Properties

Label 5610.2.a.bw.1.1
Level $5610$
Weight $2$
Character 5610.1
Self dual yes
Analytic conductor $44.796$
Analytic rank $1$
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] = [5610,2,Mod(1,5610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5610, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5610.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5610.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: \(44.7960755339\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 5610.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -7.12311 q^{13} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +3.12311 q^{19} -1.00000 q^{20} -1.00000 q^{22} -0.876894 q^{23} +1.00000 q^{24} +1.00000 q^{25} -7.12311 q^{26} +1.00000 q^{27} -0.876894 q^{29} -1.00000 q^{30} +1.12311 q^{31} +1.00000 q^{32} -1.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -7.12311 q^{37} +3.12311 q^{38} -7.12311 q^{39} -1.00000 q^{40} -5.12311 q^{41} -1.00000 q^{44} -1.00000 q^{45} -0.876894 q^{46} -1.12311 q^{47} +1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -1.00000 q^{51} -7.12311 q^{52} +4.00000 q^{53} +1.00000 q^{54} +1.00000 q^{55} +3.12311 q^{57} -0.876894 q^{58} -6.24621 q^{59} -1.00000 q^{60} +8.24621 q^{61} +1.12311 q^{62} +1.00000 q^{64} +7.12311 q^{65} -1.00000 q^{66} -15.1231 q^{67} -1.00000 q^{68} -0.876894 q^{69} -8.00000 q^{71} +1.00000 q^{72} +0.246211 q^{73} -7.12311 q^{74} +1.00000 q^{75} +3.12311 q^{76} -7.12311 q^{78} +3.12311 q^{79} -1.00000 q^{80} +1.00000 q^{81} -5.12311 q^{82} -10.2462 q^{83} +1.00000 q^{85} -0.876894 q^{87} -1.00000 q^{88} -4.24621 q^{89} -1.00000 q^{90} -0.876894 q^{92} +1.12311 q^{93} -1.12311 q^{94} -3.12311 q^{95} +1.00000 q^{96} +2.87689 q^{97} -7.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} + 2 q^{12} - 6 q^{13} - 2 q^{15} + 2 q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 2 q^{22} - 10 q^{23} + 2 q^{24} + 2 q^{25} - 6 q^{26} + 2 q^{27} - 10 q^{29} - 2 q^{30} - 6 q^{31} + 2 q^{32} - 2 q^{33} - 2 q^{34} + 2 q^{36} - 6 q^{37} - 2 q^{38} - 6 q^{39} - 2 q^{40} - 2 q^{41} - 2 q^{44} - 2 q^{45} - 10 q^{46} + 6 q^{47} + 2 q^{48} - 14 q^{49} + 2 q^{50} - 2 q^{51} - 6 q^{52} + 8 q^{53} + 2 q^{54} + 2 q^{55} - 2 q^{57} - 10 q^{58} + 4 q^{59} - 2 q^{60} - 6 q^{62} + 2 q^{64} + 6 q^{65} - 2 q^{66} - 22 q^{67} - 2 q^{68} - 10 q^{69} - 16 q^{71} + 2 q^{72} - 16 q^{73} - 6 q^{74} + 2 q^{75} - 2 q^{76} - 6 q^{78} - 2 q^{79} - 2 q^{80} + 2 q^{81} - 2 q^{82} - 4 q^{83} + 2 q^{85} - 10 q^{87} - 2 q^{88} + 8 q^{89} - 2 q^{90} - 10 q^{92} - 6 q^{93} + 6 q^{94} + 2 q^{95} + 2 q^{96} + 14 q^{97} - 14 q^{98} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) −7.12311 −1.97559 −0.987797 0.155747i \(-0.950222\pi\)
−0.987797 + 0.155747i \(0.950222\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) 1.00000 0.235702
\(19\) 3.12311 0.716490 0.358245 0.933628i \(-0.383375\pi\)
0.358245 + 0.933628i \(0.383375\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −0.876894 −0.182845 −0.0914226 0.995812i \(-0.529141\pi\)
−0.0914226 + 0.995812i \(0.529141\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −7.12311 −1.39696
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −0.876894 −0.162835 −0.0814176 0.996680i \(-0.525945\pi\)
−0.0814176 + 0.996680i \(0.525945\pi\)
\(30\) −1.00000 −0.182574
\(31\) 1.12311 0.201716 0.100858 0.994901i \(-0.467841\pi\)
0.100858 + 0.994901i \(0.467841\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −7.12311 −1.17103 −0.585516 0.810661i \(-0.699108\pi\)
−0.585516 + 0.810661i \(0.699108\pi\)
\(38\) 3.12311 0.506635
\(39\) −7.12311 −1.14061
\(40\) −1.00000 −0.158114
\(41\) −5.12311 −0.800095 −0.400047 0.916494i \(-0.631006\pi\)
−0.400047 + 0.916494i \(0.631006\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.00000 −0.149071
\(46\) −0.876894 −0.129291
\(47\) −1.12311 −0.163822 −0.0819109 0.996640i \(-0.526102\pi\)
−0.0819109 + 0.996640i \(0.526102\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) −1.00000 −0.140028
\(52\) −7.12311 −0.987797
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 1.00000 0.136083
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 3.12311 0.413665
\(58\) −0.876894 −0.115142
\(59\) −6.24621 −0.813187 −0.406594 0.913609i \(-0.633284\pi\)
−0.406594 + 0.913609i \(0.633284\pi\)
\(60\) −1.00000 −0.129099
\(61\) 8.24621 1.05582 0.527910 0.849301i \(-0.322976\pi\)
0.527910 + 0.849301i \(0.322976\pi\)
\(62\) 1.12311 0.142635
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 7.12311 0.883513
\(66\) −1.00000 −0.123091
\(67\) −15.1231 −1.84758 −0.923791 0.382898i \(-0.874926\pi\)
−0.923791 + 0.382898i \(0.874926\pi\)
\(68\) −1.00000 −0.121268
\(69\) −0.876894 −0.105566
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) 0.246211 0.0288168 0.0144084 0.999896i \(-0.495413\pi\)
0.0144084 + 0.999896i \(0.495413\pi\)
\(74\) −7.12311 −0.828044
\(75\) 1.00000 0.115470
\(76\) 3.12311 0.358245
\(77\) 0 0
\(78\) −7.12311 −0.806533
\(79\) 3.12311 0.351377 0.175688 0.984446i \(-0.443785\pi\)
0.175688 + 0.984446i \(0.443785\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −5.12311 −0.565752
\(83\) −10.2462 −1.12467 −0.562334 0.826910i \(-0.690096\pi\)
−0.562334 + 0.826910i \(0.690096\pi\)
\(84\) 0 0
\(85\) 1.00000 0.108465
\(86\) 0 0
\(87\) −0.876894 −0.0940129
\(88\) −1.00000 −0.106600
\(89\) −4.24621 −0.450097 −0.225049 0.974348i \(-0.572254\pi\)
−0.225049 + 0.974348i \(0.572254\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −0.876894 −0.0914226
\(93\) 1.12311 0.116461
\(94\) −1.12311 −0.115840
\(95\) −3.12311 −0.320424
\(96\) 1.00000 0.102062
\(97\) 2.87689 0.292104 0.146052 0.989277i \(-0.453343\pi\)
0.146052 + 0.989277i \(0.453343\pi\)
\(98\) −7.00000 −0.707107
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) 15.3693 1.52930 0.764652 0.644443i \(-0.222911\pi\)
0.764652 + 0.644443i \(0.222911\pi\)
\(102\) −1.00000 −0.0990148
\(103\) −14.2462 −1.40372 −0.701860 0.712314i \(-0.747647\pi\)
−0.701860 + 0.712314i \(0.747647\pi\)
\(104\) −7.12311 −0.698478
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 1.00000 0.0953463
\(111\) −7.12311 −0.676095
\(112\) 0 0
\(113\) −15.3693 −1.44582 −0.722912 0.690940i \(-0.757197\pi\)
−0.722912 + 0.690940i \(0.757197\pi\)
\(114\) 3.12311 0.292506
\(115\) 0.876894 0.0817708
\(116\) −0.876894 −0.0814176
\(117\) −7.12311 −0.658531
\(118\) −6.24621 −0.575010
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) 1.00000 0.0909091
\(122\) 8.24621 0.746577
\(123\) −5.12311 −0.461935
\(124\) 1.12311 0.100858
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −15.3693 −1.36381 −0.681903 0.731442i \(-0.738847\pi\)
−0.681903 + 0.731442i \(0.738847\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 7.12311 0.624738
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) −15.1231 −1.30644
\(135\) −1.00000 −0.0860663
\(136\) −1.00000 −0.0857493
\(137\) 12.2462 1.04626 0.523132 0.852252i \(-0.324763\pi\)
0.523132 + 0.852252i \(0.324763\pi\)
\(138\) −0.876894 −0.0746462
\(139\) −2.24621 −0.190521 −0.0952606 0.995452i \(-0.530368\pi\)
−0.0952606 + 0.995452i \(0.530368\pi\)
\(140\) 0 0
\(141\) −1.12311 −0.0945826
\(142\) −8.00000 −0.671345
\(143\) 7.12311 0.595664
\(144\) 1.00000 0.0833333
\(145\) 0.876894 0.0728221
\(146\) 0.246211 0.0203766
\(147\) −7.00000 −0.577350
\(148\) −7.12311 −0.585516
\(149\) −19.3693 −1.58680 −0.793398 0.608703i \(-0.791690\pi\)
−0.793398 + 0.608703i \(0.791690\pi\)
\(150\) 1.00000 0.0816497
\(151\) −15.3693 −1.25074 −0.625369 0.780329i \(-0.715051\pi\)
−0.625369 + 0.780329i \(0.715051\pi\)
\(152\) 3.12311 0.253317
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) −1.12311 −0.0902100
\(156\) −7.12311 −0.570305
\(157\) 17.6155 1.40587 0.702936 0.711253i \(-0.251872\pi\)
0.702936 + 0.711253i \(0.251872\pi\)
\(158\) 3.12311 0.248461
\(159\) 4.00000 0.317221
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −24.4924 −1.91839 −0.959197 0.282738i \(-0.908757\pi\)
−0.959197 + 0.282738i \(0.908757\pi\)
\(164\) −5.12311 −0.400047
\(165\) 1.00000 0.0778499
\(166\) −10.2462 −0.795260
\(167\) −14.8769 −1.15121 −0.575604 0.817728i \(-0.695233\pi\)
−0.575604 + 0.817728i \(0.695233\pi\)
\(168\) 0 0
\(169\) 37.7386 2.90297
\(170\) 1.00000 0.0766965
\(171\) 3.12311 0.238830
\(172\) 0 0
\(173\) −23.1231 −1.75802 −0.879009 0.476806i \(-0.841794\pi\)
−0.879009 + 0.476806i \(0.841794\pi\)
\(174\) −0.876894 −0.0664772
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −6.24621 −0.469494
\(178\) −4.24621 −0.318267
\(179\) 16.4924 1.23270 0.616351 0.787472i \(-0.288610\pi\)
0.616351 + 0.787472i \(0.288610\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 23.6155 1.75533 0.877664 0.479276i \(-0.159101\pi\)
0.877664 + 0.479276i \(0.159101\pi\)
\(182\) 0 0
\(183\) 8.24621 0.609577
\(184\) −0.876894 −0.0646455
\(185\) 7.12311 0.523701
\(186\) 1.12311 0.0823501
\(187\) 1.00000 0.0731272
\(188\) −1.12311 −0.0819109
\(189\) 0 0
\(190\) −3.12311 −0.226574
\(191\) 21.1231 1.52841 0.764207 0.644971i \(-0.223131\pi\)
0.764207 + 0.644971i \(0.223131\pi\)
\(192\) 1.00000 0.0721688
\(193\) 10.4924 0.755261 0.377631 0.925956i \(-0.376739\pi\)
0.377631 + 0.925956i \(0.376739\pi\)
\(194\) 2.87689 0.206549
\(195\) 7.12311 0.510096
\(196\) −7.00000 −0.500000
\(197\) −23.6155 −1.68254 −0.841268 0.540618i \(-0.818191\pi\)
−0.841268 + 0.540618i \(0.818191\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −6.87689 −0.487490 −0.243745 0.969839i \(-0.578376\pi\)
−0.243745 + 0.969839i \(0.578376\pi\)
\(200\) 1.00000 0.0707107
\(201\) −15.1231 −1.06670
\(202\) 15.3693 1.08138
\(203\) 0 0
\(204\) −1.00000 −0.0700140
\(205\) 5.12311 0.357813
\(206\) −14.2462 −0.992581
\(207\) −0.876894 −0.0609484
\(208\) −7.12311 −0.493899
\(209\) −3.12311 −0.216030
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 4.00000 0.274721
\(213\) −8.00000 −0.548151
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) 0.246211 0.0166374
\(220\) 1.00000 0.0674200
\(221\) 7.12311 0.479152
\(222\) −7.12311 −0.478072
\(223\) 26.2462 1.75758 0.878788 0.477212i \(-0.158353\pi\)
0.878788 + 0.477212i \(0.158353\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) −15.3693 −1.02235
\(227\) −1.75379 −0.116403 −0.0582015 0.998305i \(-0.518537\pi\)
−0.0582015 + 0.998305i \(0.518537\pi\)
\(228\) 3.12311 0.206833
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0.876894 0.0578207
\(231\) 0 0
\(232\) −0.876894 −0.0575709
\(233\) 22.4924 1.47353 0.736764 0.676150i \(-0.236353\pi\)
0.736764 + 0.676150i \(0.236353\pi\)
\(234\) −7.12311 −0.465652
\(235\) 1.12311 0.0732633
\(236\) −6.24621 −0.406594
\(237\) 3.12311 0.202868
\(238\) 0 0
\(239\) 9.75379 0.630920 0.315460 0.948939i \(-0.397841\pi\)
0.315460 + 0.948939i \(0.397841\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −25.6155 −1.65004 −0.825021 0.565103i \(-0.808837\pi\)
−0.825021 + 0.565103i \(0.808837\pi\)
\(242\) 1.00000 0.0642824
\(243\) 1.00000 0.0641500
\(244\) 8.24621 0.527910
\(245\) 7.00000 0.447214
\(246\) −5.12311 −0.326637
\(247\) −22.2462 −1.41549
\(248\) 1.12311 0.0713173
\(249\) −10.2462 −0.649327
\(250\) −1.00000 −0.0632456
\(251\) 22.7386 1.43525 0.717625 0.696430i \(-0.245229\pi\)
0.717625 + 0.696430i \(0.245229\pi\)
\(252\) 0 0
\(253\) 0.876894 0.0551299
\(254\) −15.3693 −0.964357
\(255\) 1.00000 0.0626224
\(256\) 1.00000 0.0625000
\(257\) 22.4924 1.40304 0.701519 0.712650i \(-0.252505\pi\)
0.701519 + 0.712650i \(0.252505\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 7.12311 0.441756
\(261\) −0.876894 −0.0542784
\(262\) 0 0
\(263\) 13.7538 0.848095 0.424047 0.905640i \(-0.360609\pi\)
0.424047 + 0.905640i \(0.360609\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) −4.24621 −0.259864
\(268\) −15.1231 −0.923791
\(269\) 24.2462 1.47832 0.739159 0.673531i \(-0.235223\pi\)
0.739159 + 0.673531i \(0.235223\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −27.3693 −1.66257 −0.831284 0.555848i \(-0.812394\pi\)
−0.831284 + 0.555848i \(0.812394\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) 12.2462 0.739821
\(275\) −1.00000 −0.0603023
\(276\) −0.876894 −0.0527828
\(277\) 4.24621 0.255130 0.127565 0.991830i \(-0.459284\pi\)
0.127565 + 0.991830i \(0.459284\pi\)
\(278\) −2.24621 −0.134719
\(279\) 1.12311 0.0672386
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −1.12311 −0.0668800
\(283\) −24.4924 −1.45592 −0.727962 0.685618i \(-0.759532\pi\)
−0.727962 + 0.685618i \(0.759532\pi\)
\(284\) −8.00000 −0.474713
\(285\) −3.12311 −0.184997
\(286\) 7.12311 0.421198
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 1.00000 0.0588235
\(290\) 0.876894 0.0514930
\(291\) 2.87689 0.168647
\(292\) 0.246211 0.0144084
\(293\) 18.4924 1.08034 0.540169 0.841556i \(-0.318360\pi\)
0.540169 + 0.841556i \(0.318360\pi\)
\(294\) −7.00000 −0.408248
\(295\) 6.24621 0.363668
\(296\) −7.12311 −0.414022
\(297\) −1.00000 −0.0580259
\(298\) −19.3693 −1.12203
\(299\) 6.24621 0.361228
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) −15.3693 −0.884405
\(303\) 15.3693 0.882944
\(304\) 3.12311 0.179122
\(305\) −8.24621 −0.472177
\(306\) −1.00000 −0.0571662
\(307\) 16.4924 0.941272 0.470636 0.882327i \(-0.344024\pi\)
0.470636 + 0.882327i \(0.344024\pi\)
\(308\) 0 0
\(309\) −14.2462 −0.810439
\(310\) −1.12311 −0.0637881
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) −7.12311 −0.403266
\(313\) −5.12311 −0.289575 −0.144788 0.989463i \(-0.546250\pi\)
−0.144788 + 0.989463i \(0.546250\pi\)
\(314\) 17.6155 0.994102
\(315\) 0 0
\(316\) 3.12311 0.175688
\(317\) 11.7538 0.660159 0.330079 0.943953i \(-0.392925\pi\)
0.330079 + 0.943953i \(0.392925\pi\)
\(318\) 4.00000 0.224309
\(319\) 0.876894 0.0490967
\(320\) −1.00000 −0.0559017
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −3.12311 −0.173774
\(324\) 1.00000 0.0555556
\(325\) −7.12311 −0.395119
\(326\) −24.4924 −1.35651
\(327\) −6.00000 −0.331801
\(328\) −5.12311 −0.282876
\(329\) 0 0
\(330\) 1.00000 0.0550482
\(331\) −13.7538 −0.755977 −0.377988 0.925810i \(-0.623384\pi\)
−0.377988 + 0.925810i \(0.623384\pi\)
\(332\) −10.2462 −0.562334
\(333\) −7.12311 −0.390344
\(334\) −14.8769 −0.814027
\(335\) 15.1231 0.826264
\(336\) 0 0
\(337\) −22.4924 −1.22524 −0.612620 0.790377i \(-0.709884\pi\)
−0.612620 + 0.790377i \(0.709884\pi\)
\(338\) 37.7386 2.05271
\(339\) −15.3693 −0.834747
\(340\) 1.00000 0.0542326
\(341\) −1.12311 −0.0608196
\(342\) 3.12311 0.168878
\(343\) 0 0
\(344\) 0 0
\(345\) 0.876894 0.0472104
\(346\) −23.1231 −1.24311
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) −0.876894 −0.0470065
\(349\) 4.00000 0.214115 0.107058 0.994253i \(-0.465857\pi\)
0.107058 + 0.994253i \(0.465857\pi\)
\(350\) 0 0
\(351\) −7.12311 −0.380203
\(352\) −1.00000 −0.0533002
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) −6.24621 −0.331982
\(355\) 8.00000 0.424596
\(356\) −4.24621 −0.225049
\(357\) 0 0
\(358\) 16.4924 0.871652
\(359\) −5.75379 −0.303673 −0.151837 0.988406i \(-0.548519\pi\)
−0.151837 + 0.988406i \(0.548519\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −9.24621 −0.486643
\(362\) 23.6155 1.24120
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −0.246211 −0.0128873
\(366\) 8.24621 0.431036
\(367\) 18.4924 0.965297 0.482648 0.875814i \(-0.339675\pi\)
0.482648 + 0.875814i \(0.339675\pi\)
\(368\) −0.876894 −0.0457113
\(369\) −5.12311 −0.266698
\(370\) 7.12311 0.370313
\(371\) 0 0
\(372\) 1.12311 0.0582303
\(373\) 3.61553 0.187205 0.0936025 0.995610i \(-0.470162\pi\)
0.0936025 + 0.995610i \(0.470162\pi\)
\(374\) 1.00000 0.0517088
\(375\) −1.00000 −0.0516398
\(376\) −1.12311 −0.0579198
\(377\) 6.24621 0.321696
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) −3.12311 −0.160212
\(381\) −15.3693 −0.787394
\(382\) 21.1231 1.08075
\(383\) −19.3693 −0.989726 −0.494863 0.868971i \(-0.664782\pi\)
−0.494863 + 0.868971i \(0.664782\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 10.4924 0.534050
\(387\) 0 0
\(388\) 2.87689 0.146052
\(389\) −7.61553 −0.386123 −0.193061 0.981187i \(-0.561842\pi\)
−0.193061 + 0.981187i \(0.561842\pi\)
\(390\) 7.12311 0.360692
\(391\) 0.876894 0.0443465
\(392\) −7.00000 −0.353553
\(393\) 0 0
\(394\) −23.6155 −1.18973
\(395\) −3.12311 −0.157140
\(396\) −1.00000 −0.0502519
\(397\) −21.8617 −1.09721 −0.548605 0.836082i \(-0.684841\pi\)
−0.548605 + 0.836082i \(0.684841\pi\)
\(398\) −6.87689 −0.344708
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) −15.1231 −0.754272
\(403\) −8.00000 −0.398508
\(404\) 15.3693 0.764652
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 7.12311 0.353079
\(408\) −1.00000 −0.0495074
\(409\) 10.4924 0.518817 0.259408 0.965768i \(-0.416472\pi\)
0.259408 + 0.965768i \(0.416472\pi\)
\(410\) 5.12311 0.253012
\(411\) 12.2462 0.604061
\(412\) −14.2462 −0.701860
\(413\) 0 0
\(414\) −0.876894 −0.0430970
\(415\) 10.2462 0.502967
\(416\) −7.12311 −0.349239
\(417\) −2.24621 −0.109997
\(418\) −3.12311 −0.152756
\(419\) 16.4924 0.805708 0.402854 0.915264i \(-0.368018\pi\)
0.402854 + 0.915264i \(0.368018\pi\)
\(420\) 0 0
\(421\) −40.7386 −1.98548 −0.992740 0.120282i \(-0.961620\pi\)
−0.992740 + 0.120282i \(0.961620\pi\)
\(422\) 20.0000 0.973585
\(423\) −1.12311 −0.0546073
\(424\) 4.00000 0.194257
\(425\) −1.00000 −0.0485071
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 7.12311 0.343907
\(430\) 0 0
\(431\) −30.9848 −1.49249 −0.746244 0.665673i \(-0.768145\pi\)
−0.746244 + 0.665673i \(0.768145\pi\)
\(432\) 1.00000 0.0481125
\(433\) −30.4924 −1.46537 −0.732686 0.680567i \(-0.761734\pi\)
−0.732686 + 0.680567i \(0.761734\pi\)
\(434\) 0 0
\(435\) 0.876894 0.0420439
\(436\) −6.00000 −0.287348
\(437\) −2.73863 −0.131007
\(438\) 0.246211 0.0117644
\(439\) −3.61553 −0.172560 −0.0862799 0.996271i \(-0.527498\pi\)
−0.0862799 + 0.996271i \(0.527498\pi\)
\(440\) 1.00000 0.0476731
\(441\) −7.00000 −0.333333
\(442\) 7.12311 0.338812
\(443\) −8.87689 −0.421754 −0.210877 0.977513i \(-0.567632\pi\)
−0.210877 + 0.977513i \(0.567632\pi\)
\(444\) −7.12311 −0.338048
\(445\) 4.24621 0.201290
\(446\) 26.2462 1.24279
\(447\) −19.3693 −0.916137
\(448\) 0 0
\(449\) 16.7386 0.789945 0.394972 0.918693i \(-0.370754\pi\)
0.394972 + 0.918693i \(0.370754\pi\)
\(450\) 1.00000 0.0471405
\(451\) 5.12311 0.241238
\(452\) −15.3693 −0.722912
\(453\) −15.3693 −0.722113
\(454\) −1.75379 −0.0823094
\(455\) 0 0
\(456\) 3.12311 0.146253
\(457\) −14.4924 −0.677927 −0.338963 0.940800i \(-0.610076\pi\)
−0.338963 + 0.940800i \(0.610076\pi\)
\(458\) −6.00000 −0.280362
\(459\) −1.00000 −0.0466760
\(460\) 0.876894 0.0408854
\(461\) 5.12311 0.238607 0.119303 0.992858i \(-0.461934\pi\)
0.119303 + 0.992858i \(0.461934\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −0.876894 −0.0407088
\(465\) −1.12311 −0.0520828
\(466\) 22.4924 1.04194
\(467\) 14.6307 0.677027 0.338514 0.940961i \(-0.390076\pi\)
0.338514 + 0.940961i \(0.390076\pi\)
\(468\) −7.12311 −0.329266
\(469\) 0 0
\(470\) 1.12311 0.0518050
\(471\) 17.6155 0.811681
\(472\) −6.24621 −0.287505
\(473\) 0 0
\(474\) 3.12311 0.143449
\(475\) 3.12311 0.143298
\(476\) 0 0
\(477\) 4.00000 0.183147
\(478\) 9.75379 0.446128
\(479\) 4.24621 0.194014 0.0970072 0.995284i \(-0.469073\pi\)
0.0970072 + 0.995284i \(0.469073\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 50.7386 2.31348
\(482\) −25.6155 −1.16676
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −2.87689 −0.130633
\(486\) 1.00000 0.0453609
\(487\) −9.50758 −0.430829 −0.215415 0.976523i \(-0.569110\pi\)
−0.215415 + 0.976523i \(0.569110\pi\)
\(488\) 8.24621 0.373288
\(489\) −24.4924 −1.10759
\(490\) 7.00000 0.316228
\(491\) 11.6155 0.524201 0.262101 0.965041i \(-0.415585\pi\)
0.262101 + 0.965041i \(0.415585\pi\)
\(492\) −5.12311 −0.230967
\(493\) 0.876894 0.0394933
\(494\) −22.2462 −1.00090
\(495\) 1.00000 0.0449467
\(496\) 1.12311 0.0504289
\(497\) 0 0
\(498\) −10.2462 −0.459144
\(499\) 17.7538 0.794769 0.397384 0.917652i \(-0.369918\pi\)
0.397384 + 0.917652i \(0.369918\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −14.8769 −0.664651
\(502\) 22.7386 1.01487
\(503\) −30.8769 −1.37673 −0.688366 0.725363i \(-0.741672\pi\)
−0.688366 + 0.725363i \(0.741672\pi\)
\(504\) 0 0
\(505\) −15.3693 −0.683926
\(506\) 0.876894 0.0389827
\(507\) 37.7386 1.67603
\(508\) −15.3693 −0.681903
\(509\) −36.1080 −1.60046 −0.800228 0.599695i \(-0.795288\pi\)
−0.800228 + 0.599695i \(0.795288\pi\)
\(510\) 1.00000 0.0442807
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 3.12311 0.137888
\(514\) 22.4924 0.992098
\(515\) 14.2462 0.627763
\(516\) 0 0
\(517\) 1.12311 0.0493941
\(518\) 0 0
\(519\) −23.1231 −1.01499
\(520\) 7.12311 0.312369
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) −0.876894 −0.0383806
\(523\) 30.7386 1.34411 0.672053 0.740503i \(-0.265413\pi\)
0.672053 + 0.740503i \(0.265413\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 13.7538 0.599694
\(527\) −1.12311 −0.0489232
\(528\) −1.00000 −0.0435194
\(529\) −22.2311 −0.966568
\(530\) −4.00000 −0.173749
\(531\) −6.24621 −0.271062
\(532\) 0 0
\(533\) 36.4924 1.58066
\(534\) −4.24621 −0.183752
\(535\) −12.0000 −0.518805
\(536\) −15.1231 −0.653219
\(537\) 16.4924 0.711701
\(538\) 24.2462 1.04533
\(539\) 7.00000 0.301511
\(540\) −1.00000 −0.0430331
\(541\) −14.4924 −0.623078 −0.311539 0.950233i \(-0.600844\pi\)
−0.311539 + 0.950233i \(0.600844\pi\)
\(542\) −27.3693 −1.17561
\(543\) 23.6155 1.01344
\(544\) −1.00000 −0.0428746
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) −7.50758 −0.321001 −0.160500 0.987036i \(-0.551311\pi\)
−0.160500 + 0.987036i \(0.551311\pi\)
\(548\) 12.2462 0.523132
\(549\) 8.24621 0.351940
\(550\) −1.00000 −0.0426401
\(551\) −2.73863 −0.116670
\(552\) −0.876894 −0.0373231
\(553\) 0 0
\(554\) 4.24621 0.180404
\(555\) 7.12311 0.302359
\(556\) −2.24621 −0.0952606
\(557\) −1.50758 −0.0638781 −0.0319391 0.999490i \(-0.510168\pi\)
−0.0319391 + 0.999490i \(0.510168\pi\)
\(558\) 1.12311 0.0475449
\(559\) 0 0
\(560\) 0 0
\(561\) 1.00000 0.0422200
\(562\) −18.0000 −0.759284
\(563\) −22.7386 −0.958319 −0.479160 0.877728i \(-0.659058\pi\)
−0.479160 + 0.877728i \(0.659058\pi\)
\(564\) −1.12311 −0.0472913
\(565\) 15.3693 0.646592
\(566\) −24.4924 −1.02949
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) −3.12311 −0.130812
\(571\) −10.2462 −0.428791 −0.214395 0.976747i \(-0.568778\pi\)
−0.214395 + 0.976747i \(0.568778\pi\)
\(572\) 7.12311 0.297832
\(573\) 21.1231 0.882430
\(574\) 0 0
\(575\) −0.876894 −0.0365690
\(576\) 1.00000 0.0416667
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 1.00000 0.0415945
\(579\) 10.4924 0.436050
\(580\) 0.876894 0.0364111
\(581\) 0 0
\(582\) 2.87689 0.119251
\(583\) −4.00000 −0.165663
\(584\) 0.246211 0.0101883
\(585\) 7.12311 0.294504
\(586\) 18.4924 0.763915
\(587\) 10.6307 0.438775 0.219388 0.975638i \(-0.429594\pi\)
0.219388 + 0.975638i \(0.429594\pi\)
\(588\) −7.00000 −0.288675
\(589\) 3.50758 0.144527
\(590\) 6.24621 0.257152
\(591\) −23.6155 −0.971413
\(592\) −7.12311 −0.292758
\(593\) 24.2462 0.995673 0.497836 0.867271i \(-0.334128\pi\)
0.497836 + 0.867271i \(0.334128\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) −19.3693 −0.793398
\(597\) −6.87689 −0.281453
\(598\) 6.24621 0.255427
\(599\) 8.63068 0.352640 0.176320 0.984333i \(-0.443581\pi\)
0.176320 + 0.984333i \(0.443581\pi\)
\(600\) 1.00000 0.0408248
\(601\) −10.8769 −0.443678 −0.221839 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(602\) 0 0
\(603\) −15.1231 −0.615860
\(604\) −15.3693 −0.625369
\(605\) −1.00000 −0.0406558
\(606\) 15.3693 0.624336
\(607\) 3.50758 0.142368 0.0711841 0.997463i \(-0.477322\pi\)
0.0711841 + 0.997463i \(0.477322\pi\)
\(608\) 3.12311 0.126659
\(609\) 0 0
\(610\) −8.24621 −0.333879
\(611\) 8.00000 0.323645
\(612\) −1.00000 −0.0404226
\(613\) 19.6155 0.792264 0.396132 0.918194i \(-0.370352\pi\)
0.396132 + 0.918194i \(0.370352\pi\)
\(614\) 16.4924 0.665580
\(615\) 5.12311 0.206584
\(616\) 0 0
\(617\) 31.3693 1.26288 0.631441 0.775424i \(-0.282464\pi\)
0.631441 + 0.775424i \(0.282464\pi\)
\(618\) −14.2462 −0.573067
\(619\) 9.75379 0.392038 0.196019 0.980600i \(-0.437199\pi\)
0.196019 + 0.980600i \(0.437199\pi\)
\(620\) −1.12311 −0.0451050
\(621\) −0.876894 −0.0351886
\(622\) −4.00000 −0.160385
\(623\) 0 0
\(624\) −7.12311 −0.285152
\(625\) 1.00000 0.0400000
\(626\) −5.12311 −0.204760
\(627\) −3.12311 −0.124725
\(628\) 17.6155 0.702936
\(629\) 7.12311 0.284017
\(630\) 0 0
\(631\) −34.2462 −1.36332 −0.681660 0.731669i \(-0.738742\pi\)
−0.681660 + 0.731669i \(0.738742\pi\)
\(632\) 3.12311 0.124230
\(633\) 20.0000 0.794929
\(634\) 11.7538 0.466803
\(635\) 15.3693 0.609913
\(636\) 4.00000 0.158610
\(637\) 49.8617 1.97559
\(638\) 0.876894 0.0347166
\(639\) −8.00000 −0.316475
\(640\) −1.00000 −0.0395285
\(641\) 36.7386 1.45109 0.725544 0.688175i \(-0.241588\pi\)
0.725544 + 0.688175i \(0.241588\pi\)
\(642\) 12.0000 0.473602
\(643\) 40.0000 1.57745 0.788723 0.614749i \(-0.210743\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −3.12311 −0.122877
\(647\) 31.3693 1.23326 0.616628 0.787255i \(-0.288498\pi\)
0.616628 + 0.787255i \(0.288498\pi\)
\(648\) 1.00000 0.0392837
\(649\) 6.24621 0.245185
\(650\) −7.12311 −0.279391
\(651\) 0 0
\(652\) −24.4924 −0.959197
\(653\) −11.7538 −0.459961 −0.229981 0.973195i \(-0.573866\pi\)
−0.229981 + 0.973195i \(0.573866\pi\)
\(654\) −6.00000 −0.234619
\(655\) 0 0
\(656\) −5.12311 −0.200024
\(657\) 0.246211 0.00960562
\(658\) 0 0
\(659\) 32.1080 1.25075 0.625374 0.780325i \(-0.284946\pi\)
0.625374 + 0.780325i \(0.284946\pi\)
\(660\) 1.00000 0.0389249
\(661\) 30.9848 1.20517 0.602585 0.798054i \(-0.294137\pi\)
0.602585 + 0.798054i \(0.294137\pi\)
\(662\) −13.7538 −0.534556
\(663\) 7.12311 0.276638
\(664\) −10.2462 −0.397630
\(665\) 0 0
\(666\) −7.12311 −0.276015
\(667\) 0.768944 0.0297736
\(668\) −14.8769 −0.575604
\(669\) 26.2462 1.01474
\(670\) 15.1231 0.584257
\(671\) −8.24621 −0.318341
\(672\) 0 0
\(673\) 22.9848 0.886001 0.443000 0.896521i \(-0.353914\pi\)
0.443000 + 0.896521i \(0.353914\pi\)
\(674\) −22.4924 −0.866376
\(675\) 1.00000 0.0384900
\(676\) 37.7386 1.45149
\(677\) −36.1080 −1.38774 −0.693871 0.720100i \(-0.744096\pi\)
−0.693871 + 0.720100i \(0.744096\pi\)
\(678\) −15.3693 −0.590255
\(679\) 0 0
\(680\) 1.00000 0.0383482
\(681\) −1.75379 −0.0672053
\(682\) −1.12311 −0.0430059
\(683\) 0.492423 0.0188420 0.00942101 0.999956i \(-0.497001\pi\)
0.00942101 + 0.999956i \(0.497001\pi\)
\(684\) 3.12311 0.119415
\(685\) −12.2462 −0.467904
\(686\) 0 0
\(687\) −6.00000 −0.228914
\(688\) 0 0
\(689\) −28.4924 −1.08547
\(690\) 0.876894 0.0333828
\(691\) 16.4924 0.627401 0.313701 0.949522i \(-0.398431\pi\)
0.313701 + 0.949522i \(0.398431\pi\)
\(692\) −23.1231 −0.879009
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) 2.24621 0.0852036
\(696\) −0.876894 −0.0332386
\(697\) 5.12311 0.194051
\(698\) 4.00000 0.151402
\(699\) 22.4924 0.850742
\(700\) 0 0
\(701\) −23.3693 −0.882647 −0.441323 0.897348i \(-0.645491\pi\)
−0.441323 + 0.897348i \(0.645491\pi\)
\(702\) −7.12311 −0.268844
\(703\) −22.2462 −0.839032
\(704\) −1.00000 −0.0376889
\(705\) 1.12311 0.0422986
\(706\) 26.0000 0.978523
\(707\) 0 0
\(708\) −6.24621 −0.234747
\(709\) 5.36932 0.201649 0.100824 0.994904i \(-0.467852\pi\)
0.100824 + 0.994904i \(0.467852\pi\)
\(710\) 8.00000 0.300235
\(711\) 3.12311 0.117126
\(712\) −4.24621 −0.159133
\(713\) −0.984845 −0.0368827
\(714\) 0 0
\(715\) −7.12311 −0.266389
\(716\) 16.4924 0.616351
\(717\) 9.75379 0.364262
\(718\) −5.75379 −0.214729
\(719\) −18.7386 −0.698833 −0.349417 0.936967i \(-0.613620\pi\)
−0.349417 + 0.936967i \(0.613620\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) −9.24621 −0.344108
\(723\) −25.6155 −0.952652
\(724\) 23.6155 0.877664
\(725\) −0.876894 −0.0325670
\(726\) 1.00000 0.0371135
\(727\) −7.50758 −0.278441 −0.139220 0.990261i \(-0.544460\pi\)
−0.139220 + 0.990261i \(0.544460\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −0.246211 −0.00911269
\(731\) 0 0
\(732\) 8.24621 0.304789
\(733\) −24.8769 −0.918849 −0.459425 0.888217i \(-0.651944\pi\)
−0.459425 + 0.888217i \(0.651944\pi\)
\(734\) 18.4924 0.682568
\(735\) 7.00000 0.258199
\(736\) −0.876894 −0.0323228
\(737\) 15.1231 0.557067
\(738\) −5.12311 −0.188584
\(739\) 47.1231 1.73345 0.866726 0.498785i \(-0.166220\pi\)
0.866726 + 0.498785i \(0.166220\pi\)
\(740\) 7.12311 0.261851
\(741\) −22.2462 −0.817235
\(742\) 0 0
\(743\) −41.6155 −1.52673 −0.763363 0.645970i \(-0.776453\pi\)
−0.763363 + 0.645970i \(0.776453\pi\)
\(744\) 1.12311 0.0411750
\(745\) 19.3693 0.709637
\(746\) 3.61553 0.132374
\(747\) −10.2462 −0.374889
\(748\) 1.00000 0.0365636
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) −1.12311 −0.0409827 −0.0204914 0.999790i \(-0.506523\pi\)
−0.0204914 + 0.999790i \(0.506523\pi\)
\(752\) −1.12311 −0.0409554
\(753\) 22.7386 0.828642
\(754\) 6.24621 0.227474
\(755\) 15.3693 0.559347
\(756\) 0 0
\(757\) 1.12311 0.0408200 0.0204100 0.999792i \(-0.493503\pi\)
0.0204100 + 0.999792i \(0.493503\pi\)
\(758\) 28.0000 1.01701
\(759\) 0.876894 0.0318292
\(760\) −3.12311 −0.113287
\(761\) 46.4924 1.68535 0.842674 0.538423i \(-0.180980\pi\)
0.842674 + 0.538423i \(0.180980\pi\)
\(762\) −15.3693 −0.556772
\(763\) 0 0
\(764\) 21.1231 0.764207
\(765\) 1.00000 0.0361551
\(766\) −19.3693 −0.699842
\(767\) 44.4924 1.60653
\(768\) 1.00000 0.0360844
\(769\) −11.7538 −0.423852 −0.211926 0.977286i \(-0.567974\pi\)
−0.211926 + 0.977286i \(0.567974\pi\)
\(770\) 0 0
\(771\) 22.4924 0.810045
\(772\) 10.4924 0.377631
\(773\) −12.4924 −0.449321 −0.224661 0.974437i \(-0.572127\pi\)
−0.224661 + 0.974437i \(0.572127\pi\)
\(774\) 0 0
\(775\) 1.12311 0.0403431
\(776\) 2.87689 0.103274
\(777\) 0 0
\(778\) −7.61553 −0.273030
\(779\) −16.0000 −0.573259
\(780\) 7.12311 0.255048
\(781\) 8.00000 0.286263
\(782\) 0.876894 0.0313577
\(783\) −0.876894 −0.0313376
\(784\) −7.00000 −0.250000
\(785\) −17.6155 −0.628725
\(786\) 0 0
\(787\) 16.4924 0.587891 0.293946 0.955822i \(-0.405032\pi\)
0.293946 + 0.955822i \(0.405032\pi\)
\(788\) −23.6155 −0.841268
\(789\) 13.7538 0.489648
\(790\) −3.12311 −0.111115
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) −58.7386 −2.08587
\(794\) −21.8617 −0.775844
\(795\) −4.00000 −0.141865
\(796\) −6.87689 −0.243745
\(797\) 42.2462 1.49644 0.748219 0.663452i \(-0.230909\pi\)
0.748219 + 0.663452i \(0.230909\pi\)
\(798\) 0 0
\(799\) 1.12311 0.0397326
\(800\) 1.00000 0.0353553
\(801\) −4.24621 −0.150032
\(802\) −18.0000 −0.635602
\(803\) −0.246211 −0.00868861
\(804\) −15.1231 −0.533351
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 24.2462 0.853507
\(808\) 15.3693 0.540691
\(809\) −43.3693 −1.52478 −0.762392 0.647115i \(-0.775975\pi\)
−0.762392 + 0.647115i \(0.775975\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −20.9848 −0.736878 −0.368439 0.929652i \(-0.620108\pi\)
−0.368439 + 0.929652i \(0.620108\pi\)
\(812\) 0 0
\(813\) −27.3693 −0.959884
\(814\) 7.12311 0.249665
\(815\) 24.4924 0.857932
\(816\) −1.00000 −0.0350070
\(817\) 0 0
\(818\) 10.4924 0.366859
\(819\) 0 0
\(820\) 5.12311 0.178907
\(821\) −41.3693 −1.44380 −0.721900 0.691998i \(-0.756731\pi\)
−0.721900 + 0.691998i \(0.756731\pi\)
\(822\) 12.2462 0.427136
\(823\) −49.2311 −1.71609 −0.858043 0.513577i \(-0.828320\pi\)
−0.858043 + 0.513577i \(0.828320\pi\)
\(824\) −14.2462 −0.496290
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) 33.7538 1.17373 0.586867 0.809683i \(-0.300361\pi\)
0.586867 + 0.809683i \(0.300361\pi\)
\(828\) −0.876894 −0.0304742
\(829\) −3.75379 −0.130374 −0.0651872 0.997873i \(-0.520764\pi\)
−0.0651872 + 0.997873i \(0.520764\pi\)
\(830\) 10.2462 0.355651
\(831\) 4.24621 0.147299
\(832\) −7.12311 −0.246949
\(833\) 7.00000 0.242536
\(834\) −2.24621 −0.0777799
\(835\) 14.8769 0.514836
\(836\) −3.12311 −0.108015
\(837\) 1.12311 0.0388202
\(838\) 16.4924 0.569721
\(839\) 52.9848 1.82924 0.914620 0.404315i \(-0.132490\pi\)
0.914620 + 0.404315i \(0.132490\pi\)
\(840\) 0 0
\(841\) −28.2311 −0.973485
\(842\) −40.7386 −1.40395
\(843\) −18.0000 −0.619953
\(844\) 20.0000 0.688428
\(845\) −37.7386 −1.29825
\(846\) −1.12311 −0.0386132
\(847\) 0 0
\(848\) 4.00000 0.137361
\(849\) −24.4924 −0.840578
\(850\) −1.00000 −0.0342997
\(851\) 6.24621 0.214117
\(852\) −8.00000 −0.274075
\(853\) −22.4924 −0.770126 −0.385063 0.922890i \(-0.625820\pi\)
−0.385063 + 0.922890i \(0.625820\pi\)
\(854\) 0 0
\(855\) −3.12311 −0.106808
\(856\) 12.0000 0.410152
\(857\) −26.0000 −0.888143 −0.444072 0.895991i \(-0.646466\pi\)
−0.444072 + 0.895991i \(0.646466\pi\)
\(858\) 7.12311 0.243179
\(859\) 43.2311 1.47502 0.737512 0.675334i \(-0.236000\pi\)
0.737512 + 0.675334i \(0.236000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −30.9848 −1.05535
\(863\) 10.8769 0.370254 0.185127 0.982715i \(-0.440730\pi\)
0.185127 + 0.982715i \(0.440730\pi\)
\(864\) 1.00000 0.0340207
\(865\) 23.1231 0.786209
\(866\) −30.4924 −1.03617
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) −3.12311 −0.105944
\(870\) 0.876894 0.0297295
\(871\) 107.723 3.65007
\(872\) −6.00000 −0.203186
\(873\) 2.87689 0.0973681
\(874\) −2.73863 −0.0926357
\(875\) 0 0
\(876\) 0.246211 0.00831871
\(877\) 36.7386 1.24058 0.620288 0.784374i \(-0.287016\pi\)
0.620288 + 0.784374i \(0.287016\pi\)
\(878\) −3.61553 −0.122018
\(879\) 18.4924 0.623734
\(880\) 1.00000 0.0337100
\(881\) 12.2462 0.412585 0.206293 0.978490i \(-0.433860\pi\)
0.206293 + 0.978490i \(0.433860\pi\)
\(882\) −7.00000 −0.235702
\(883\) −7.12311 −0.239712 −0.119856 0.992791i \(-0.538243\pi\)
−0.119856 + 0.992791i \(0.538243\pi\)
\(884\) 7.12311 0.239576
\(885\) 6.24621 0.209964
\(886\) −8.87689 −0.298225
\(887\) −31.3693 −1.05328 −0.526639 0.850089i \(-0.676548\pi\)
−0.526639 + 0.850089i \(0.676548\pi\)
\(888\) −7.12311 −0.239036
\(889\) 0 0
\(890\) 4.24621 0.142333
\(891\) −1.00000 −0.0335013
\(892\) 26.2462 0.878788
\(893\) −3.50758 −0.117377
\(894\) −19.3693 −0.647807
\(895\) −16.4924 −0.551281
\(896\) 0 0
\(897\) 6.24621 0.208555
\(898\) 16.7386 0.558575
\(899\) −0.984845 −0.0328464
\(900\) 1.00000 0.0333333
\(901\) −4.00000 −0.133259
\(902\) 5.12311 0.170581
\(903\) 0 0
\(904\) −15.3693 −0.511176
\(905\) −23.6155 −0.785007
\(906\) −15.3693 −0.510611
\(907\) −54.7386 −1.81757 −0.908783 0.417269i \(-0.862987\pi\)
−0.908783 + 0.417269i \(0.862987\pi\)
\(908\) −1.75379 −0.0582015
\(909\) 15.3693 0.509768
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 3.12311 0.103416
\(913\) 10.2462 0.339100
\(914\) −14.4924 −0.479367
\(915\) −8.24621 −0.272611
\(916\) −6.00000 −0.198246
\(917\) 0 0
\(918\) −1.00000 −0.0330049
\(919\) −38.1080 −1.25707 −0.628533 0.777783i \(-0.716344\pi\)
−0.628533 + 0.777783i \(0.716344\pi\)
\(920\) 0.876894 0.0289104
\(921\) 16.4924 0.543444
\(922\) 5.12311 0.168720
\(923\) 56.9848 1.87568
\(924\) 0 0
\(925\) −7.12311 −0.234206
\(926\) −24.0000 −0.788689
\(927\) −14.2462 −0.467907
\(928\) −0.876894 −0.0287855
\(929\) 3.75379 0.123158 0.0615789 0.998102i \(-0.480386\pi\)
0.0615789 + 0.998102i \(0.480386\pi\)
\(930\) −1.12311 −0.0368281
\(931\) −21.8617 −0.716490
\(932\) 22.4924 0.736764
\(933\) −4.00000 −0.130954
\(934\) 14.6307 0.478731
\(935\) −1.00000 −0.0327035
\(936\) −7.12311 −0.232826
\(937\) −10.4924 −0.342773 −0.171386 0.985204i \(-0.554825\pi\)
−0.171386 + 0.985204i \(0.554825\pi\)
\(938\) 0 0
\(939\) −5.12311 −0.167186
\(940\) 1.12311 0.0366317
\(941\) −8.38447 −0.273326 −0.136663 0.990618i \(-0.543638\pi\)
−0.136663 + 0.990618i \(0.543638\pi\)
\(942\) 17.6155 0.573945
\(943\) 4.49242 0.146293
\(944\) −6.24621 −0.203297
\(945\) 0 0
\(946\) 0 0
\(947\) 23.5076 0.763894 0.381947 0.924184i \(-0.375254\pi\)
0.381947 + 0.924184i \(0.375254\pi\)
\(948\) 3.12311 0.101434
\(949\) −1.75379 −0.0569304
\(950\) 3.12311 0.101327
\(951\) 11.7538 0.381143
\(952\) 0 0
\(953\) −58.4924 −1.89476 −0.947378 0.320118i \(-0.896277\pi\)
−0.947378 + 0.320118i \(0.896277\pi\)
\(954\) 4.00000 0.129505
\(955\) −21.1231 −0.683528
\(956\) 9.75379 0.315460
\(957\) 0.876894 0.0283460
\(958\) 4.24621 0.137189
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) −29.7386 −0.959311
\(962\) 50.7386 1.63588
\(963\) 12.0000 0.386695
\(964\) −25.6155 −0.825021
\(965\) −10.4924 −0.337763
\(966\) 0 0
\(967\) −24.6307 −0.792069 −0.396035 0.918236i \(-0.629614\pi\)
−0.396035 + 0.918236i \(0.629614\pi\)
\(968\) 1.00000 0.0321412
\(969\) −3.12311 −0.100329
\(970\) −2.87689 −0.0923715
\(971\) −53.4773 −1.71617 −0.858084 0.513510i \(-0.828345\pi\)
−0.858084 + 0.513510i \(0.828345\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −9.50758 −0.304642
\(975\) −7.12311 −0.228122
\(976\) 8.24621 0.263955
\(977\) 61.2311 1.95896 0.979478 0.201553i \(-0.0645990\pi\)
0.979478 + 0.201553i \(0.0645990\pi\)
\(978\) −24.4924 −0.783181
\(979\) 4.24621 0.135710
\(980\) 7.00000 0.223607
\(981\) −6.00000 −0.191565
\(982\) 11.6155 0.370666
\(983\) −20.8769 −0.665870 −0.332935 0.942950i \(-0.608039\pi\)
−0.332935 + 0.942950i \(0.608039\pi\)
\(984\) −5.12311 −0.163319
\(985\) 23.6155 0.752453
\(986\) 0.876894 0.0279260
\(987\) 0 0
\(988\) −22.2462 −0.707746
\(989\) 0 0
\(990\) 1.00000 0.0317821
\(991\) −2.87689 −0.0913876 −0.0456938 0.998955i \(-0.514550\pi\)
−0.0456938 + 0.998955i \(0.514550\pi\)
\(992\) 1.12311 0.0356586
\(993\) −13.7538 −0.436463
\(994\) 0 0
\(995\) 6.87689 0.218012
\(996\) −10.2462 −0.324664
\(997\) −20.7386 −0.656799 −0.328400 0.944539i \(-0.606509\pi\)
−0.328400 + 0.944539i \(0.606509\pi\)
\(998\) 17.7538 0.561986
\(999\) −7.12311 −0.225365
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 5610.2.a.bw.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5610.2.a.bw.1.1 2 1.1 even 1 trivial