Properties

Label 8281.2.a.t.1.1
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
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] = [8281,2,Mod(1,8281)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8281.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.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: \(66.1241179138\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{2} -2.73205 q^{3} +1.00000 q^{4} +1.73205 q^{5} +4.73205 q^{6} +1.73205 q^{8} +4.46410 q^{9} +O(q^{10})\) \(q-1.73205 q^{2} -2.73205 q^{3} +1.00000 q^{4} +1.73205 q^{5} +4.73205 q^{6} +1.73205 q^{8} +4.46410 q^{9} -3.00000 q^{10} -1.26795 q^{11} -2.73205 q^{12} -4.73205 q^{15} -5.00000 q^{16} +7.73205 q^{17} -7.73205 q^{18} +2.00000 q^{19} +1.73205 q^{20} +2.19615 q^{22} +4.73205 q^{23} -4.73205 q^{24} -2.00000 q^{25} -4.00000 q^{27} -3.00000 q^{29} +8.19615 q^{30} +4.19615 q^{31} +5.19615 q^{32} +3.46410 q^{33} -13.3923 q^{34} +4.46410 q^{36} +7.00000 q^{37} -3.46410 q^{38} +3.00000 q^{40} -5.19615 q^{41} -0.196152 q^{43} -1.26795 q^{44} +7.73205 q^{45} -8.19615 q^{46} +12.9282 q^{47} +13.6603 q^{48} +3.46410 q^{50} -21.1244 q^{51} -9.92820 q^{53} +6.92820 q^{54} -2.19615 q^{55} -5.46410 q^{57} +5.19615 q^{58} +7.26795 q^{59} -4.73205 q^{60} +4.80385 q^{61} -7.26795 q^{62} +1.00000 q^{64} -6.00000 q^{66} +6.19615 q^{67} +7.73205 q^{68} -12.9282 q^{69} -6.00000 q^{71} +7.73205 q^{72} -3.19615 q^{73} -12.1244 q^{74} +5.46410 q^{75} +2.00000 q^{76} +16.1962 q^{79} -8.66025 q^{80} -2.46410 q^{81} +9.00000 q^{82} -2.19615 q^{83} +13.3923 q^{85} +0.339746 q^{86} +8.19615 q^{87} -2.19615 q^{88} +12.9282 q^{89} -13.3923 q^{90} +4.73205 q^{92} -11.4641 q^{93} -22.3923 q^{94} +3.46410 q^{95} -14.1962 q^{96} +6.39230 q^{97} -5.66025 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} + 6 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{4} + 6 q^{6} + 2 q^{9} - 6 q^{10} - 6 q^{11} - 2 q^{12} - 6 q^{15} - 10 q^{16} + 12 q^{17} - 12 q^{18} + 4 q^{19} - 6 q^{22} + 6 q^{23} - 6 q^{24} - 4 q^{25} - 8 q^{27} - 6 q^{29} + 6 q^{30} - 2 q^{31} - 6 q^{34} + 2 q^{36} + 14 q^{37} + 6 q^{40} + 10 q^{43} - 6 q^{44} + 12 q^{45} - 6 q^{46} + 12 q^{47} + 10 q^{48} - 18 q^{51} - 6 q^{53} + 6 q^{55} - 4 q^{57} + 18 q^{59} - 6 q^{60} + 20 q^{61} - 18 q^{62} + 2 q^{64} - 12 q^{66} + 2 q^{67} + 12 q^{68} - 12 q^{69} - 12 q^{71} + 12 q^{72} + 4 q^{73} + 4 q^{75} + 4 q^{76} + 22 q^{79} + 2 q^{81} + 18 q^{82} + 6 q^{83} + 6 q^{85} + 18 q^{86} + 6 q^{87} + 6 q^{88} + 12 q^{89} - 6 q^{90} + 6 q^{92} - 16 q^{93} - 24 q^{94} - 18 q^{96} - 8 q^{97} + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 −1.22474 −0.612372 0.790569i \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) −2.73205 −1.57735 −0.788675 0.614810i \(-0.789233\pi\)
−0.788675 + 0.614810i \(0.789233\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.73205 0.774597 0.387298 0.921954i \(-0.373408\pi\)
0.387298 + 0.921954i \(0.373408\pi\)
\(6\) 4.73205 1.93185
\(7\) 0 0
\(8\) 1.73205 0.612372
\(9\) 4.46410 1.48803
\(10\) −3.00000 −0.948683
\(11\) −1.26795 −0.382301 −0.191151 0.981561i \(-0.561222\pi\)
−0.191151 + 0.981561i \(0.561222\pi\)
\(12\) −2.73205 −0.788675
\(13\) 0 0
\(14\) 0 0
\(15\) −4.73205 −1.22181
\(16\) −5.00000 −1.25000
\(17\) 7.73205 1.87530 0.937649 0.347584i \(-0.112998\pi\)
0.937649 + 0.347584i \(0.112998\pi\)
\(18\) −7.73205 −1.82246
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 1.73205 0.387298
\(21\) 0 0
\(22\) 2.19615 0.468221
\(23\) 4.73205 0.986701 0.493350 0.869831i \(-0.335772\pi\)
0.493350 + 0.869831i \(0.335772\pi\)
\(24\) −4.73205 −0.965926
\(25\) −2.00000 −0.400000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 8.19615 1.49641
\(31\) 4.19615 0.753651 0.376826 0.926284i \(-0.377016\pi\)
0.376826 + 0.926284i \(0.377016\pi\)
\(32\) 5.19615 0.918559
\(33\) 3.46410 0.603023
\(34\) −13.3923 −2.29676
\(35\) 0 0
\(36\) 4.46410 0.744017
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) −3.46410 −0.561951
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) −5.19615 −0.811503 −0.405751 0.913984i \(-0.632990\pi\)
−0.405751 + 0.913984i \(0.632990\pi\)
\(42\) 0 0
\(43\) −0.196152 −0.0299130 −0.0149565 0.999888i \(-0.504761\pi\)
−0.0149565 + 0.999888i \(0.504761\pi\)
\(44\) −1.26795 −0.191151
\(45\) 7.73205 1.15263
\(46\) −8.19615 −1.20846
\(47\) 12.9282 1.88577 0.942886 0.333115i \(-0.108100\pi\)
0.942886 + 0.333115i \(0.108100\pi\)
\(48\) 13.6603 1.97169
\(49\) 0 0
\(50\) 3.46410 0.489898
\(51\) −21.1244 −2.95800
\(52\) 0 0
\(53\) −9.92820 −1.36374 −0.681872 0.731472i \(-0.738834\pi\)
−0.681872 + 0.731472i \(0.738834\pi\)
\(54\) 6.92820 0.942809
\(55\) −2.19615 −0.296129
\(56\) 0 0
\(57\) −5.46410 −0.723738
\(58\) 5.19615 0.682288
\(59\) 7.26795 0.946206 0.473103 0.881007i \(-0.343134\pi\)
0.473103 + 0.881007i \(0.343134\pi\)
\(60\) −4.73205 −0.610905
\(61\) 4.80385 0.615070 0.307535 0.951537i \(-0.400496\pi\)
0.307535 + 0.951537i \(0.400496\pi\)
\(62\) −7.26795 −0.923030
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 6.19615 0.756980 0.378490 0.925605i \(-0.376443\pi\)
0.378490 + 0.925605i \(0.376443\pi\)
\(68\) 7.73205 0.937649
\(69\) −12.9282 −1.55637
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 7.73205 0.911231
\(73\) −3.19615 −0.374081 −0.187041 0.982352i \(-0.559890\pi\)
−0.187041 + 0.982352i \(0.559890\pi\)
\(74\) −12.1244 −1.40943
\(75\) 5.46410 0.630940
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 0 0
\(79\) 16.1962 1.82221 0.911105 0.412175i \(-0.135231\pi\)
0.911105 + 0.412175i \(0.135231\pi\)
\(80\) −8.66025 −0.968246
\(81\) −2.46410 −0.273789
\(82\) 9.00000 0.993884
\(83\) −2.19615 −0.241059 −0.120530 0.992710i \(-0.538459\pi\)
−0.120530 + 0.992710i \(0.538459\pi\)
\(84\) 0 0
\(85\) 13.3923 1.45260
\(86\) 0.339746 0.0366357
\(87\) 8.19615 0.878720
\(88\) −2.19615 −0.234111
\(89\) 12.9282 1.37039 0.685193 0.728361i \(-0.259718\pi\)
0.685193 + 0.728361i \(0.259718\pi\)
\(90\) −13.3923 −1.41167
\(91\) 0 0
\(92\) 4.73205 0.493350
\(93\) −11.4641 −1.18877
\(94\) −22.3923 −2.30959
\(95\) 3.46410 0.355409
\(96\) −14.1962 −1.44889
\(97\) 6.39230 0.649040 0.324520 0.945879i \(-0.394797\pi\)
0.324520 + 0.945879i \(0.394797\pi\)
\(98\) 0 0
\(99\) −5.66025 −0.568877
\(100\) −2.00000 −0.200000
\(101\) 7.73205 0.769368 0.384684 0.923048i \(-0.374310\pi\)
0.384684 + 0.923048i \(0.374310\pi\)
\(102\) 36.5885 3.62280
\(103\) 14.3923 1.41812 0.709058 0.705150i \(-0.249120\pi\)
0.709058 + 0.705150i \(0.249120\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 17.1962 1.67024
\(107\) −7.85641 −0.759507 −0.379754 0.925088i \(-0.623991\pi\)
−0.379754 + 0.925088i \(0.623991\pi\)
\(108\) −4.00000 −0.384900
\(109\) 8.39230 0.803837 0.401919 0.915675i \(-0.368344\pi\)
0.401919 + 0.915675i \(0.368344\pi\)
\(110\) 3.80385 0.362683
\(111\) −19.1244 −1.81520
\(112\) 0 0
\(113\) 13.3923 1.25984 0.629921 0.776659i \(-0.283087\pi\)
0.629921 + 0.776659i \(0.283087\pi\)
\(114\) 9.46410 0.886394
\(115\) 8.19615 0.764295
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) −12.5885 −1.15886
\(119\) 0 0
\(120\) −8.19615 −0.748203
\(121\) −9.39230 −0.853846
\(122\) −8.32051 −0.753303
\(123\) 14.1962 1.28002
\(124\) 4.19615 0.376826
\(125\) −12.1244 −1.08444
\(126\) 0 0
\(127\) 18.3923 1.63205 0.816027 0.578014i \(-0.196172\pi\)
0.816027 + 0.578014i \(0.196172\pi\)
\(128\) −12.1244 −1.07165
\(129\) 0.535898 0.0471832
\(130\) 0 0
\(131\) 3.46410 0.302660 0.151330 0.988483i \(-0.451644\pi\)
0.151330 + 0.988483i \(0.451644\pi\)
\(132\) 3.46410 0.301511
\(133\) 0 0
\(134\) −10.7321 −0.927108
\(135\) −6.92820 −0.596285
\(136\) 13.3923 1.14838
\(137\) −8.07180 −0.689620 −0.344810 0.938672i \(-0.612057\pi\)
−0.344810 + 0.938672i \(0.612057\pi\)
\(138\) 22.3923 1.90616
\(139\) 10.5885 0.898101 0.449051 0.893506i \(-0.351762\pi\)
0.449051 + 0.893506i \(0.351762\pi\)
\(140\) 0 0
\(141\) −35.3205 −2.97452
\(142\) 10.3923 0.872103
\(143\) 0 0
\(144\) −22.3205 −1.86004
\(145\) −5.19615 −0.431517
\(146\) 5.53590 0.458154
\(147\) 0 0
\(148\) 7.00000 0.575396
\(149\) 6.46410 0.529560 0.264780 0.964309i \(-0.414701\pi\)
0.264780 + 0.964309i \(0.414701\pi\)
\(150\) −9.46410 −0.772741
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 3.46410 0.280976
\(153\) 34.5167 2.79051
\(154\) 0 0
\(155\) 7.26795 0.583776
\(156\) 0 0
\(157\) −1.19615 −0.0954634 −0.0477317 0.998860i \(-0.515199\pi\)
−0.0477317 + 0.998860i \(0.515199\pi\)
\(158\) −28.0526 −2.23174
\(159\) 27.1244 2.15110
\(160\) 9.00000 0.711512
\(161\) 0 0
\(162\) 4.26795 0.335322
\(163\) −16.1962 −1.26858 −0.634290 0.773095i \(-0.718708\pi\)
−0.634290 + 0.773095i \(0.718708\pi\)
\(164\) −5.19615 −0.405751
\(165\) 6.00000 0.467099
\(166\) 3.80385 0.295236
\(167\) −6.58846 −0.509830 −0.254915 0.966963i \(-0.582048\pi\)
−0.254915 + 0.966963i \(0.582048\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −23.1962 −1.77906
\(171\) 8.92820 0.682757
\(172\) −0.196152 −0.0149565
\(173\) −8.53590 −0.648972 −0.324486 0.945890i \(-0.605191\pi\)
−0.324486 + 0.945890i \(0.605191\pi\)
\(174\) −14.1962 −1.07621
\(175\) 0 0
\(176\) 6.33975 0.477876
\(177\) −19.8564 −1.49250
\(178\) −22.3923 −1.67837
\(179\) −6.92820 −0.517838 −0.258919 0.965899i \(-0.583366\pi\)
−0.258919 + 0.965899i \(0.583366\pi\)
\(180\) 7.73205 0.576313
\(181\) −5.58846 −0.415387 −0.207693 0.978194i \(-0.566596\pi\)
−0.207693 + 0.978194i \(0.566596\pi\)
\(182\) 0 0
\(183\) −13.1244 −0.970180
\(184\) 8.19615 0.604228
\(185\) 12.1244 0.891400
\(186\) 19.8564 1.45594
\(187\) −9.80385 −0.716928
\(188\) 12.9282 0.942886
\(189\) 0 0
\(190\) −6.00000 −0.435286
\(191\) 4.73205 0.342399 0.171200 0.985236i \(-0.445236\pi\)
0.171200 + 0.985236i \(0.445236\pi\)
\(192\) −2.73205 −0.197169
\(193\) −5.00000 −0.359908 −0.179954 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(194\) −11.0718 −0.794909
\(195\) 0 0
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 9.80385 0.696729
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) −3.46410 −0.244949
\(201\) −16.9282 −1.19402
\(202\) −13.3923 −0.942279
\(203\) 0 0
\(204\) −21.1244 −1.47900
\(205\) −9.00000 −0.628587
\(206\) −24.9282 −1.73683
\(207\) 21.1244 1.46824
\(208\) 0 0
\(209\) −2.53590 −0.175412
\(210\) 0 0
\(211\) −1.80385 −0.124182 −0.0620910 0.998070i \(-0.519777\pi\)
−0.0620910 + 0.998070i \(0.519777\pi\)
\(212\) −9.92820 −0.681872
\(213\) 16.3923 1.12318
\(214\) 13.6077 0.930203
\(215\) −0.339746 −0.0231705
\(216\) −6.92820 −0.471405
\(217\) 0 0
\(218\) −14.5359 −0.984495
\(219\) 8.73205 0.590057
\(220\) −2.19615 −0.148065
\(221\) 0 0
\(222\) 33.1244 2.22316
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 0 0
\(225\) −8.92820 −0.595214
\(226\) −23.1962 −1.54299
\(227\) 5.66025 0.375684 0.187842 0.982199i \(-0.439851\pi\)
0.187842 + 0.982199i \(0.439851\pi\)
\(228\) −5.46410 −0.361869
\(229\) −14.3923 −0.951070 −0.475535 0.879697i \(-0.657746\pi\)
−0.475535 + 0.879697i \(0.657746\pi\)
\(230\) −14.1962 −0.936067
\(231\) 0 0
\(232\) −5.19615 −0.341144
\(233\) −1.85641 −0.121617 −0.0608086 0.998149i \(-0.519368\pi\)
−0.0608086 + 0.998149i \(0.519368\pi\)
\(234\) 0 0
\(235\) 22.3923 1.46071
\(236\) 7.26795 0.473103
\(237\) −44.2487 −2.87426
\(238\) 0 0
\(239\) 15.8038 1.02227 0.511133 0.859502i \(-0.329226\pi\)
0.511133 + 0.859502i \(0.329226\pi\)
\(240\) 23.6603 1.52726
\(241\) −21.1962 −1.36536 −0.682682 0.730716i \(-0.739187\pi\)
−0.682682 + 0.730716i \(0.739187\pi\)
\(242\) 16.2679 1.04574
\(243\) 18.7321 1.20166
\(244\) 4.80385 0.307535
\(245\) 0 0
\(246\) −24.5885 −1.56770
\(247\) 0 0
\(248\) 7.26795 0.461515
\(249\) 6.00000 0.380235
\(250\) 21.0000 1.32816
\(251\) 1.60770 0.101477 0.0507384 0.998712i \(-0.483843\pi\)
0.0507384 + 0.998712i \(0.483843\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) −31.8564 −1.99885
\(255\) −36.5885 −2.29126
\(256\) 19.0000 1.18750
\(257\) −6.12436 −0.382027 −0.191013 0.981587i \(-0.561177\pi\)
−0.191013 + 0.981587i \(0.561177\pi\)
\(258\) −0.928203 −0.0577874
\(259\) 0 0
\(260\) 0 0
\(261\) −13.3923 −0.828963
\(262\) −6.00000 −0.370681
\(263\) 1.26795 0.0781851 0.0390925 0.999236i \(-0.487553\pi\)
0.0390925 + 0.999236i \(0.487553\pi\)
\(264\) 6.00000 0.369274
\(265\) −17.1962 −1.05635
\(266\) 0 0
\(267\) −35.3205 −2.16158
\(268\) 6.19615 0.378490
\(269\) −5.07180 −0.309233 −0.154616 0.987975i \(-0.549414\pi\)
−0.154616 + 0.987975i \(0.549414\pi\)
\(270\) 12.0000 0.730297
\(271\) 5.80385 0.352559 0.176279 0.984340i \(-0.443594\pi\)
0.176279 + 0.984340i \(0.443594\pi\)
\(272\) −38.6603 −2.34412
\(273\) 0 0
\(274\) 13.9808 0.844609
\(275\) 2.53590 0.152920
\(276\) −12.9282 −0.778186
\(277\) 17.0000 1.02143 0.510716 0.859750i \(-0.329381\pi\)
0.510716 + 0.859750i \(0.329381\pi\)
\(278\) −18.3397 −1.09994
\(279\) 18.7321 1.12146
\(280\) 0 0
\(281\) −13.3923 −0.798918 −0.399459 0.916751i \(-0.630802\pi\)
−0.399459 + 0.916751i \(0.630802\pi\)
\(282\) 61.1769 3.64303
\(283\) −10.1962 −0.606098 −0.303049 0.952975i \(-0.598005\pi\)
−0.303049 + 0.952975i \(0.598005\pi\)
\(284\) −6.00000 −0.356034
\(285\) −9.46410 −0.560605
\(286\) 0 0
\(287\) 0 0
\(288\) 23.1962 1.36685
\(289\) 42.7846 2.51674
\(290\) 9.00000 0.528498
\(291\) −17.4641 −1.02376
\(292\) −3.19615 −0.187041
\(293\) −0.803848 −0.0469613 −0.0234806 0.999724i \(-0.507475\pi\)
−0.0234806 + 0.999724i \(0.507475\pi\)
\(294\) 0 0
\(295\) 12.5885 0.732928
\(296\) 12.1244 0.704714
\(297\) 5.07180 0.294295
\(298\) −11.1962 −0.648576
\(299\) 0 0
\(300\) 5.46410 0.315470
\(301\) 0 0
\(302\) 3.46410 0.199337
\(303\) −21.1244 −1.21356
\(304\) −10.0000 −0.573539
\(305\) 8.32051 0.476431
\(306\) −59.7846 −3.41766
\(307\) −4.58846 −0.261877 −0.130939 0.991390i \(-0.541799\pi\)
−0.130939 + 0.991390i \(0.541799\pi\)
\(308\) 0 0
\(309\) −39.3205 −2.23687
\(310\) −12.5885 −0.714976
\(311\) −1.26795 −0.0718988 −0.0359494 0.999354i \(-0.511446\pi\)
−0.0359494 + 0.999354i \(0.511446\pi\)
\(312\) 0 0
\(313\) −28.7846 −1.62700 −0.813501 0.581563i \(-0.802441\pi\)
−0.813501 + 0.581563i \(0.802441\pi\)
\(314\) 2.07180 0.116918
\(315\) 0 0
\(316\) 16.1962 0.911105
\(317\) 6.46410 0.363060 0.181530 0.983385i \(-0.441895\pi\)
0.181530 + 0.983385i \(0.441895\pi\)
\(318\) −46.9808 −2.63455
\(319\) 3.80385 0.212975
\(320\) 1.73205 0.0968246
\(321\) 21.4641 1.19801
\(322\) 0 0
\(323\) 15.4641 0.860446
\(324\) −2.46410 −0.136895
\(325\) 0 0
\(326\) 28.0526 1.55369
\(327\) −22.9282 −1.26793
\(328\) −9.00000 −0.496942
\(329\) 0 0
\(330\) −10.3923 −0.572078
\(331\) −24.9808 −1.37307 −0.686533 0.727098i \(-0.740868\pi\)
−0.686533 + 0.727098i \(0.740868\pi\)
\(332\) −2.19615 −0.120530
\(333\) 31.2487 1.71242
\(334\) 11.4115 0.624412
\(335\) 10.7321 0.586355
\(336\) 0 0
\(337\) 11.0000 0.599208 0.299604 0.954064i \(-0.403145\pi\)
0.299604 + 0.954064i \(0.403145\pi\)
\(338\) 0 0
\(339\) −36.5885 −1.98721
\(340\) 13.3923 0.726300
\(341\) −5.32051 −0.288122
\(342\) −15.4641 −0.836203
\(343\) 0 0
\(344\) −0.339746 −0.0183179
\(345\) −22.3923 −1.20556
\(346\) 14.7846 0.794826
\(347\) 7.26795 0.390164 0.195082 0.980787i \(-0.437503\pi\)
0.195082 + 0.980787i \(0.437503\pi\)
\(348\) 8.19615 0.439360
\(349\) −24.7846 −1.32669 −0.663345 0.748314i \(-0.730864\pi\)
−0.663345 + 0.748314i \(0.730864\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −6.58846 −0.351166
\(353\) 20.6603 1.09963 0.549817 0.835285i \(-0.314697\pi\)
0.549817 + 0.835285i \(0.314697\pi\)
\(354\) 34.3923 1.82793
\(355\) −10.3923 −0.551566
\(356\) 12.9282 0.685193
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −18.9282 −0.998992 −0.499496 0.866316i \(-0.666482\pi\)
−0.499496 + 0.866316i \(0.666482\pi\)
\(360\) 13.3923 0.705836
\(361\) −15.0000 −0.789474
\(362\) 9.67949 0.508743
\(363\) 25.6603 1.34681
\(364\) 0 0
\(365\) −5.53590 −0.289762
\(366\) 22.7321 1.18822
\(367\) −4.19615 −0.219037 −0.109519 0.993985i \(-0.534931\pi\)
−0.109519 + 0.993985i \(0.534931\pi\)
\(368\) −23.6603 −1.23338
\(369\) −23.1962 −1.20754
\(370\) −21.0000 −1.09174
\(371\) 0 0
\(372\) −11.4641 −0.594386
\(373\) −11.3923 −0.589871 −0.294936 0.955517i \(-0.595298\pi\)
−0.294936 + 0.955517i \(0.595298\pi\)
\(374\) 16.9808 0.878054
\(375\) 33.1244 1.71053
\(376\) 22.3923 1.15479
\(377\) 0 0
\(378\) 0 0
\(379\) −26.5885 −1.36576 −0.682879 0.730532i \(-0.739272\pi\)
−0.682879 + 0.730532i \(0.739272\pi\)
\(380\) 3.46410 0.177705
\(381\) −50.2487 −2.57432
\(382\) −8.19615 −0.419352
\(383\) −11.6603 −0.595811 −0.297906 0.954595i \(-0.596288\pi\)
−0.297906 + 0.954595i \(0.596288\pi\)
\(384\) 33.1244 1.69037
\(385\) 0 0
\(386\) 8.66025 0.440795
\(387\) −0.875644 −0.0445115
\(388\) 6.39230 0.324520
\(389\) −23.5359 −1.19332 −0.596659 0.802495i \(-0.703505\pi\)
−0.596659 + 0.802495i \(0.703505\pi\)
\(390\) 0 0
\(391\) 36.5885 1.85036
\(392\) 0 0
\(393\) −9.46410 −0.477401
\(394\) 20.7846 1.04711
\(395\) 28.0526 1.41148
\(396\) −5.66025 −0.284438
\(397\) −18.7846 −0.942773 −0.471386 0.881927i \(-0.656246\pi\)
−0.471386 + 0.881927i \(0.656246\pi\)
\(398\) 3.46410 0.173640
\(399\) 0 0
\(400\) 10.0000 0.500000
\(401\) 10.8564 0.542143 0.271072 0.962559i \(-0.412622\pi\)
0.271072 + 0.962559i \(0.412622\pi\)
\(402\) 29.3205 1.46237
\(403\) 0 0
\(404\) 7.73205 0.384684
\(405\) −4.26795 −0.212076
\(406\) 0 0
\(407\) −8.87564 −0.439949
\(408\) −36.5885 −1.81140
\(409\) −16.8038 −0.830897 −0.415448 0.909617i \(-0.636375\pi\)
−0.415448 + 0.909617i \(0.636375\pi\)
\(410\) 15.5885 0.769859
\(411\) 22.0526 1.08777
\(412\) 14.3923 0.709058
\(413\) 0 0
\(414\) −36.5885 −1.79822
\(415\) −3.80385 −0.186724
\(416\) 0 0
\(417\) −28.9282 −1.41662
\(418\) 4.39230 0.214835
\(419\) 32.1962 1.57288 0.786442 0.617664i \(-0.211921\pi\)
0.786442 + 0.617664i \(0.211921\pi\)
\(420\) 0 0
\(421\) 32.1769 1.56821 0.784103 0.620630i \(-0.213123\pi\)
0.784103 + 0.620630i \(0.213123\pi\)
\(422\) 3.12436 0.152091
\(423\) 57.7128 2.80609
\(424\) −17.1962 −0.835119
\(425\) −15.4641 −0.750119
\(426\) −28.3923 −1.37561
\(427\) 0 0
\(428\) −7.85641 −0.379754
\(429\) 0 0
\(430\) 0.588457 0.0283779
\(431\) −0.679492 −0.0327300 −0.0163650 0.999866i \(-0.505209\pi\)
−0.0163650 + 0.999866i \(0.505209\pi\)
\(432\) 20.0000 0.962250
\(433\) 13.5885 0.653020 0.326510 0.945194i \(-0.394127\pi\)
0.326510 + 0.945194i \(0.394127\pi\)
\(434\) 0 0
\(435\) 14.1962 0.680653
\(436\) 8.39230 0.401919
\(437\) 9.46410 0.452729
\(438\) −15.1244 −0.722670
\(439\) −14.5885 −0.696269 −0.348135 0.937445i \(-0.613185\pi\)
−0.348135 + 0.937445i \(0.613185\pi\)
\(440\) −3.80385 −0.181341
\(441\) 0 0
\(442\) 0 0
\(443\) 23.3205 1.10799 0.553995 0.832520i \(-0.313103\pi\)
0.553995 + 0.832520i \(0.313103\pi\)
\(444\) −19.1244 −0.907602
\(445\) 22.3923 1.06150
\(446\) 17.3205 0.820150
\(447\) −17.6603 −0.835301
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 15.4641 0.728985
\(451\) 6.58846 0.310238
\(452\) 13.3923 0.629921
\(453\) 5.46410 0.256726
\(454\) −9.80385 −0.460117
\(455\) 0 0
\(456\) −9.46410 −0.443197
\(457\) −11.0000 −0.514558 −0.257279 0.966337i \(-0.582826\pi\)
−0.257279 + 0.966337i \(0.582826\pi\)
\(458\) 24.9282 1.16482
\(459\) −30.9282 −1.44360
\(460\) 8.19615 0.382148
\(461\) −15.5885 −0.726027 −0.363013 0.931784i \(-0.618252\pi\)
−0.363013 + 0.931784i \(0.618252\pi\)
\(462\) 0 0
\(463\) 4.58846 0.213244 0.106622 0.994300i \(-0.465997\pi\)
0.106622 + 0.994300i \(0.465997\pi\)
\(464\) 15.0000 0.696358
\(465\) −19.8564 −0.920819
\(466\) 3.21539 0.148950
\(467\) −25.5167 −1.18077 −0.590385 0.807122i \(-0.701024\pi\)
−0.590385 + 0.807122i \(0.701024\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −38.7846 −1.78900
\(471\) 3.26795 0.150579
\(472\) 12.5885 0.579431
\(473\) 0.248711 0.0114358
\(474\) 76.6410 3.52024
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −44.3205 −2.02930
\(478\) −27.3731 −1.25201
\(479\) 1.26795 0.0579341 0.0289670 0.999580i \(-0.490778\pi\)
0.0289670 + 0.999580i \(0.490778\pi\)
\(480\) −24.5885 −1.12230
\(481\) 0 0
\(482\) 36.7128 1.67222
\(483\) 0 0
\(484\) −9.39230 −0.426923
\(485\) 11.0718 0.502744
\(486\) −32.4449 −1.47173
\(487\) −40.7846 −1.84813 −0.924064 0.382239i \(-0.875153\pi\)
−0.924064 + 0.382239i \(0.875153\pi\)
\(488\) 8.32051 0.376652
\(489\) 44.2487 2.00100
\(490\) 0 0
\(491\) 7.60770 0.343330 0.171665 0.985155i \(-0.445085\pi\)
0.171665 + 0.985155i \(0.445085\pi\)
\(492\) 14.1962 0.640012
\(493\) −23.1962 −1.04470
\(494\) 0 0
\(495\) −9.80385 −0.440650
\(496\) −20.9808 −0.942064
\(497\) 0 0
\(498\) −10.3923 −0.465690
\(499\) 38.9808 1.74502 0.872509 0.488598i \(-0.162491\pi\)
0.872509 + 0.488598i \(0.162491\pi\)
\(500\) −12.1244 −0.542218
\(501\) 18.0000 0.804181
\(502\) −2.78461 −0.124283
\(503\) 18.5885 0.828818 0.414409 0.910091i \(-0.363988\pi\)
0.414409 + 0.910091i \(0.363988\pi\)
\(504\) 0 0
\(505\) 13.3923 0.595950
\(506\) 10.3923 0.461994
\(507\) 0 0
\(508\) 18.3923 0.816027
\(509\) 13.7321 0.608662 0.304331 0.952566i \(-0.401567\pi\)
0.304331 + 0.952566i \(0.401567\pi\)
\(510\) 63.3731 2.80621
\(511\) 0 0
\(512\) −8.66025 −0.382733
\(513\) −8.00000 −0.353209
\(514\) 10.6077 0.467885
\(515\) 24.9282 1.09847
\(516\) 0.535898 0.0235916
\(517\) −16.3923 −0.720933
\(518\) 0 0
\(519\) 23.3205 1.02366
\(520\) 0 0
\(521\) 24.1244 1.05691 0.528454 0.848962i \(-0.322772\pi\)
0.528454 + 0.848962i \(0.322772\pi\)
\(522\) 23.1962 1.01527
\(523\) 29.1769 1.27582 0.637909 0.770112i \(-0.279800\pi\)
0.637909 + 0.770112i \(0.279800\pi\)
\(524\) 3.46410 0.151330
\(525\) 0 0
\(526\) −2.19615 −0.0957568
\(527\) 32.4449 1.41332
\(528\) −17.3205 −0.753778
\(529\) −0.607695 −0.0264215
\(530\) 29.7846 1.29376
\(531\) 32.4449 1.40799
\(532\) 0 0
\(533\) 0 0
\(534\) 61.1769 2.64738
\(535\) −13.6077 −0.588312
\(536\) 10.7321 0.463554
\(537\) 18.9282 0.816812
\(538\) 8.78461 0.378731
\(539\) 0 0
\(540\) −6.92820 −0.298142
\(541\) 14.6077 0.628034 0.314017 0.949417i \(-0.398325\pi\)
0.314017 + 0.949417i \(0.398325\pi\)
\(542\) −10.0526 −0.431794
\(543\) 15.2679 0.655210
\(544\) 40.1769 1.72257
\(545\) 14.5359 0.622649
\(546\) 0 0
\(547\) 17.8038 0.761238 0.380619 0.924732i \(-0.375711\pi\)
0.380619 + 0.924732i \(0.375711\pi\)
\(548\) −8.07180 −0.344810
\(549\) 21.4449 0.915244
\(550\) −4.39230 −0.187289
\(551\) −6.00000 −0.255609
\(552\) −22.3923 −0.953080
\(553\) 0 0
\(554\) −29.4449 −1.25099
\(555\) −33.1244 −1.40605
\(556\) 10.5885 0.449051
\(557\) −43.6410 −1.84913 −0.924565 0.381025i \(-0.875571\pi\)
−0.924565 + 0.381025i \(0.875571\pi\)
\(558\) −32.4449 −1.37350
\(559\) 0 0
\(560\) 0 0
\(561\) 26.7846 1.13085
\(562\) 23.1962 0.978471
\(563\) −28.0526 −1.18227 −0.591137 0.806571i \(-0.701321\pi\)
−0.591137 + 0.806571i \(0.701321\pi\)
\(564\) −35.3205 −1.48726
\(565\) 23.1962 0.975869
\(566\) 17.6603 0.742316
\(567\) 0 0
\(568\) −10.3923 −0.436051
\(569\) 42.9282 1.79964 0.899822 0.436257i \(-0.143696\pi\)
0.899822 + 0.436257i \(0.143696\pi\)
\(570\) 16.3923 0.686598
\(571\) 16.7846 0.702414 0.351207 0.936298i \(-0.385771\pi\)
0.351207 + 0.936298i \(0.385771\pi\)
\(572\) 0 0
\(573\) −12.9282 −0.540083
\(574\) 0 0
\(575\) −9.46410 −0.394680
\(576\) 4.46410 0.186004
\(577\) 43.1962 1.79828 0.899140 0.437662i \(-0.144193\pi\)
0.899140 + 0.437662i \(0.144193\pi\)
\(578\) −74.1051 −3.08237
\(579\) 13.6603 0.567701
\(580\) −5.19615 −0.215758
\(581\) 0 0
\(582\) 30.2487 1.25385
\(583\) 12.5885 0.521361
\(584\) −5.53590 −0.229077
\(585\) 0 0
\(586\) 1.39230 0.0575156
\(587\) 16.3923 0.676583 0.338291 0.941041i \(-0.390151\pi\)
0.338291 + 0.941041i \(0.390151\pi\)
\(588\) 0 0
\(589\) 8.39230 0.345799
\(590\) −21.8038 −0.897650
\(591\) 32.7846 1.34858
\(592\) −35.0000 −1.43849
\(593\) −17.4449 −0.716375 −0.358187 0.933650i \(-0.616605\pi\)
−0.358187 + 0.933650i \(0.616605\pi\)
\(594\) −8.78461 −0.360437
\(595\) 0 0
\(596\) 6.46410 0.264780
\(597\) 5.46410 0.223631
\(598\) 0 0
\(599\) 43.8564 1.79192 0.895962 0.444131i \(-0.146487\pi\)
0.895962 + 0.444131i \(0.146487\pi\)
\(600\) 9.46410 0.386370
\(601\) 29.9808 1.22294 0.611470 0.791267i \(-0.290578\pi\)
0.611470 + 0.791267i \(0.290578\pi\)
\(602\) 0 0
\(603\) 27.6603 1.12641
\(604\) −2.00000 −0.0813788
\(605\) −16.2679 −0.661386
\(606\) 36.5885 1.48630
\(607\) 14.3923 0.584166 0.292083 0.956393i \(-0.405652\pi\)
0.292083 + 0.956393i \(0.405652\pi\)
\(608\) 10.3923 0.421464
\(609\) 0 0
\(610\) −14.4115 −0.583506
\(611\) 0 0
\(612\) 34.5167 1.39525
\(613\) −3.39230 −0.137014 −0.0685070 0.997651i \(-0.521824\pi\)
−0.0685070 + 0.997651i \(0.521824\pi\)
\(614\) 7.94744 0.320733
\(615\) 24.5885 0.991502
\(616\) 0 0
\(617\) −49.3923 −1.98846 −0.994230 0.107272i \(-0.965788\pi\)
−0.994230 + 0.107272i \(0.965788\pi\)
\(618\) 68.1051 2.73959
\(619\) 35.3731 1.42176 0.710882 0.703311i \(-0.248296\pi\)
0.710882 + 0.703311i \(0.248296\pi\)
\(620\) 7.26795 0.291888
\(621\) −18.9282 −0.759563
\(622\) 2.19615 0.0880577
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 49.8564 1.99266
\(627\) 6.92820 0.276686
\(628\) −1.19615 −0.0477317
\(629\) 54.1244 2.15808
\(630\) 0 0
\(631\) 12.7846 0.508947 0.254474 0.967080i \(-0.418098\pi\)
0.254474 + 0.967080i \(0.418098\pi\)
\(632\) 28.0526 1.11587
\(633\) 4.92820 0.195878
\(634\) −11.1962 −0.444656
\(635\) 31.8564 1.26418
\(636\) 27.1244 1.07555
\(637\) 0 0
\(638\) −6.58846 −0.260840
\(639\) −26.7846 −1.05958
\(640\) −21.0000 −0.830098
\(641\) 28.8564 1.13976 0.569880 0.821728i \(-0.306990\pi\)
0.569880 + 0.821728i \(0.306990\pi\)
\(642\) −37.1769 −1.46726
\(643\) −0.784610 −0.0309420 −0.0154710 0.999880i \(-0.504925\pi\)
−0.0154710 + 0.999880i \(0.504925\pi\)
\(644\) 0 0
\(645\) 0.928203 0.0365480
\(646\) −26.7846 −1.05383
\(647\) −45.0333 −1.77044 −0.885221 0.465170i \(-0.845993\pi\)
−0.885221 + 0.465170i \(0.845993\pi\)
\(648\) −4.26795 −0.167661
\(649\) −9.21539 −0.361736
\(650\) 0 0
\(651\) 0 0
\(652\) −16.1962 −0.634290
\(653\) −37.8564 −1.48144 −0.740718 0.671816i \(-0.765514\pi\)
−0.740718 + 0.671816i \(0.765514\pi\)
\(654\) 39.7128 1.55289
\(655\) 6.00000 0.234439
\(656\) 25.9808 1.01438
\(657\) −14.2679 −0.556646
\(658\) 0 0
\(659\) 28.3923 1.10601 0.553004 0.833179i \(-0.313482\pi\)
0.553004 + 0.833179i \(0.313482\pi\)
\(660\) 6.00000 0.233550
\(661\) −33.1962 −1.29118 −0.645590 0.763684i \(-0.723389\pi\)
−0.645590 + 0.763684i \(0.723389\pi\)
\(662\) 43.2679 1.68166
\(663\) 0 0
\(664\) −3.80385 −0.147618
\(665\) 0 0
\(666\) −54.1244 −2.09728
\(667\) −14.1962 −0.549677
\(668\) −6.58846 −0.254915
\(669\) 27.3205 1.05627
\(670\) −18.5885 −0.718135
\(671\) −6.09103 −0.235142
\(672\) 0 0
\(673\) −44.1769 −1.70289 −0.851447 0.524440i \(-0.824275\pi\)
−0.851447 + 0.524440i \(0.824275\pi\)
\(674\) −19.0526 −0.733877
\(675\) 8.00000 0.307920
\(676\) 0 0
\(677\) 23.0718 0.886721 0.443361 0.896343i \(-0.353786\pi\)
0.443361 + 0.896343i \(0.353786\pi\)
\(678\) 63.3731 2.43383
\(679\) 0 0
\(680\) 23.1962 0.889532
\(681\) −15.4641 −0.592586
\(682\) 9.21539 0.352876
\(683\) −15.4641 −0.591717 −0.295859 0.955232i \(-0.595606\pi\)
−0.295859 + 0.955232i \(0.595606\pi\)
\(684\) 8.92820 0.341378
\(685\) −13.9808 −0.534177
\(686\) 0 0
\(687\) 39.3205 1.50017
\(688\) 0.980762 0.0373912
\(689\) 0 0
\(690\) 38.7846 1.47650
\(691\) 0.392305 0.0149240 0.00746199 0.999972i \(-0.497625\pi\)
0.00746199 + 0.999972i \(0.497625\pi\)
\(692\) −8.53590 −0.324486
\(693\) 0 0
\(694\) −12.5885 −0.477851
\(695\) 18.3397 0.695666
\(696\) 14.1962 0.538104
\(697\) −40.1769 −1.52181
\(698\) 42.9282 1.62486
\(699\) 5.07180 0.191833
\(700\) 0 0
\(701\) 20.7846 0.785024 0.392512 0.919747i \(-0.371606\pi\)
0.392512 + 0.919747i \(0.371606\pi\)
\(702\) 0 0
\(703\) 14.0000 0.528020
\(704\) −1.26795 −0.0477876
\(705\) −61.1769 −2.30406
\(706\) −35.7846 −1.34677
\(707\) 0 0
\(708\) −19.8564 −0.746249
\(709\) −30.1769 −1.13332 −0.566659 0.823952i \(-0.691764\pi\)
−0.566659 + 0.823952i \(0.691764\pi\)
\(710\) 18.0000 0.675528
\(711\) 72.3013 2.71151
\(712\) 22.3923 0.839187
\(713\) 19.8564 0.743628
\(714\) 0 0
\(715\) 0 0
\(716\) −6.92820 −0.258919
\(717\) −43.1769 −1.61247
\(718\) 32.7846 1.22351
\(719\) 7.26795 0.271049 0.135524 0.990774i \(-0.456728\pi\)
0.135524 + 0.990774i \(0.456728\pi\)
\(720\) −38.6603 −1.44078
\(721\) 0 0
\(722\) 25.9808 0.966904
\(723\) 57.9090 2.15366
\(724\) −5.58846 −0.207693
\(725\) 6.00000 0.222834
\(726\) −44.4449 −1.64950
\(727\) 41.1769 1.52717 0.763584 0.645709i \(-0.223438\pi\)
0.763584 + 0.645709i \(0.223438\pi\)
\(728\) 0 0
\(729\) −43.7846 −1.62165
\(730\) 9.58846 0.354885
\(731\) −1.51666 −0.0560957
\(732\) −13.1244 −0.485090
\(733\) 23.5885 0.871260 0.435630 0.900126i \(-0.356526\pi\)
0.435630 + 0.900126i \(0.356526\pi\)
\(734\) 7.26795 0.268265
\(735\) 0 0
\(736\) 24.5885 0.906343
\(737\) −7.85641 −0.289394
\(738\) 40.1769 1.47893
\(739\) −40.7846 −1.50029 −0.750143 0.661276i \(-0.770015\pi\)
−0.750143 + 0.661276i \(0.770015\pi\)
\(740\) 12.1244 0.445700
\(741\) 0 0
\(742\) 0 0
\(743\) 7.60770 0.279099 0.139550 0.990215i \(-0.455435\pi\)
0.139550 + 0.990215i \(0.455435\pi\)
\(744\) −19.8564 −0.727971
\(745\) 11.1962 0.410195
\(746\) 19.7321 0.722442
\(747\) −9.80385 −0.358704
\(748\) −9.80385 −0.358464
\(749\) 0 0
\(750\) −57.3731 −2.09497
\(751\) 35.8038 1.30650 0.653250 0.757142i \(-0.273405\pi\)
0.653250 + 0.757142i \(0.273405\pi\)
\(752\) −64.6410 −2.35722
\(753\) −4.39230 −0.160064
\(754\) 0 0
\(755\) −3.46410 −0.126072
\(756\) 0 0
\(757\) −16.0000 −0.581530 −0.290765 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) 46.0526 1.67270
\(759\) 16.3923 0.595003
\(760\) 6.00000 0.217643
\(761\) 41.3205 1.49787 0.748934 0.662645i \(-0.230566\pi\)
0.748934 + 0.662645i \(0.230566\pi\)
\(762\) 87.0333 3.15288
\(763\) 0 0
\(764\) 4.73205 0.171200
\(765\) 59.7846 2.16152
\(766\) 20.1962 0.729717
\(767\) 0 0
\(768\) −51.9090 −1.87310
\(769\) 15.1769 0.547294 0.273647 0.961830i \(-0.411770\pi\)
0.273647 + 0.961830i \(0.411770\pi\)
\(770\) 0 0
\(771\) 16.7321 0.602590
\(772\) −5.00000 −0.179954
\(773\) −12.9282 −0.464995 −0.232498 0.972597i \(-0.574690\pi\)
−0.232498 + 0.972597i \(0.574690\pi\)
\(774\) 1.51666 0.0545152
\(775\) −8.39230 −0.301460
\(776\) 11.0718 0.397454
\(777\) 0 0
\(778\) 40.7654 1.46151
\(779\) −10.3923 −0.372343
\(780\) 0 0
\(781\) 7.60770 0.272225
\(782\) −63.3731 −2.26622
\(783\) 12.0000 0.428845
\(784\) 0 0
\(785\) −2.07180 −0.0739456
\(786\) 16.3923 0.584694
\(787\) 12.9808 0.462714 0.231357 0.972869i \(-0.425683\pi\)
0.231357 + 0.972869i \(0.425683\pi\)
\(788\) −12.0000 −0.427482
\(789\) −3.46410 −0.123325
\(790\) −48.5885 −1.72870
\(791\) 0 0
\(792\) −9.80385 −0.348365
\(793\) 0 0
\(794\) 32.5359 1.15466
\(795\) 46.9808 1.66624
\(796\) −2.00000 −0.0708881
\(797\) 34.3923 1.21824 0.609119 0.793079i \(-0.291523\pi\)
0.609119 + 0.793079i \(0.291523\pi\)
\(798\) 0 0
\(799\) 99.9615 3.53638
\(800\) −10.3923 −0.367423
\(801\) 57.7128 2.03918
\(802\) −18.8038 −0.663987
\(803\) 4.05256 0.143012
\(804\) −16.9282 −0.597012
\(805\) 0 0
\(806\) 0 0
\(807\) 13.8564 0.487769
\(808\) 13.3923 0.471140
\(809\) 2.07180 0.0728405 0.0364202 0.999337i \(-0.488405\pi\)
0.0364202 + 0.999337i \(0.488405\pi\)
\(810\) 7.39230 0.259739
\(811\) −16.5885 −0.582500 −0.291250 0.956647i \(-0.594071\pi\)
−0.291250 + 0.956647i \(0.594071\pi\)
\(812\) 0 0
\(813\) −15.8564 −0.556108
\(814\) 15.3731 0.538826
\(815\) −28.0526 −0.982638
\(816\) 105.622 3.69750
\(817\) −0.392305 −0.0137250
\(818\) 29.1051 1.01764
\(819\) 0 0
\(820\) −9.00000 −0.314294
\(821\) −4.14359 −0.144612 −0.0723062 0.997382i \(-0.523036\pi\)
−0.0723062 + 0.997382i \(0.523036\pi\)
\(822\) −38.1962 −1.33224
\(823\) −41.1769 −1.43534 −0.717669 0.696385i \(-0.754791\pi\)
−0.717669 + 0.696385i \(0.754791\pi\)
\(824\) 24.9282 0.868415
\(825\) −6.92820 −0.241209
\(826\) 0 0
\(827\) 16.9808 0.590479 0.295239 0.955423i \(-0.404601\pi\)
0.295239 + 0.955423i \(0.404601\pi\)
\(828\) 21.1244 0.734122
\(829\) 0.411543 0.0142935 0.00714673 0.999974i \(-0.497725\pi\)
0.00714673 + 0.999974i \(0.497725\pi\)
\(830\) 6.58846 0.228689
\(831\) −46.4449 −1.61115
\(832\) 0 0
\(833\) 0 0
\(834\) 50.1051 1.73500
\(835\) −11.4115 −0.394913
\(836\) −2.53590 −0.0877059
\(837\) −16.7846 −0.580161
\(838\) −55.7654 −1.92638
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −55.7321 −1.92065
\(843\) 36.5885 1.26017
\(844\) −1.80385 −0.0620910
\(845\) 0 0
\(846\) −99.9615 −3.43675
\(847\) 0 0
\(848\) 49.6410 1.70468
\(849\) 27.8564 0.956029
\(850\) 26.7846 0.918705
\(851\) 33.1244 1.13549
\(852\) 16.3923 0.561591
\(853\) 5.58846 0.191345 0.0956726 0.995413i \(-0.469500\pi\)
0.0956726 + 0.995413i \(0.469500\pi\)
\(854\) 0 0
\(855\) 15.4641 0.528861
\(856\) −13.6077 −0.465101
\(857\) 30.1244 1.02903 0.514514 0.857482i \(-0.327972\pi\)
0.514514 + 0.857482i \(0.327972\pi\)
\(858\) 0 0
\(859\) 7.80385 0.266264 0.133132 0.991098i \(-0.457497\pi\)
0.133132 + 0.991098i \(0.457497\pi\)
\(860\) −0.339746 −0.0115852
\(861\) 0 0
\(862\) 1.17691 0.0400859
\(863\) −7.51666 −0.255870 −0.127935 0.991783i \(-0.540835\pi\)
−0.127935 + 0.991783i \(0.540835\pi\)
\(864\) −20.7846 −0.707107
\(865\) −14.7846 −0.502692
\(866\) −23.5359 −0.799782
\(867\) −116.890 −3.96978
\(868\) 0 0
\(869\) −20.5359 −0.696633
\(870\) −24.5885 −0.833627
\(871\) 0 0
\(872\) 14.5359 0.492248
\(873\) 28.5359 0.965794
\(874\) −16.3923 −0.554478
\(875\) 0 0
\(876\) 8.73205 0.295029
\(877\) −19.7846 −0.668079 −0.334039 0.942559i \(-0.608412\pi\)
−0.334039 + 0.942559i \(0.608412\pi\)
\(878\) 25.2679 0.852752
\(879\) 2.19615 0.0740744
\(880\) 10.9808 0.370161
\(881\) 8.41154 0.283392 0.141696 0.989910i \(-0.454744\pi\)
0.141696 + 0.989910i \(0.454744\pi\)
\(882\) 0 0
\(883\) −47.7654 −1.60743 −0.803716 0.595013i \(-0.797147\pi\)
−0.803716 + 0.595013i \(0.797147\pi\)
\(884\) 0 0
\(885\) −34.3923 −1.15608
\(886\) −40.3923 −1.35701
\(887\) −11.3205 −0.380105 −0.190053 0.981774i \(-0.560866\pi\)
−0.190053 + 0.981774i \(0.560866\pi\)
\(888\) −33.1244 −1.11158
\(889\) 0 0
\(890\) −38.7846 −1.30006
\(891\) 3.12436 0.104670
\(892\) −10.0000 −0.334825
\(893\) 25.8564 0.865252
\(894\) 30.5885 1.02303
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) −20.7846 −0.693591
\(899\) −12.5885 −0.419849
\(900\) −8.92820 −0.297607
\(901\) −76.7654 −2.55743
\(902\) −11.4115 −0.379963
\(903\) 0 0
\(904\) 23.1962 0.771493
\(905\) −9.67949 −0.321757
\(906\) −9.46410 −0.314424
\(907\) −16.5885 −0.550811 −0.275405 0.961328i \(-0.588812\pi\)
−0.275405 + 0.961328i \(0.588812\pi\)
\(908\) 5.66025 0.187842
\(909\) 34.5167 1.14485
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 27.3205 0.904672
\(913\) 2.78461 0.0921571
\(914\) 19.0526 0.630203
\(915\) −22.7321 −0.751498
\(916\) −14.3923 −0.475535
\(917\) 0 0
\(918\) 53.5692 1.76805
\(919\) −39.5692 −1.30527 −0.652634 0.757673i \(-0.726336\pi\)
−0.652634 + 0.757673i \(0.726336\pi\)
\(920\) 14.1962 0.468033
\(921\) 12.5359 0.413072
\(922\) 27.0000 0.889198
\(923\) 0 0
\(924\) 0 0
\(925\) −14.0000 −0.460317
\(926\) −7.94744 −0.261169
\(927\) 64.2487 2.11020
\(928\) −15.5885 −0.511716
\(929\) 52.5167 1.72302 0.861508 0.507744i \(-0.169520\pi\)
0.861508 + 0.507744i \(0.169520\pi\)
\(930\) 34.3923 1.12777
\(931\) 0 0
\(932\) −1.85641 −0.0608086
\(933\) 3.46410 0.113410
\(934\) 44.1962 1.44614
\(935\) −16.9808 −0.555330
\(936\) 0 0
\(937\) 51.1962 1.67251 0.836253 0.548344i \(-0.184742\pi\)
0.836253 + 0.548344i \(0.184742\pi\)
\(938\) 0 0
\(939\) 78.6410 2.56635
\(940\) 22.3923 0.730356
\(941\) −28.1436 −0.917455 −0.458727 0.888577i \(-0.651695\pi\)
−0.458727 + 0.888577i \(0.651695\pi\)
\(942\) −5.66025 −0.184421
\(943\) −24.5885 −0.800710
\(944\) −36.3397 −1.18276
\(945\) 0 0
\(946\) −0.430781 −0.0140059
\(947\) 7.26795 0.236177 0.118088 0.993003i \(-0.462323\pi\)
0.118088 + 0.993003i \(0.462323\pi\)
\(948\) −44.2487 −1.43713
\(949\) 0 0
\(950\) 6.92820 0.224781
\(951\) −17.6603 −0.572673
\(952\) 0 0
\(953\) −25.1769 −0.815560 −0.407780 0.913080i \(-0.633697\pi\)
−0.407780 + 0.913080i \(0.633697\pi\)
\(954\) 76.7654 2.48537
\(955\) 8.19615 0.265221
\(956\) 15.8038 0.511133
\(957\) −10.3923 −0.335936
\(958\) −2.19615 −0.0709545
\(959\) 0 0
\(960\) −4.73205 −0.152726
\(961\) −13.3923 −0.432010
\(962\) 0 0
\(963\) −35.0718 −1.13017
\(964\) −21.1962 −0.682682
\(965\) −8.66025 −0.278783
\(966\) 0 0
\(967\) −54.9808 −1.76806 −0.884031 0.467428i \(-0.845181\pi\)
−0.884031 + 0.467428i \(0.845181\pi\)
\(968\) −16.2679 −0.522872
\(969\) −42.2487 −1.35722
\(970\) −19.1769 −0.615734
\(971\) 52.6410 1.68933 0.844665 0.535295i \(-0.179799\pi\)
0.844665 + 0.535295i \(0.179799\pi\)
\(972\) 18.7321 0.600831
\(973\) 0 0
\(974\) 70.6410 2.26348
\(975\) 0 0
\(976\) −24.0192 −0.768837
\(977\) −31.6410 −1.01229 −0.506143 0.862450i \(-0.668929\pi\)
−0.506143 + 0.862450i \(0.668929\pi\)
\(978\) −76.6410 −2.45071
\(979\) −16.3923 −0.523900
\(980\) 0 0
\(981\) 37.4641 1.19614
\(982\) −13.1769 −0.420492
\(983\) −17.3205 −0.552438 −0.276219 0.961095i \(-0.589082\pi\)
−0.276219 + 0.961095i \(0.589082\pi\)
\(984\) 24.5885 0.783851
\(985\) −20.7846 −0.662253
\(986\) 40.1769 1.27949
\(987\) 0 0
\(988\) 0 0
\(989\) −0.928203 −0.0295151
\(990\) 16.9808 0.539684
\(991\) 18.9808 0.602944 0.301472 0.953475i \(-0.402522\pi\)
0.301472 + 0.953475i \(0.402522\pi\)
\(992\) 21.8038 0.692273
\(993\) 68.2487 2.16581
\(994\) 0 0
\(995\) −3.46410 −0.109819
\(996\) 6.00000 0.190117
\(997\) 4.80385 0.152139 0.0760697 0.997103i \(-0.475763\pi\)
0.0760697 + 0.997103i \(0.475763\pi\)
\(998\) −67.5167 −2.13720
\(999\) −28.0000 −0.885881
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 8281.2.a.t.1.1 2
7.6 odd 2 1183.2.a.e.1.1 2
13.4 even 6 637.2.f.d.393.1 4
13.10 even 6 637.2.f.d.295.1 4
13.12 even 2 8281.2.a.r.1.2 2
91.4 even 6 637.2.h.e.471.2 4
91.10 odd 6 637.2.g.e.373.1 4
91.17 odd 6 637.2.h.d.471.2 4
91.23 even 6 637.2.h.e.165.2 4
91.30 even 6 637.2.g.d.263.1 4
91.34 even 4 1183.2.c.e.337.2 4
91.62 odd 6 91.2.f.b.22.1 4
91.69 odd 6 91.2.f.b.29.1 yes 4
91.75 odd 6 637.2.h.d.165.2 4
91.82 odd 6 637.2.g.e.263.1 4
91.83 even 4 1183.2.c.e.337.4 4
91.88 even 6 637.2.g.d.373.1 4
91.90 odd 2 1183.2.a.f.1.2 2
273.62 even 6 819.2.o.b.568.2 4
273.251 even 6 819.2.o.b.757.2 4
364.251 even 6 1456.2.s.o.1121.2 4
364.335 even 6 1456.2.s.o.113.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.b.22.1 4 91.62 odd 6
91.2.f.b.29.1 yes 4 91.69 odd 6
637.2.f.d.295.1 4 13.10 even 6
637.2.f.d.393.1 4 13.4 even 6
637.2.g.d.263.1 4 91.30 even 6
637.2.g.d.373.1 4 91.88 even 6
637.2.g.e.263.1 4 91.82 odd 6
637.2.g.e.373.1 4 91.10 odd 6
637.2.h.d.165.2 4 91.75 odd 6
637.2.h.d.471.2 4 91.17 odd 6
637.2.h.e.165.2 4 91.23 even 6
637.2.h.e.471.2 4 91.4 even 6
819.2.o.b.568.2 4 273.62 even 6
819.2.o.b.757.2 4 273.251 even 6
1183.2.a.e.1.1 2 7.6 odd 2
1183.2.a.f.1.2 2 91.90 odd 2
1183.2.c.e.337.2 4 91.34 even 4
1183.2.c.e.337.4 4 91.83 even 4
1456.2.s.o.113.2 4 364.335 even 6
1456.2.s.o.1121.2 4 364.251 even 6
8281.2.a.r.1.2 2 13.12 even 2
8281.2.a.t.1.1 2 1.1 even 1 trivial