 0.472136 0.0276771
\(292\) 32.8328 1.92140
\(293\) 5.70820 0.333477 0.166738 0.986001i \(-0.446676\pi\)
0.166738 + 0.986001i \(0.446676\pi\)
\(294\) 0 0
\(295\) −41.8885 −2.43885
\(296\) −10.0000 −0.581238
\(297\) 4.00000 0.232104
\(298\) 3.81966 0.221267
\(299\) −4.47214 −0.258630
\(300\) 16.4164 0.947802
\(301\) 0 0
\(302\) −35.7771 −2.05874
\(303\) −10.9443 −0.628732
\(304\) 2.76393 0.158522
\(305\) 14.4721 0.828672
\(306\) 16.1803 0.924968
\(307\) 6.47214 0.369384 0.184692 0.982796i \(-0.440871\pi\)
0.184692 + 0.982796i \(0.440871\pi\)
\(308\) 0 0
\(309\) 6.76393 0.384787
\(310\) −17.8885 −1.01600
\(311\) −24.9443 −1.41446 −0.707230 0.706984i \(-0.750055\pi\)
−0.707230 + 0.706984i \(0.750055\pi\)
\(312\) −10.0000 −0.566139
\(313\) −22.9443 −1.29689 −0.648443 0.761263i \(-0.724580\pi\)
−0.648443 + 0.761263i \(0.724580\pi\)
\(314\) 27.8885 1.57384
\(315\) 0 0
\(316\) −11.1246 −0.625808
\(317\) 29.4164 1.65219 0.826095 0.563531i \(-0.190557\pi\)
0.826095 + 0.563531i \(0.190557\pi\)
\(318\) −1.70820 −0.0957913
\(319\) −17.8885 −1.00157
\(320\) −42.0689 −2.35172
\(321\) −0.944272 −0.0527041
\(322\) 0 0
\(323\) −20.0000 −1.11283
\(324\) 3.00000 0.166667
\(325\) −24.4721 −1.35747
\(326\) −43.4164 −2.40461
\(327\) 2.94427 0.162819
\(328\) −15.5279 −0.857383
\(329\) 0 0
\(330\) 28.9443 1.59333
\(331\) −3.41641 −0.187783 −0.0938914 0.995582i \(-0.529931\pi\)
−0.0938914 + 0.995582i \(0.529931\pi\)
\(332\) −12.0000 −0.658586
\(333\) −4.47214 −0.245072
\(334\) 11.0557 0.604943
\(335\) 16.9443 0.925764
\(336\) 0 0
\(337\) 30.3607 1.65385 0.826926 0.562311i \(-0.190088\pi\)
0.826926 + 0.562311i \(0.190088\pi\)
\(338\) 15.6525 0.851382
\(339\) 16.1803 0.878795
\(340\) 70.2492 3.80980
\(341\) −9.88854 −0.535495
\(342\) −6.18034 −0.334195
\(343\) 0 0
\(344\) 17.2361 0.929307
\(345\) 3.23607 0.174224
\(346\) 10.0000 0.537603
\(347\) 25.8885 1.38977 0.694885 0.719121i \(-0.255455\pi\)
0.694885 + 0.719121i \(0.255455\pi\)
\(348\) −13.4164 −0.719195
\(349\) 32.4721 1.73819 0.869097 0.494642i \(-0.164701\pi\)
0.869097 + 0.494642i \(0.164701\pi\)
\(350\) 0 0
\(351\) −4.47214 −0.238705
\(352\) −26.8328 −1.43019
\(353\) −9.41641 −0.501185 −0.250592 0.968093i \(-0.580625\pi\)
−0.250592 + 0.968093i \(0.580625\pi\)
\(354\) −28.9443 −1.53837
\(355\) −25.8885 −1.37402
\(356\) −9.70820 −0.514534
\(357\) 0 0
\(358\) 6.83282 0.361126
\(359\) −2.47214 −0.130474 −0.0652372 0.997870i \(-0.520780\pi\)
−0.0652372 + 0.997870i \(0.520780\pi\)
\(360\) 7.23607 0.381374
\(361\) −11.3607 −0.597931
\(362\) −53.4164 −2.80750
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 35.4164 1.85378
\(366\) 10.0000 0.522708
\(367\) −12.2918 −0.641627 −0.320813 0.947142i \(-0.603956\pi\)
−0.320813 + 0.947142i \(0.603956\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −6.94427 −0.361504
\(370\) −32.3607 −1.68235
\(371\) 0 0
\(372\) −7.41641 −0.384523
\(373\) 11.5279 0.596890 0.298445 0.954427i \(-0.403532\pi\)
0.298445 + 0.954427i \(0.403532\pi\)
\(374\) 64.7214 3.34666
\(375\) 1.52786 0.0788986
\(376\) 8.94427 0.461266
\(377\) 20.0000 1.03005
\(378\) 0 0
\(379\) −28.0689 −1.44180 −0.720901 0.693038i \(-0.756272\pi\)
−0.720901 + 0.693038i \(0.756272\pi\)
\(380\) −26.8328 −1.37649
\(381\) 10.4721 0.536504
\(382\) 23.4164 1.19809
\(383\) −34.4721 −1.76144 −0.880722 0.473634i \(-0.842942\pi\)
−0.880722 + 0.473634i \(0.842942\pi\)
\(384\) −15.6525 −0.798762
\(385\) 0 0
\(386\) −21.0557 −1.07171
\(387\) 7.70820 0.391830
\(388\) 1.41641 0.0719072
\(389\) −10.6525 −0.540102 −0.270051 0.962846i \(-0.587041\pi\)
−0.270051 + 0.962846i \(0.587041\pi\)
\(390\) −32.3607 −1.63865
\(391\) 7.23607 0.365944
\(392\) 0 0
\(393\) 0.944272 0.0476322
\(394\) −21.0557 −1.06077
\(395\) −12.0000 −0.603786
\(396\) 12.0000 0.603023
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 8.29180 0.415630
\(399\) 0 0
\(400\) −5.47214 −0.273607
\(401\) −38.0689 −1.90107 −0.950535 0.310618i \(-0.899464\pi\)
−0.950535 + 0.310618i \(0.899464\pi\)
\(402\) 11.7082 0.583952
\(403\) 11.0557 0.550725
\(404\) −32.8328 −1.63349
\(405\) 3.23607 0.160802
\(406\) 0 0
\(407\) −17.8885 −0.886702
\(408\) 16.1803 0.801046
\(409\) 9.41641 0.465611 0.232806 0.972523i \(-0.425209\pi\)
0.232806 + 0.972523i \(0.425209\pi\)
\(410\) −50.2492 −2.48163
\(411\) 3.23607 0.159623
\(412\) 20.2918 0.999705
\(413\) 0 0
\(414\) 2.23607 0.109897
\(415\) −12.9443 −0.635409
\(416\) 30.0000 1.47087
\(417\) −8.94427 −0.438003
\(418\) −24.7214 −1.20916
\(419\) 21.8885 1.06933 0.534663 0.845066i \(-0.320439\pi\)
0.534663 + 0.845066i \(0.320439\pi\)
\(420\) 0 0
\(421\) −13.0557 −0.636297 −0.318149 0.948041i \(-0.603061\pi\)
−0.318149 + 0.948041i \(0.603061\pi\)
\(422\) 50.2492 2.44609
\(423\) 4.00000 0.194487
\(424\) −1.70820 −0.0829577
\(425\) 39.5967 1.92072
\(426\) −17.8885 −0.866703
\(427\) 0 0
\(428\) −2.83282 −0.136929
\(429\) −17.8885 −0.863667
\(430\) 55.7771 2.68981
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 7.88854 0.379099 0.189550 0.981871i \(-0.439297\pi\)
0.189550 + 0.981871i \(0.439297\pi\)
\(434\) 0 0
\(435\) −14.4721 −0.693886
\(436\) 8.83282 0.423015
\(437\) −2.76393 −0.132217
\(438\) 24.4721 1.16932
\(439\) −25.8885 −1.23559 −0.617796 0.786338i \(-0.711974\pi\)
−0.617796 + 0.786338i \(0.711974\pi\)
\(440\) 28.9443 1.37986
\(441\) 0 0
\(442\) −72.3607 −3.44185
\(443\) 18.8328 0.894774 0.447387 0.894340i \(-0.352355\pi\)
0.447387 + 0.894340i \(0.352355\pi\)
\(444\) −13.4164 −0.636715
\(445\) −10.4721 −0.496427
\(446\) −22.1115 −1.04701
\(447\) 1.70820 0.0807953
\(448\) 0 0
\(449\) −2.94427 −0.138949 −0.0694744 0.997584i \(-0.522132\pi\)
−0.0694744 + 0.997584i \(0.522132\pi\)
\(450\) 12.2361 0.576814
\(451\) −27.7771 −1.30797
\(452\) 48.5410 2.28318
\(453\) −16.0000 −0.751746
\(454\) 50.2492 2.35831
\(455\) 0 0
\(456\) −6.18034 −0.289421
\(457\) −0.472136 −0.0220856 −0.0110428 0.999939i \(-0.503515\pi\)
−0.0110428 + 0.999939i \(0.503515\pi\)
\(458\) 33.4164 1.56145
\(459\) 7.23607 0.337751
\(460\) 9.70820 0.452647
\(461\) −21.4164 −0.997462 −0.498731 0.866757i \(-0.666200\pi\)
−0.498731 + 0.866757i \(0.666200\pi\)
\(462\) 0 0
\(463\) 4.94427 0.229780 0.114890 0.993378i \(-0.463348\pi\)
0.114890 + 0.993378i \(0.463348\pi\)
\(464\) 4.47214 0.207614
\(465\) −8.00000 −0.370991
\(466\) −31.3050 −1.45017
\(467\) 24.3607 1.12728 0.563639 0.826021i \(-0.309401\pi\)
0.563639 + 0.826021i \(0.309401\pi\)
\(468\) −13.4164 −0.620174
\(469\) 0 0
\(470\) 28.9443 1.33510
\(471\) 12.4721 0.574686
\(472\) −28.9443 −1.33227
\(473\) 30.8328 1.41769
\(474\) −8.29180 −0.380855
\(475\) −15.1246 −0.693965
\(476\) 0 0
\(477\) −0.763932 −0.0349780
\(478\) −28.9443 −1.32388
\(479\) −1.88854 −0.0862898 −0.0431449 0.999069i \(-0.513738\pi\)
−0.0431449 + 0.999069i \(0.513738\pi\)
\(480\) −21.7082 −0.990839
\(481\) 20.0000 0.911922
\(482\) −63.6656 −2.89989
\(483\) 0 0
\(484\) 15.0000 0.681818
\(485\) 1.52786 0.0693767
\(486\) 2.23607 0.101430
\(487\) 24.0000 1.08754 0.543772 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) 10.0000 0.452679
\(489\) −19.4164 −0.878040
\(490\) 0 0
\(491\) 40.0000 1.80517 0.902587 0.430507i \(-0.141665\pi\)
0.902587 + 0.430507i \(0.141665\pi\)
\(492\) −20.8328 −0.939216
\(493\) −32.3607 −1.45745
\(494\) 27.6393 1.24355
\(495\) 12.9443 0.581802
\(496\) 2.47214 0.111002
\(497\) 0 0
\(498\) −8.94427 −0.400802
\(499\) 32.3607 1.44866 0.724331 0.689452i \(-0.242149\pi\)
0.724331 + 0.689452i \(0.242149\pi\)
\(500\) 4.58359 0.204984
\(501\) 4.94427 0.220894
\(502\) 3.41641 0.152482
\(503\) 36.9443 1.64726 0.823632 0.567125i \(-0.191944\pi\)
0.823632 + 0.567125i \(0.191944\pi\)
\(504\) 0 0
\(505\) −35.4164 −1.57601
\(506\) 8.94427 0.397621
\(507\) 7.00000 0.310881
\(508\) 31.4164 1.39388
\(509\) 7.52786 0.333667 0.166833 0.985985i \(-0.446646\pi\)
0.166833 + 0.985985i \(0.446646\pi\)
\(510\) 52.3607 2.31857
\(511\) 0 0
\(512\) 11.1803 0.494106
\(513\) −2.76393 −0.122031
\(514\) 49.1935 2.16983
\(515\) 21.8885 0.964524
\(516\) 23.1246 1.01800
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) 4.47214 0.196305
\(520\) −32.3607 −1.41911
\(521\) 28.7639 1.26017 0.630085 0.776526i \(-0.283020\pi\)
0.630085 + 0.776526i \(0.283020\pi\)
\(522\) −10.0000 −0.437688
\(523\) 33.5967 1.46908 0.734542 0.678564i \(-0.237397\pi\)
0.734542 + 0.678564i \(0.237397\pi\)
\(524\) 2.83282 0.123752
\(525\) 0 0
\(526\) −5.52786 −0.241026
\(527\) −17.8885 −0.779237
\(528\) −4.00000 −0.174078
\(529\) 1.00000 0.0434783
\(530\) −5.52786 −0.240115
\(531\) −12.9443 −0.561734
\(532\) 0 0
\(533\) 31.0557 1.34517
\(534\) −7.23607 −0.313135
\(535\) −3.05573 −0.132111
\(536\) 11.7082 0.505717
\(537\) 3.05573 0.131864
\(538\) −18.9443 −0.816746
\(539\) 0 0
\(540\) 9.70820 0.417775
\(541\) −24.4721 −1.05214 −0.526070 0.850441i \(-0.676335\pi\)
−0.526070 + 0.850441i \(0.676335\pi\)
\(542\) 59.1935 2.54258
\(543\) −23.8885 −1.02516
\(544\) −48.5410 −2.08118
\(545\) 9.52786 0.408129
\(546\) 0 0
\(547\) 22.4721 0.960839 0.480420 0.877039i \(-0.340484\pi\)
0.480420 + 0.877039i \(0.340484\pi\)
\(548\) 9.70820 0.414714
\(549\) 4.47214 0.190866
\(550\) 48.9443 2.08699
\(551\) 12.3607 0.526583
\(552\) 2.23607 0.0951734
\(553\) 0 0
\(554\) −44.4721 −1.88944
\(555\) −14.4721 −0.614308
\(556\) −26.8328 −1.13796
\(557\) −19.8197 −0.839786 −0.419893 0.907574i \(-0.637932\pi\)
−0.419893 + 0.907574i \(0.637932\pi\)
\(558\) −5.52786 −0.234013
\(559\) −34.4721 −1.45802
\(560\) 0 0
\(561\) 28.9443 1.22203
\(562\) −14.8754 −0.627480
\(563\) −19.4164 −0.818304 −0.409152 0.912466i \(-0.634175\pi\)
−0.409152 + 0.912466i \(0.634175\pi\)
\(564\) 12.0000 0.505291
\(565\) 52.3607 2.20283
\(566\) 17.2361 0.724486
\(567\) 0 0
\(568\) −17.8885 −0.750587
\(569\) −15.2361 −0.638729 −0.319365 0.947632i \(-0.603469\pi\)
−0.319365 + 0.947632i \(0.603469\pi\)
\(570\) −20.0000 −0.837708
\(571\) 16.2918 0.681790 0.340895 0.940101i \(-0.389270\pi\)
0.340895 + 0.940101i \(0.389270\pi\)
\(572\) −53.6656 −2.24387
\(573\) 10.4721 0.437480
\(574\) 0 0
\(575\) 5.47214 0.228204
\(576\) −13.0000 −0.541667
\(577\) −16.4721 −0.685744 −0.342872 0.939382i \(-0.611400\pi\)
−0.342872 + 0.939382i \(0.611400\pi\)
\(578\) 79.0689 3.28883
\(579\) −9.41641 −0.391333
\(580\) −43.4164 −1.80277
\(581\) 0 0
\(582\) 1.05573 0.0437613
\(583\) −3.05573 −0.126555
\(584\) 24.4721 1.01266
\(585\) −14.4721 −0.598349
\(586\) 12.7639 0.527273
\(587\) −16.9443 −0.699365 −0.349682 0.936868i \(-0.613711\pi\)
−0.349682 + 0.936868i \(0.613711\pi\)
\(588\) 0 0
\(589\) 6.83282 0.281541
\(590\) −93.6656 −3.85615
\(591\) −9.41641 −0.387339
\(592\) 4.47214 0.183804
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 8.94427 0.366988
\(595\) 0 0
\(596\) 5.12461 0.209912
\(597\) 3.70820 0.151767
\(598\) −10.0000 −0.408930
\(599\) −20.9443 −0.855760 −0.427880 0.903836i \(-0.640739\pi\)
−0.427880 + 0.903836i \(0.640739\pi\)
\(600\) 12.2361 0.499535
\(601\) −2.36068 −0.0962941 −0.0481471 0.998840i \(-0.515332\pi\)
−0.0481471 + 0.998840i \(0.515332\pi\)
\(602\) 0 0
\(603\) 5.23607 0.213229
\(604\) −48.0000 −1.95309
\(605\) 16.1803 0.657824
\(606\) −24.4721 −0.994113
\(607\) −38.8328 −1.57618 −0.788088 0.615563i \(-0.788929\pi\)
−0.788088 + 0.615563i \(0.788929\pi\)
\(608\) 18.5410 0.751938
\(609\) 0 0
\(610\) 32.3607 1.31025
\(611\) −17.8885 −0.723693
\(612\) 21.7082 0.877502
\(613\) −31.5279 −1.27340 −0.636699 0.771112i \(-0.719701\pi\)
−0.636699 + 0.771112i \(0.719701\pi\)
\(614\) 14.4721 0.584048
\(615\) −22.4721 −0.906164
\(616\) 0 0
\(617\) −33.7082 −1.35704 −0.678521 0.734581i \(-0.737379\pi\)
−0.678521 + 0.734581i \(0.737379\pi\)
\(618\) 15.1246 0.608401
\(619\) −31.1246 −1.25100 −0.625502 0.780223i \(-0.715106\pi\)
−0.625502 + 0.780223i \(0.715106\pi\)
\(620\) −24.0000 −0.963863
\(621\) 1.00000 0.0401286
\(622\) −55.7771 −2.23646
\(623\) 0 0
\(624\) 4.47214 0.179029
\(625\) −22.4164 −0.896656
\(626\) −51.3050 −2.05056
\(627\) −11.0557 −0.441523
\(628\) 37.4164 1.49308
\(629\) −32.3607 −1.29030
\(630\) 0 0
\(631\) 6.18034 0.246035 0.123018 0.992404i \(-0.460743\pi\)
0.123018 + 0.992404i \(0.460743\pi\)
\(632\) −8.29180 −0.329830
\(633\) 22.4721 0.893187
\(634\) 65.7771 2.61234
\(635\) 33.8885 1.34483
\(636\) −2.29180 −0.0908756
\(637\) 0 0
\(638\) −40.0000 −1.58362
\(639\) −8.00000 −0.316475
\(640\) −50.6525 −2.00221
\(641\) −27.0132 −1.06696 −0.533478 0.845814i \(-0.679115\pi\)
−0.533478 + 0.845814i \(0.679115\pi\)
\(642\) −2.11146 −0.0833325
\(643\) 20.0689 0.791440 0.395720 0.918371i \(-0.370495\pi\)
0.395720 + 0.918371i \(0.370495\pi\)
\(644\) 0 0
\(645\) 24.9443 0.982180
\(646\) −44.7214 −1.75954
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 2.23607 0.0878410
\(649\) −51.7771 −2.03243
\(650\) −54.7214 −2.14635
\(651\) 0 0
\(652\) −58.2492 −2.28122
\(653\) −5.05573 −0.197846 −0.0989230 0.995095i \(-0.531540\pi\)
−0.0989230 + 0.995095i \(0.531540\pi\)
\(654\) 6.58359 0.257439
\(655\) 3.05573 0.119397
\(656\) 6.94427 0.271128
\(657\) 10.9443 0.426977
\(658\) 0 0
\(659\) −8.36068 −0.325686 −0.162843 0.986652i \(-0.552066\pi\)
−0.162843 + 0.986652i \(0.552066\pi\)
\(660\) 38.8328 1.51157
\(661\) 20.4721 0.796274 0.398137 0.917326i \(-0.369657\pi\)
0.398137 + 0.917326i \(0.369657\pi\)
\(662\) −7.63932 −0.296911
\(663\) −32.3607 −1.25678
\(664\) −8.94427 −0.347105
\(665\) 0 0
\(666\) −10.0000 −0.387492
\(667\) −4.47214 −0.173162
\(668\) 14.8328 0.573899
\(669\) −9.88854 −0.382313
\(670\) 37.8885 1.46376
\(671\) 17.8885 0.690580
\(672\) 0 0
\(673\) 29.4164 1.13392 0.566960 0.823746i \(-0.308120\pi\)
0.566960 + 0.823746i \(0.308120\pi\)
\(674\) 67.8885 2.61497
\(675\) 5.47214 0.210623
\(676\) 21.0000 0.807692
\(677\) −37.4853 −1.44068 −0.720338 0.693623i \(-0.756013\pi\)
−0.720338 + 0.693623i \(0.756013\pi\)
\(678\) 36.1803 1.38950
\(679\) 0 0
\(680\) 52.3607 2.00794
\(681\) 22.4721 0.861134
\(682\) −22.1115 −0.846691
\(683\) −20.0000 −0.765279 −0.382639 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) −8.29180 −0.317045
\(685\) 10.4721 0.400120
\(686\) 0 0
\(687\) 14.9443 0.570160
\(688\) −7.70820 −0.293873
\(689\) 3.41641 0.130155
\(690\) 7.23607 0.275472
\(691\) 40.3607 1.53539 0.767696 0.640814i \(-0.221403\pi\)
0.767696 + 0.640814i \(0.221403\pi\)
\(692\) 13.4164 0.510015
\(693\) 0 0
\(694\) 57.8885 2.19742
\(695\) −28.9443 −1.09792
\(696\) −10.0000 −0.379049
\(697\) −50.2492 −1.90333
\(698\) 72.6099 2.74833
\(699\) −14.0000 −0.529529
\(700\) 0 0
\(701\) 20.7639 0.784243 0.392121 0.919913i \(-0.371741\pi\)
0.392121 + 0.919913i \(0.371741\pi\)
\(702\) −10.0000 −0.377426
\(703\) 12.3607 0.466192
\(704\) −52.0000 −1.95982
\(705\) 12.9443 0.487509
\(706\) −21.0557 −0.792443
\(707\) 0 0
\(708\) −38.8328 −1.45943
\(709\) 20.8328 0.782393 0.391196 0.920307i \(-0.372061\pi\)
0.391196 + 0.920307i \(0.372061\pi\)
\(710\) −57.8885 −2.17252
\(711\) −3.70820 −0.139069
\(712\) −7.23607 −0.271183
\(713\) −2.47214 −0.0925822
\(714\) 0 0
\(715\) −57.8885 −2.16491
\(716\) 9.16718 0.342594
\(717\) −12.9443 −0.483413
\(718\) −5.52786 −0.206298
\(719\) −4.00000 −0.149175 −0.0745874 0.997214i \(-0.523764\pi\)
−0.0745874 + 0.997214i \(0.523764\pi\)
\(720\) −3.23607 −0.120601
\(721\) 0 0
\(722\) −25.4033 −0.945411
\(723\) −28.4721 −1.05889
\(724\) −71.6656 −2.66343
\(725\) −24.4721 −0.908872
\(726\) 11.1803 0.414941
\(727\) −32.6525 −1.21101 −0.605507 0.795840i \(-0.707029\pi\)
−0.605507 + 0.795840i \(0.707029\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 79.1935 2.93108
\(731\) 55.7771 2.06299
\(732\) 13.4164 0.495885
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) −27.4853 −1.01450
\(735\) 0 0
\(736\) −6.70820 −0.247268
\(737\) 20.9443 0.771492
\(738\) −15.5279 −0.571589
\(739\) 26.8328 0.987061 0.493531 0.869728i \(-0.335706\pi\)
0.493531 + 0.869728i \(0.335706\pi\)
\(740\) −43.4164 −1.59602
\(741\) 12.3607 0.454081
\(742\) 0 0
\(743\) 20.3607 0.746961 0.373480 0.927638i \(-0.378164\pi\)
0.373480 + 0.927638i \(0.378164\pi\)
\(744\) −5.52786 −0.202661
\(745\) 5.52786 0.202525
\(746\) 25.7771 0.943766
\(747\) −4.00000 −0.146352
\(748\) 86.8328 3.17492
\(749\) 0 0
\(750\) 3.41641 0.124750
\(751\) −8.06888 −0.294438 −0.147219 0.989104i \(-0.547032\pi\)
−0.147219 + 0.989104i \(0.547032\pi\)
\(752\) −4.00000 −0.145865
\(753\) 1.52786 0.0556785
\(754\) 44.7214 1.62866
\(755\) −51.7771 −1.88436
\(756\) 0 0
\(757\) 4.11146 0.149433 0.0747167 0.997205i \(-0.476195\pi\)
0.0747167 + 0.997205i \(0.476195\pi\)
\(758\) −62.7639 −2.27969
\(759\) 4.00000 0.145191
\(760\) −20.0000 −0.725476
\(761\) 2.36068 0.0855746 0.0427873 0.999084i \(-0.486376\pi\)
0.0427873 + 0.999084i \(0.486376\pi\)
\(762\) 23.4164 0.848287
\(763\) 0 0
\(764\) 31.4164 1.13661
\(765\) 23.4164 0.846622
\(766\) −77.0820 −2.78509
\(767\) 57.8885 2.09023
\(768\) −9.00000 −0.324760
\(769\) 36.8328 1.32823 0.664113 0.747633i \(-0.268810\pi\)
0.664113 + 0.747633i \(0.268810\pi\)
\(770\) 0 0
\(771\) 22.0000 0.792311
\(772\) −28.2492 −1.01671
\(773\) −49.7082 −1.78788 −0.893940 0.448187i \(-0.852070\pi\)
−0.893940 + 0.448187i \(0.852070\pi\)
\(774\) 17.2361 0.619538
\(775\) −13.5279 −0.485935
\(776\) 1.05573 0.0378984
\(777\) 0 0
\(778\) −23.8197 −0.853976
\(779\) 19.1935 0.687678
\(780\) −43.4164 −1.55456
\(781\) −32.0000 −1.14505
\(782\) 16.1803 0.578608
\(783\) −4.47214 −0.159821
\(784\) 0 0
\(785\) 40.3607 1.44053
\(786\) 2.11146 0.0753131
\(787\) 0.291796 0.0104014 0.00520070 0.999986i \(-0.498345\pi\)
0.00520070 + 0.999986i \(0.498345\pi\)
\(788\) −28.2492 −1.00634
\(789\) −2.47214 −0.0880104
\(790\) −26.8328 −0.954669
\(791\) 0 0
\(792\) 8.94427 0.317821
\(793\) −20.0000 −0.710221
\(794\) 4.47214 0.158710
\(795\) −2.47214 −0.0876776
\(796\) 11.1246 0.394301
\(797\) 18.6525 0.660705 0.330352 0.943858i \(-0.392832\pi\)
0.330352 + 0.943858i \(0.392832\pi\)
\(798\) 0 0
\(799\) 28.9443 1.02397
\(800\) −36.7082 −1.29783
\(801\) −3.23607 −0.114341
\(802\) −85.1246 −3.00585
\(803\) 43.7771 1.54486
\(804\) 15.7082 0.553986
\(805\) 0 0
\(806\) 24.7214 0.870773
\(807\) −8.47214 −0.298233
\(808\) −24.4721 −0.860927
\(809\) 19.3050 0.678726 0.339363 0.940655i \(-0.389789\pi\)
0.339363 + 0.940655i \(0.389789\pi\)
\(810\) 7.23607 0.254250
\(811\) −3.41641 −0.119966 −0.0599832 0.998199i \(-0.519105\pi\)
−0.0599832 + 0.998199i \(0.519105\pi\)
\(812\) 0 0
\(813\) 26.4721 0.928418
\(814\) −40.0000 −1.40200
\(815\) −62.8328 −2.20094
\(816\) −7.23607 −0.253313
\(817\) −21.3050 −0.745366
\(818\) 21.0557 0.736196
\(819\) 0 0
\(820\) −67.4164 −2.35428
\(821\) 47.8885 1.67132 0.835661 0.549246i \(-0.185085\pi\)
0.835661 + 0.549246i \(0.185085\pi\)
\(822\) 7.23607 0.252387
\(823\) −34.4721 −1.20162 −0.600812 0.799391i \(-0.705156\pi\)
−0.600812 + 0.799391i \(0.705156\pi\)
\(824\) 15.1246 0.526891
\(825\) 21.8885 0.762061
\(826\) 0 0
\(827\) 35.4164 1.23155 0.615775 0.787922i \(-0.288843\pi\)
0.615775 + 0.787922i \(0.288843\pi\)
\(828\) 3.00000 0.104257
\(829\) 34.3607 1.19340 0.596698 0.802466i \(-0.296479\pi\)
0.596698 + 0.802466i \(0.296479\pi\)
\(830\) −28.9443 −1.00467
\(831\) −19.8885 −0.689926
\(832\) 58.1378 2.01556
\(833\) 0 0
\(834\) −20.0000 −0.692543
\(835\) 16.0000 0.553703
\(836\) −33.1672 −1.14711
\(837\) −2.47214 −0.0854495
\(838\) 48.9443 1.69075
\(839\) 39.4164 1.36081 0.680403 0.732838i \(-0.261805\pi\)
0.680403 + 0.732838i \(0.261805\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) −29.1935 −1.00607
\(843\) −6.65248 −0.229123
\(844\) 67.4164 2.32057
\(845\) 22.6525 0.779269
\(846\) 8.94427 0.307510
\(847\) 0 0
\(848\) 0.763932 0.0262335
\(849\) 7.70820 0.264545
\(850\) 88.5410 3.03693
\(851\) −4.47214 −0.153303
\(852\) −24.0000 −0.822226
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) 0 0
\(855\) −8.94427 −0.305888
\(856\) −2.11146 −0.0721681
\(857\) 25.0557 0.855887 0.427944 0.903805i \(-0.359238\pi\)
0.427944 + 0.903805i \(0.359238\pi\)
\(858\) −40.0000 −1.36558
\(859\) 24.9443 0.851088 0.425544 0.904938i \(-0.360083\pi\)
0.425544 + 0.904938i \(0.360083\pi\)
\(860\) 74.8328 2.55178
\(861\) 0 0
\(862\) 17.8885 0.609286
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) −6.70820 −0.228218
\(865\) 14.4721 0.492067
\(866\) 17.6393 0.599409
\(867\) 35.3607 1.20091
\(868\) 0 0
\(869\) −14.8328 −0.503169
\(870\) −32.3607 −1.09713
\(871\) −23.4164 −0.793435
\(872\) 6.58359 0.222949
\(873\) 0.472136 0.0159794
\(874\) −6.18034 −0.209053
\(875\) 0 0
\(876\) 32.8328 1.10932
\(877\) 50.9443 1.72027 0.860133 0.510070i \(-0.170381\pi\)
0.860133 + 0.510070i \(0.170381\pi\)
\(878\) −57.8885 −1.95364
\(879\) 5.70820 0.192533
\(880\) −12.9443 −0.436351
\(881\) 40.5410 1.36586 0.682931 0.730483i \(-0.260705\pi\)
0.682931 + 0.730483i \(0.260705\pi\)
\(882\) 0 0
\(883\) 19.4164 0.653414 0.326707 0.945126i \(-0.394061\pi\)
0.326707 + 0.945126i \(0.394061\pi\)
\(884\) −97.0820 −3.26522
\(885\) −41.8885 −1.40807
\(886\) 42.1115 1.41476
\(887\) −48.9443 −1.64339 −0.821694 0.569929i \(-0.806971\pi\)
−0.821694 + 0.569929i \(0.806971\pi\)
\(888\) −10.0000 −0.335578
\(889\) 0 0
\(890\) −23.4164 −0.784920
\(891\) 4.00000 0.134005
\(892\) −29.6656 −0.993279
\(893\) −11.0557 −0.369966
\(894\) 3.81966 0.127749
\(895\) 9.88854 0.330538
\(896\) 0 0
\(897\) −4.47214 −0.149320
\(898\) −6.58359 −0.219697
\(899\) 11.0557 0.368729
\(900\) 16.4164 0.547214
\(901\) −5.52786 −0.184160
\(902\) −62.1115 −2.06809
\(903\) 0 0
\(904\) 36.1803 1.20334
\(905\) −77.3050 −2.56970
\(906\) −35.7771 −1.18861
\(907\) 39.1246 1.29911 0.649556 0.760314i \(-0.274955\pi\)
0.649556 + 0.760314i \(0.274955\pi\)
\(908\) 67.4164 2.23729
\(909\) −10.9443 −0.362999
\(910\) 0 0
\(911\) 48.7214 1.61421 0.807105 0.590407i \(-0.201033\pi\)
0.807105 + 0.590407i \(0.201033\pi\)
\(912\) 2.76393 0.0915229
\(913\) −16.0000 −0.529523
\(914\) −1.05573 −0.0349204
\(915\) 14.4721 0.478434
\(916\) 44.8328 1.48132
\(917\) 0 0
\(918\) 16.1803 0.534031
\(919\) 53.0132 1.74874 0.874371 0.485257i \(-0.161274\pi\)
0.874371 + 0.485257i \(0.161274\pi\)
\(920\) 7.23607 0.238566
\(921\) 6.47214 0.213264
\(922\) −47.8885 −1.57713
\(923\) 35.7771 1.17762
\(924\) 0 0
\(925\) −24.4721 −0.804639
\(926\) 11.0557 0.363314
\(927\) 6.76393 0.222157
\(928\) 30.0000 0.984798
\(929\) −24.8328 −0.814738 −0.407369 0.913264i \(-0.633554\pi\)
−0.407369 + 0.913264i \(0.633554\pi\)
\(930\) −17.8885 −0.586588
\(931\) 0 0
\(932\) −42.0000 −1.37576
\(933\) −24.9443 −0.816639
\(934\) 54.4721 1.78238
\(935\) 93.6656 3.06319
\(936\) −10.0000 −0.326860
\(937\) −40.8328 −1.33395 −0.666975 0.745080i \(-0.732411\pi\)
−0.666975 + 0.745080i \(0.732411\pi\)
\(938\) 0 0
\(939\) −22.9443 −0.748758
\(940\) 38.8328 1.26659
\(941\) −24.5410 −0.800014 −0.400007 0.916512i \(-0.630992\pi\)
−0.400007 + 0.916512i \(0.630992\pi\)
\(942\) 27.8885 0.908658
\(943\) −6.94427 −0.226137
\(944\) 12.9443 0.421300
\(945\) 0 0
\(946\) 68.9443 2.24157
\(947\) −54.8328 −1.78183 −0.890914 0.454173i \(-0.849935\pi\)
−0.890914 + 0.454173i \(0.849935\pi\)
\(948\) −11.1246 −0.361311
\(949\) −48.9443 −1.58880
\(950\) −33.8197 −1.09725
\(951\) 29.4164 0.953892
\(952\) 0 0
\(953\) −46.0689 −1.49232 −0.746159 0.665768i \(-0.768104\pi\)
−0.746159 + 0.665768i \(0.768104\pi\)
\(954\) −1.70820 −0.0553051
\(955\) 33.8885 1.09661
\(956\) −38.8328 −1.25594
\(957\) −17.8885 −0.578254
\(958\) −4.22291 −0.136436
\(959\) 0 0
\(960\) −42.0689 −1.35777
\(961\) −24.8885 −0.802856
\(962\) 44.7214 1.44187
\(963\) −0.944272 −0.0304287
\(964\) −85.4164 −2.75108
\(965\) −30.4721 −0.980933
\(966\) 0 0
\(967\) −52.3607 −1.68381 −0.841903 0.539629i \(-0.818565\pi\)
−0.841903 + 0.539629i \(0.818565\pi\)
\(968\) 11.1803 0.359350
\(969\) −20.0000 −0.642493
\(970\) 3.41641 0.109694
\(971\) 45.3050 1.45391 0.726953 0.686688i \(-0.240936\pi\)
0.726953 + 0.686688i \(0.240936\pi\)
\(972\) 3.00000 0.0962250
\(973\) 0 0
\(974\) 53.6656 1.71956
\(975\) −24.4721 −0.783736
\(976\) −4.47214 −0.143150
\(977\) 26.0689 0.834017 0.417009 0.908902i \(-0.363078\pi\)
0.417009 + 0.908902i \(0.363078\pi\)
\(978\) −43.4164 −1.38830
\(979\) −12.9443 −0.413701
\(980\) 0 0
\(981\) 2.94427 0.0940034
\(982\) 89.4427 2.85423
\(983\) 30.8328 0.983414 0.491707 0.870761i \(-0.336373\pi\)
0.491707 + 0.870761i \(0.336373\pi\)
\(984\) −15.5279 −0.495010
\(985\) −30.4721 −0.970923
\(986\) −72.3607 −2.30443
\(987\) 0 0
\(988\) 37.0820 1.17974
\(989\) 7.70820 0.245107
\(990\) 28.9443 0.919909
\(991\) −10.4721 −0.332658 −0.166329 0.986070i \(-0.553191\pi\)
−0.166329 + 0.986070i \(0.553191\pi\)
\(992\) 16.5836 0.526530
\(993\) −3.41641 −0.108416
\(994\) 0 0
\(995\) 12.0000 0.380426
\(996\) −12.0000 −0.380235
\(997\) 2.36068 0.0747635 0.0373817 0.999301i \(-0.488098\pi\)
0.0373817 + 0.999301i \(0.488098\pi\)
\(998\) 72.3607 2.29054
\(999\) −4.47214 −0.141492
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.t.1.2 2
7.6 odd 2 69.2.a.b.1.2 2
21.20 even 2 207.2.a.c.1.1 2
28.27 even 2 1104.2.a.m.1.1 2
35.13 even 4 1725.2.b.o.1174.1 4
35.27 even 4 1725.2.b.o.1174.4 4
35.34 odd 2 1725.2.a.ba.1.1 2
56.13 odd 2 4416.2.a.bm.1.2 2
56.27 even 2 4416.2.a.bg.1.2 2
77.76 even 2 8349.2.a.i.1.1 2
84.83 odd 2 3312.2.a.bb.1.2 2
105.104 even 2 5175.2.a.bk.1.2 2
161.160 even 2 1587.2.a.i.1.2 2
483.482 odd 2 4761.2.a.v.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.2.a.b.1.2 2 7.6 odd 2
207.2.a.c.1.1 2 21.20 even 2
1104.2.a.m.1.1 2 28.27 even 2
1587.2.a.i.1.2 2 161.160 even 2
1725.2.a.ba.1.1 2 35.34 odd 2
1725.2.b.o.1174.1 4 35.13 even 4
1725.2.b.o.1174.4 4 35.27 even 4
3312.2.a.bb.1.2 2 84.83 odd 2
3381.2.a.t.1.2 2 1.1 even 1 trivial
4416.2.a.bg.1.2 2 56.27 even 2
4416.2.a.bm.1.2 2 56.13 odd 2
4761.2.a.v.1.1 2 483.482 odd 2
5175.2.a.bk.1.2 2 105.104 even 2
8349.2.a.i.1.1 2 77.76 even 2