Properties

Label 2842.2.a.u.1.1
Level $2842$
Weight $2$
Character 2842.1
Self dual yes
Analytic conductor $22.693$
Analytic rank $0$
Dimension $5$
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] = [2842,2,Mod(1,2842)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2842, 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("2842.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2842.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: \(22.6934842544\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.974241.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 9x^{3} - 2x^{2} + 11x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 406)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.76647\) of defining polynomial
Character \(\chi\) \(=\) 2842.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.42629 q^{3} +1.00000 q^{4} +0.260736 q^{5} +2.42629 q^{6} -1.00000 q^{8} +2.88687 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.42629 q^{3} +1.00000 q^{4} +0.260736 q^{5} +2.42629 q^{6} -1.00000 q^{8} +2.88687 q^{9} -0.260736 q^{10} +2.14760 q^{11} -2.42629 q^{12} +6.34036 q^{13} -0.632619 q^{15} +1.00000 q^{16} +2.22705 q^{17} -2.88687 q^{18} +8.43775 q^{19} +0.260736 q^{20} -2.14760 q^{22} +4.38533 q^{23} +2.42629 q^{24} -4.93202 q^{25} -6.34036 q^{26} +0.274492 q^{27} -1.00000 q^{29} +0.632619 q^{30} -3.04515 q^{31} -1.00000 q^{32} -5.21070 q^{33} -2.22705 q^{34} +2.88687 q^{36} -7.24517 q^{37} -8.43775 q^{38} -15.3835 q^{39} -0.260736 q^{40} +10.7302 q^{41} +0.195049 q^{43} +2.14760 q^{44} +0.752709 q^{45} -4.38533 q^{46} +12.2245 q^{47} -2.42629 q^{48} +4.93202 q^{50} -5.40345 q^{51} +6.34036 q^{52} -0.390391 q^{53} -0.274492 q^{54} +0.559957 q^{55} -20.4724 q^{57} +1.00000 q^{58} -14.7537 q^{59} -0.632619 q^{60} -9.83960 q^{61} +3.04515 q^{62} +1.00000 q^{64} +1.65316 q^{65} +5.21070 q^{66} +9.53790 q^{67} +2.22705 q^{68} -10.6401 q^{69} +2.18856 q^{71} -2.88687 q^{72} +4.18350 q^{73} +7.24517 q^{74} +11.9665 q^{75} +8.43775 q^{76} +15.3835 q^{78} -4.48597 q^{79} +0.260736 q^{80} -9.32660 q^{81} -10.7302 q^{82} +1.91389 q^{83} +0.580670 q^{85} -0.195049 q^{86} +2.42629 q^{87} -2.14760 q^{88} +11.8209 q^{89} -0.752709 q^{90} +4.38533 q^{92} +7.38841 q^{93} -12.2245 q^{94} +2.20002 q^{95} +2.42629 q^{96} -0.283480 q^{97} +6.19984 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{2} + 3 q^{3} + 5 q^{4} + q^{5} - 3 q^{6} - 5 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{2} + 3 q^{3} + 5 q^{4} + q^{5} - 3 q^{6} - 5 q^{8} + 8 q^{9} - q^{10} + 4 q^{11} + 3 q^{12} - 8 q^{15} + 5 q^{16} + 10 q^{17} - 8 q^{18} + 8 q^{19} + q^{20} - 4 q^{22} + 9 q^{23} - 3 q^{24} + 9 q^{27} - 5 q^{29} + 8 q^{30} + 3 q^{31} - 5 q^{32} + 7 q^{33} - 10 q^{34} + 8 q^{36} + 10 q^{37} - 8 q^{38} - 26 q^{39} - q^{40} + 19 q^{41} + 3 q^{43} + 4 q^{44} - 14 q^{45} - 9 q^{46} + 36 q^{47} + 3 q^{48} + 31 q^{51} - 3 q^{53} - 9 q^{54} + 10 q^{55} - 18 q^{57} + 5 q^{58} - 5 q^{59} - 8 q^{60} + 3 q^{61} - 3 q^{62} + 5 q^{64} + 29 q^{65} - 7 q^{66} - 2 q^{67} + 10 q^{68} - 18 q^{69} + 2 q^{71} - 8 q^{72} - 2 q^{73} - 10 q^{74} + 22 q^{75} + 8 q^{76} + 26 q^{78} + 17 q^{79} + q^{80} - 7 q^{81} - 19 q^{82} + 30 q^{83} - 11 q^{85} - 3 q^{86} - 3 q^{87} - 4 q^{88} + 29 q^{89} + 14 q^{90} + 9 q^{92} + 37 q^{93} - 36 q^{94} - 17 q^{95} - 3 q^{96} + 40 q^{99}+O(q^{100}) \) 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
\(3\) −2.42629 −1.40082 −0.700409 0.713742i \(-0.746999\pi\)
−0.700409 + 0.713742i \(0.746999\pi\)
\(4\) 1.00000 0.500000
\(5\) 0.260736 0.116605 0.0583023 0.998299i \(-0.481431\pi\)
0.0583023 + 0.998299i \(0.481431\pi\)
\(6\) 2.42629 0.990527
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 2.88687 0.962289
\(10\) −0.260736 −0.0824519
\(11\) 2.14760 0.647527 0.323763 0.946138i \(-0.395052\pi\)
0.323763 + 0.946138i \(0.395052\pi\)
\(12\) −2.42629 −0.700409
\(13\) 6.34036 1.75850 0.879249 0.476362i \(-0.158045\pi\)
0.879249 + 0.476362i \(0.158045\pi\)
\(14\) 0 0
\(15\) −0.632619 −0.163342
\(16\) 1.00000 0.250000
\(17\) 2.22705 0.540138 0.270069 0.962841i \(-0.412953\pi\)
0.270069 + 0.962841i \(0.412953\pi\)
\(18\) −2.88687 −0.680441
\(19\) 8.43775 1.93575 0.967876 0.251428i \(-0.0809002\pi\)
0.967876 + 0.251428i \(0.0809002\pi\)
\(20\) 0.260736 0.0583023
\(21\) 0 0
\(22\) −2.14760 −0.457870
\(23\) 4.38533 0.914404 0.457202 0.889363i \(-0.348852\pi\)
0.457202 + 0.889363i \(0.348852\pi\)
\(24\) 2.42629 0.495264
\(25\) −4.93202 −0.986403
\(26\) −6.34036 −1.24345
\(27\) 0.274492 0.0528261
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0.632619 0.115500
\(31\) −3.04515 −0.546925 −0.273463 0.961883i \(-0.588169\pi\)
−0.273463 + 0.961883i \(0.588169\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.21070 −0.907067
\(34\) −2.22705 −0.381935
\(35\) 0 0
\(36\) 2.88687 0.481145
\(37\) −7.24517 −1.19110 −0.595549 0.803319i \(-0.703066\pi\)
−0.595549 + 0.803319i \(0.703066\pi\)
\(38\) −8.43775 −1.36878
\(39\) −15.3835 −2.46333
\(40\) −0.260736 −0.0412259
\(41\) 10.7302 1.67577 0.837887 0.545844i \(-0.183791\pi\)
0.837887 + 0.545844i \(0.183791\pi\)
\(42\) 0 0
\(43\) 0.195049 0.0297447 0.0148723 0.999889i \(-0.495266\pi\)
0.0148723 + 0.999889i \(0.495266\pi\)
\(44\) 2.14760 0.323763
\(45\) 0.752709 0.112207
\(46\) −4.38533 −0.646581
\(47\) 12.2245 1.78312 0.891560 0.452902i \(-0.149611\pi\)
0.891560 + 0.452902i \(0.149611\pi\)
\(48\) −2.42629 −0.350204
\(49\) 0 0
\(50\) 4.93202 0.697493
\(51\) −5.40345 −0.756635
\(52\) 6.34036 0.879249
\(53\) −0.390391 −0.0536243 −0.0268121 0.999640i \(-0.508536\pi\)
−0.0268121 + 0.999640i \(0.508536\pi\)
\(54\) −0.274492 −0.0373537
\(55\) 0.559957 0.0755045
\(56\) 0 0
\(57\) −20.4724 −2.71163
\(58\) 1.00000 0.131306
\(59\) −14.7537 −1.92076 −0.960382 0.278687i \(-0.910101\pi\)
−0.960382 + 0.278687i \(0.910101\pi\)
\(60\) −0.632619 −0.0816708
\(61\) −9.83960 −1.25983 −0.629916 0.776664i \(-0.716911\pi\)
−0.629916 + 0.776664i \(0.716911\pi\)
\(62\) 3.04515 0.386734
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.65316 0.205049
\(66\) 5.21070 0.641393
\(67\) 9.53790 1.16524 0.582620 0.812745i \(-0.302028\pi\)
0.582620 + 0.812745i \(0.302028\pi\)
\(68\) 2.22705 0.270069
\(69\) −10.6401 −1.28091
\(70\) 0 0
\(71\) 2.18856 0.259734 0.129867 0.991531i \(-0.458545\pi\)
0.129867 + 0.991531i \(0.458545\pi\)
\(72\) −2.88687 −0.340221
\(73\) 4.18350 0.489642 0.244821 0.969568i \(-0.421271\pi\)
0.244821 + 0.969568i \(0.421271\pi\)
\(74\) 7.24517 0.842234
\(75\) 11.9665 1.38177
\(76\) 8.43775 0.967876
\(77\) 0 0
\(78\) 15.3835 1.74184
\(79\) −4.48597 −0.504712 −0.252356 0.967635i \(-0.581205\pi\)
−0.252356 + 0.967635i \(0.581205\pi\)
\(80\) 0.260736 0.0291511
\(81\) −9.32660 −1.03629
\(82\) −10.7302 −1.18495
\(83\) 1.91389 0.210077 0.105038 0.994468i \(-0.466503\pi\)
0.105038 + 0.994468i \(0.466503\pi\)
\(84\) 0 0
\(85\) 0.580670 0.0629825
\(86\) −0.195049 −0.0210327
\(87\) 2.42629 0.260125
\(88\) −2.14760 −0.228935
\(89\) 11.8209 1.25301 0.626505 0.779418i \(-0.284485\pi\)
0.626505 + 0.779418i \(0.284485\pi\)
\(90\) −0.752709 −0.0793425
\(91\) 0 0
\(92\) 4.38533 0.457202
\(93\) 7.38841 0.766142
\(94\) −12.2245 −1.26086
\(95\) 2.20002 0.225717
\(96\) 2.42629 0.247632
\(97\) −0.283480 −0.0287830 −0.0143915 0.999896i \(-0.504581\pi\)
−0.0143915 + 0.999896i \(0.504581\pi\)
\(98\) 0 0
\(99\) 6.19984 0.623108
\(100\) −4.93202 −0.493202
\(101\) −10.9123 −1.08581 −0.542905 0.839794i \(-0.682676\pi\)
−0.542905 + 0.839794i \(0.682676\pi\)
\(102\) 5.40345 0.535022
\(103\) −11.5444 −1.13750 −0.568751 0.822509i \(-0.692573\pi\)
−0.568751 + 0.822509i \(0.692573\pi\)
\(104\) −6.34036 −0.621723
\(105\) 0 0
\(106\) 0.390391 0.0379181
\(107\) 9.96649 0.963497 0.481748 0.876310i \(-0.340002\pi\)
0.481748 + 0.876310i \(0.340002\pi\)
\(108\) 0.274492 0.0264130
\(109\) −9.82477 −0.941042 −0.470521 0.882389i \(-0.655934\pi\)
−0.470521 + 0.882389i \(0.655934\pi\)
\(110\) −0.559957 −0.0533898
\(111\) 17.5789 1.66851
\(112\) 0 0
\(113\) −1.20430 −0.113291 −0.0566456 0.998394i \(-0.518041\pi\)
−0.0566456 + 0.998394i \(0.518041\pi\)
\(114\) 20.4724 1.91742
\(115\) 1.14341 0.106624
\(116\) −1.00000 −0.0928477
\(117\) 18.3038 1.69218
\(118\) 14.7537 1.35819
\(119\) 0 0
\(120\) 0.632619 0.0577500
\(121\) −6.38780 −0.580709
\(122\) 9.83960 0.890835
\(123\) −26.0345 −2.34745
\(124\) −3.04515 −0.273463
\(125\) −2.58963 −0.231624
\(126\) 0 0
\(127\) 4.82334 0.428002 0.214001 0.976833i \(-0.431350\pi\)
0.214001 + 0.976833i \(0.431350\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.473244 −0.0416668
\(130\) −1.65316 −0.144991
\(131\) 14.0889 1.23095 0.615475 0.788156i \(-0.288964\pi\)
0.615475 + 0.788156i \(0.288964\pi\)
\(132\) −5.21070 −0.453533
\(133\) 0 0
\(134\) −9.53790 −0.823949
\(135\) 0.0715699 0.00615976
\(136\) −2.22705 −0.190968
\(137\) −13.0249 −1.11279 −0.556396 0.830917i \(-0.687816\pi\)
−0.556396 + 0.830917i \(0.687816\pi\)
\(138\) 10.6401 0.905742
\(139\) −16.8987 −1.43333 −0.716664 0.697419i \(-0.754332\pi\)
−0.716664 + 0.697419i \(0.754332\pi\)
\(140\) 0 0
\(141\) −29.6600 −2.49783
\(142\) −2.18856 −0.183660
\(143\) 13.6166 1.13867
\(144\) 2.88687 0.240572
\(145\) −0.260736 −0.0216529
\(146\) −4.18350 −0.346229
\(147\) 0 0
\(148\) −7.24517 −0.595549
\(149\) −0.491974 −0.0403040 −0.0201520 0.999797i \(-0.506415\pi\)
−0.0201520 + 0.999797i \(0.506415\pi\)
\(150\) −11.9665 −0.977060
\(151\) 2.10185 0.171046 0.0855230 0.996336i \(-0.472744\pi\)
0.0855230 + 0.996336i \(0.472744\pi\)
\(152\) −8.43775 −0.684392
\(153\) 6.42919 0.519769
\(154\) 0 0
\(155\) −0.793979 −0.0637739
\(156\) −15.3835 −1.23167
\(157\) 9.92354 0.791985 0.395993 0.918254i \(-0.370401\pi\)
0.395993 + 0.918254i \(0.370401\pi\)
\(158\) 4.48597 0.356885
\(159\) 0.947199 0.0751178
\(160\) −0.260736 −0.0206130
\(161\) 0 0
\(162\) 9.32660 0.732767
\(163\) 6.75259 0.528904 0.264452 0.964399i \(-0.414809\pi\)
0.264452 + 0.964399i \(0.414809\pi\)
\(164\) 10.7302 0.837887
\(165\) −1.35862 −0.105768
\(166\) −1.91389 −0.148547
\(167\) 7.22428 0.559032 0.279516 0.960141i \(-0.409826\pi\)
0.279516 + 0.960141i \(0.409826\pi\)
\(168\) 0 0
\(169\) 27.2001 2.09232
\(170\) −0.580670 −0.0445354
\(171\) 24.3587 1.86275
\(172\) 0.195049 0.0148723
\(173\) 8.60818 0.654468 0.327234 0.944943i \(-0.393883\pi\)
0.327234 + 0.944943i \(0.393883\pi\)
\(174\) −2.42629 −0.183936
\(175\) 0 0
\(176\) 2.14760 0.161882
\(177\) 35.7966 2.69064
\(178\) −11.8209 −0.886012
\(179\) 21.0797 1.57557 0.787786 0.615949i \(-0.211227\pi\)
0.787786 + 0.615949i \(0.211227\pi\)
\(180\) 0.752709 0.0561036
\(181\) −8.59244 −0.638671 −0.319336 0.947642i \(-0.603460\pi\)
−0.319336 + 0.947642i \(0.603460\pi\)
\(182\) 0 0
\(183\) 23.8737 1.76479
\(184\) −4.38533 −0.323291
\(185\) −1.88907 −0.138888
\(186\) −7.38841 −0.541744
\(187\) 4.78281 0.349754
\(188\) 12.2245 0.891560
\(189\) 0 0
\(190\) −2.20002 −0.159606
\(191\) 8.52165 0.616605 0.308302 0.951288i \(-0.400239\pi\)
0.308302 + 0.951288i \(0.400239\pi\)
\(192\) −2.42629 −0.175102
\(193\) 13.1790 0.948645 0.474322 0.880351i \(-0.342693\pi\)
0.474322 + 0.880351i \(0.342693\pi\)
\(194\) 0.283480 0.0203527
\(195\) −4.01103 −0.287236
\(196\) 0 0
\(197\) −5.96908 −0.425279 −0.212640 0.977131i \(-0.568206\pi\)
−0.212640 + 0.977131i \(0.568206\pi\)
\(198\) −6.19984 −0.440604
\(199\) 13.3315 0.945042 0.472521 0.881319i \(-0.343344\pi\)
0.472521 + 0.881319i \(0.343344\pi\)
\(200\) 4.93202 0.348746
\(201\) −23.1417 −1.63229
\(202\) 10.9123 0.767784
\(203\) 0 0
\(204\) −5.40345 −0.378317
\(205\) 2.79774 0.195403
\(206\) 11.5444 0.804336
\(207\) 12.6599 0.879921
\(208\) 6.34036 0.439625
\(209\) 18.1209 1.25345
\(210\) 0 0
\(211\) 9.91857 0.682823 0.341411 0.939914i \(-0.389095\pi\)
0.341411 + 0.939914i \(0.389095\pi\)
\(212\) −0.390391 −0.0268121
\(213\) −5.31008 −0.363840
\(214\) −9.96649 −0.681295
\(215\) 0.0508562 0.00346836
\(216\) −0.274492 −0.0186768
\(217\) 0 0
\(218\) 9.82477 0.665417
\(219\) −10.1504 −0.685898
\(220\) 0.559957 0.0377523
\(221\) 14.1203 0.949832
\(222\) −17.5789 −1.17982
\(223\) −19.5885 −1.31174 −0.655872 0.754872i \(-0.727699\pi\)
−0.655872 + 0.754872i \(0.727699\pi\)
\(224\) 0 0
\(225\) −14.2381 −0.949205
\(226\) 1.20430 0.0801090
\(227\) 15.6918 1.04150 0.520751 0.853708i \(-0.325652\pi\)
0.520751 + 0.853708i \(0.325652\pi\)
\(228\) −20.4724 −1.35582
\(229\) −14.4197 −0.952879 −0.476440 0.879207i \(-0.658073\pi\)
−0.476440 + 0.879207i \(0.658073\pi\)
\(230\) −1.14341 −0.0753943
\(231\) 0 0
\(232\) 1.00000 0.0656532
\(233\) −10.0291 −0.657029 −0.328515 0.944499i \(-0.606548\pi\)
−0.328515 + 0.944499i \(0.606548\pi\)
\(234\) −18.3038 −1.19655
\(235\) 3.18735 0.207920
\(236\) −14.7537 −0.960382
\(237\) 10.8843 0.707009
\(238\) 0 0
\(239\) −0.920380 −0.0595344 −0.0297672 0.999557i \(-0.509477\pi\)
−0.0297672 + 0.999557i \(0.509477\pi\)
\(240\) −0.632619 −0.0408354
\(241\) −7.64883 −0.492704 −0.246352 0.969180i \(-0.579232\pi\)
−0.246352 + 0.969180i \(0.579232\pi\)
\(242\) 6.38780 0.410623
\(243\) 21.8055 1.39883
\(244\) −9.83960 −0.629916
\(245\) 0 0
\(246\) 26.0345 1.65990
\(247\) 53.4983 3.40402
\(248\) 3.04515 0.193367
\(249\) −4.64365 −0.294279
\(250\) 2.58963 0.163783
\(251\) 10.3042 0.650395 0.325197 0.945646i \(-0.394569\pi\)
0.325197 + 0.945646i \(0.394569\pi\)
\(252\) 0 0
\(253\) 9.41794 0.592101
\(254\) −4.82334 −0.302643
\(255\) −1.40887 −0.0882270
\(256\) 1.00000 0.0625000
\(257\) −4.43144 −0.276425 −0.138213 0.990403i \(-0.544136\pi\)
−0.138213 + 0.990403i \(0.544136\pi\)
\(258\) 0.473244 0.0294629
\(259\) 0 0
\(260\) 1.65316 0.102524
\(261\) −2.88687 −0.178693
\(262\) −14.0889 −0.870413
\(263\) 8.75489 0.539850 0.269925 0.962881i \(-0.413001\pi\)
0.269925 + 0.962881i \(0.413001\pi\)
\(264\) 5.21070 0.320696
\(265\) −0.101789 −0.00625283
\(266\) 0 0
\(267\) −28.6808 −1.75524
\(268\) 9.53790 0.582620
\(269\) 21.7855 1.32828 0.664141 0.747607i \(-0.268797\pi\)
0.664141 + 0.747607i \(0.268797\pi\)
\(270\) −0.0715699 −0.00435561
\(271\) −16.9776 −1.03132 −0.515658 0.856794i \(-0.672453\pi\)
−0.515658 + 0.856794i \(0.672453\pi\)
\(272\) 2.22705 0.135035
\(273\) 0 0
\(274\) 13.0249 0.786863
\(275\) −10.5920 −0.638722
\(276\) −10.6401 −0.640457
\(277\) −0.608653 −0.0365704 −0.0182852 0.999833i \(-0.505821\pi\)
−0.0182852 + 0.999833i \(0.505821\pi\)
\(278\) 16.8987 1.01352
\(279\) −8.79094 −0.526300
\(280\) 0 0
\(281\) −25.5555 −1.52451 −0.762256 0.647276i \(-0.775908\pi\)
−0.762256 + 0.647276i \(0.775908\pi\)
\(282\) 29.6600 1.76623
\(283\) 24.7141 1.46910 0.734551 0.678553i \(-0.237393\pi\)
0.734551 + 0.678553i \(0.237393\pi\)
\(284\) 2.18856 0.129867
\(285\) −5.33788 −0.316189
\(286\) −13.6166 −0.805164
\(287\) 0 0
\(288\) −2.88687 −0.170110
\(289\) −12.0403 −0.708251
\(290\) 0.260736 0.0153109
\(291\) 0.687804 0.0403198
\(292\) 4.18350 0.244821
\(293\) 20.9908 1.22630 0.613148 0.789968i \(-0.289903\pi\)
0.613148 + 0.789968i \(0.289903\pi\)
\(294\) 0 0
\(295\) −3.84681 −0.223970
\(296\) 7.24517 0.421117
\(297\) 0.589500 0.0342063
\(298\) 0.491974 0.0284993
\(299\) 27.8045 1.60798
\(300\) 11.9665 0.690885
\(301\) 0 0
\(302\) −2.10185 −0.120948
\(303\) 26.4763 1.52102
\(304\) 8.43775 0.483938
\(305\) −2.56553 −0.146902
\(306\) −6.42919 −0.367532
\(307\) 24.9164 1.42205 0.711027 0.703164i \(-0.248230\pi\)
0.711027 + 0.703164i \(0.248230\pi\)
\(308\) 0 0
\(309\) 28.0100 1.59343
\(310\) 0.793979 0.0450950
\(311\) 3.38252 0.191805 0.0959025 0.995391i \(-0.469426\pi\)
0.0959025 + 0.995391i \(0.469426\pi\)
\(312\) 15.3835 0.870920
\(313\) 27.7834 1.57041 0.785206 0.619235i \(-0.212557\pi\)
0.785206 + 0.619235i \(0.212557\pi\)
\(314\) −9.92354 −0.560018
\(315\) 0 0
\(316\) −4.48597 −0.252356
\(317\) −13.6437 −0.766304 −0.383152 0.923685i \(-0.625161\pi\)
−0.383152 + 0.923685i \(0.625161\pi\)
\(318\) −0.947199 −0.0531163
\(319\) −2.14760 −0.120243
\(320\) 0.260736 0.0145756
\(321\) −24.1816 −1.34968
\(322\) 0 0
\(323\) 18.7913 1.04557
\(324\) −9.32660 −0.518144
\(325\) −31.2707 −1.73459
\(326\) −6.75259 −0.373992
\(327\) 23.8377 1.31823
\(328\) −10.7302 −0.592475
\(329\) 0 0
\(330\) 1.35862 0.0747893
\(331\) −12.1576 −0.668244 −0.334122 0.942530i \(-0.608440\pi\)
−0.334122 + 0.942530i \(0.608440\pi\)
\(332\) 1.91389 0.105038
\(333\) −20.9158 −1.14618
\(334\) −7.22428 −0.395295
\(335\) 2.48687 0.135872
\(336\) 0 0
\(337\) 12.7616 0.695168 0.347584 0.937649i \(-0.387002\pi\)
0.347584 + 0.937649i \(0.387002\pi\)
\(338\) −27.2001 −1.47949
\(339\) 2.92198 0.158700
\(340\) 0.580670 0.0314913
\(341\) −6.53977 −0.354149
\(342\) −24.3587 −1.31717
\(343\) 0 0
\(344\) −0.195049 −0.0105163
\(345\) −2.77424 −0.149360
\(346\) −8.60818 −0.462779
\(347\) −3.20642 −0.172130 −0.0860648 0.996290i \(-0.527429\pi\)
−0.0860648 + 0.996290i \(0.527429\pi\)
\(348\) 2.42629 0.130063
\(349\) 3.95168 0.211529 0.105764 0.994391i \(-0.466271\pi\)
0.105764 + 0.994391i \(0.466271\pi\)
\(350\) 0 0
\(351\) 1.74038 0.0928945
\(352\) −2.14760 −0.114468
\(353\) −15.0305 −0.799995 −0.399998 0.916516i \(-0.630989\pi\)
−0.399998 + 0.916516i \(0.630989\pi\)
\(354\) −35.7966 −1.90257
\(355\) 0.570636 0.0302862
\(356\) 11.8209 0.626505
\(357\) 0 0
\(358\) −21.0797 −1.11410
\(359\) −12.5777 −0.663823 −0.331912 0.943310i \(-0.607694\pi\)
−0.331912 + 0.943310i \(0.607694\pi\)
\(360\) −0.752709 −0.0396713
\(361\) 52.1956 2.74714
\(362\) 8.59244 0.451609
\(363\) 15.4986 0.813468
\(364\) 0 0
\(365\) 1.09079 0.0570944
\(366\) −23.8737 −1.24790
\(367\) −15.2524 −0.796166 −0.398083 0.917349i \(-0.630324\pi\)
−0.398083 + 0.917349i \(0.630324\pi\)
\(368\) 4.38533 0.228601
\(369\) 30.9766 1.61258
\(370\) 1.88907 0.0982083
\(371\) 0 0
\(372\) 7.38841 0.383071
\(373\) −14.9777 −0.775517 −0.387759 0.921761i \(-0.626751\pi\)
−0.387759 + 0.921761i \(0.626751\pi\)
\(374\) −4.78281 −0.247313
\(375\) 6.28319 0.324462
\(376\) −12.2245 −0.630428
\(377\) −6.34036 −0.326545
\(378\) 0 0
\(379\) 0.578081 0.0296940 0.0148470 0.999890i \(-0.495274\pi\)
0.0148470 + 0.999890i \(0.495274\pi\)
\(380\) 2.20002 0.112859
\(381\) −11.7028 −0.599553
\(382\) −8.52165 −0.436005
\(383\) −28.9218 −1.47783 −0.738917 0.673797i \(-0.764662\pi\)
−0.738917 + 0.673797i \(0.764662\pi\)
\(384\) 2.42629 0.123816
\(385\) 0 0
\(386\) −13.1790 −0.670793
\(387\) 0.563080 0.0286230
\(388\) −0.283480 −0.0143915
\(389\) −9.91719 −0.502821 −0.251411 0.967881i \(-0.580894\pi\)
−0.251411 + 0.967881i \(0.580894\pi\)
\(390\) 4.01103 0.203107
\(391\) 9.76633 0.493905
\(392\) 0 0
\(393\) −34.1836 −1.72434
\(394\) 5.96908 0.300718
\(395\) −1.16965 −0.0588516
\(396\) 6.19984 0.311554
\(397\) 10.3356 0.518729 0.259365 0.965779i \(-0.416487\pi\)
0.259365 + 0.965779i \(0.416487\pi\)
\(398\) −13.3315 −0.668245
\(399\) 0 0
\(400\) −4.93202 −0.246601
\(401\) −20.8127 −1.03933 −0.519667 0.854369i \(-0.673944\pi\)
−0.519667 + 0.854369i \(0.673944\pi\)
\(402\) 23.1417 1.15420
\(403\) −19.3073 −0.961767
\(404\) −10.9123 −0.542905
\(405\) −2.43178 −0.120836
\(406\) 0 0
\(407\) −15.5598 −0.771268
\(408\) 5.40345 0.267511
\(409\) 9.80067 0.484612 0.242306 0.970200i \(-0.422096\pi\)
0.242306 + 0.970200i \(0.422096\pi\)
\(410\) −2.79774 −0.138171
\(411\) 31.6022 1.55882
\(412\) −11.5444 −0.568751
\(413\) 0 0
\(414\) −12.6599 −0.622198
\(415\) 0.499020 0.0244959
\(416\) −6.34036 −0.310862
\(417\) 41.0010 2.00783
\(418\) −18.1209 −0.886324
\(419\) 0.564842 0.0275943 0.0137972 0.999905i \(-0.495608\pi\)
0.0137972 + 0.999905i \(0.495608\pi\)
\(420\) 0 0
\(421\) 28.1593 1.37240 0.686201 0.727412i \(-0.259277\pi\)
0.686201 + 0.727412i \(0.259277\pi\)
\(422\) −9.91857 −0.482829
\(423\) 35.2904 1.71588
\(424\) 0.390391 0.0189590
\(425\) −10.9838 −0.532794
\(426\) 5.31008 0.257274
\(427\) 0 0
\(428\) 9.96649 0.481748
\(429\) −33.0377 −1.59507
\(430\) −0.0508562 −0.00245250
\(431\) 28.1658 1.35670 0.678350 0.734739i \(-0.262695\pi\)
0.678350 + 0.734739i \(0.262695\pi\)
\(432\) 0.274492 0.0132065
\(433\) 17.6927 0.850259 0.425130 0.905132i \(-0.360229\pi\)
0.425130 + 0.905132i \(0.360229\pi\)
\(434\) 0 0
\(435\) 0.632619 0.0303318
\(436\) −9.82477 −0.470521
\(437\) 37.0023 1.77006
\(438\) 10.1504 0.485003
\(439\) −28.1245 −1.34231 −0.671154 0.741318i \(-0.734201\pi\)
−0.671154 + 0.741318i \(0.734201\pi\)
\(440\) −0.559957 −0.0266949
\(441\) 0 0
\(442\) −14.1203 −0.671633
\(443\) −6.21048 −0.295069 −0.147534 0.989057i \(-0.547134\pi\)
−0.147534 + 0.989057i \(0.547134\pi\)
\(444\) 17.5789 0.834256
\(445\) 3.08212 0.146107
\(446\) 19.5885 0.927543
\(447\) 1.19367 0.0564586
\(448\) 0 0
\(449\) −5.87960 −0.277475 −0.138738 0.990329i \(-0.544304\pi\)
−0.138738 + 0.990329i \(0.544304\pi\)
\(450\) 14.2381 0.671189
\(451\) 23.0442 1.08511
\(452\) −1.20430 −0.0566456
\(453\) −5.09969 −0.239604
\(454\) −15.6918 −0.736454
\(455\) 0 0
\(456\) 20.4724 0.958708
\(457\) −34.2340 −1.60140 −0.800700 0.599065i \(-0.795539\pi\)
−0.800700 + 0.599065i \(0.795539\pi\)
\(458\) 14.4197 0.673788
\(459\) 0.611307 0.0285334
\(460\) 1.14341 0.0533118
\(461\) −7.77623 −0.362175 −0.181088 0.983467i \(-0.557962\pi\)
−0.181088 + 0.983467i \(0.557962\pi\)
\(462\) 0 0
\(463\) −6.50388 −0.302261 −0.151130 0.988514i \(-0.548291\pi\)
−0.151130 + 0.988514i \(0.548291\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 1.92642 0.0893356
\(466\) 10.0291 0.464590
\(467\) −5.00733 −0.231712 −0.115856 0.993266i \(-0.536961\pi\)
−0.115856 + 0.993266i \(0.536961\pi\)
\(468\) 18.3038 0.846092
\(469\) 0 0
\(470\) −3.18735 −0.147022
\(471\) −24.0774 −1.10943
\(472\) 14.7537 0.679093
\(473\) 0.418887 0.0192605
\(474\) −10.8843 −0.499931
\(475\) −41.6151 −1.90943
\(476\) 0 0
\(477\) −1.12701 −0.0516020
\(478\) 0.920380 0.0420972
\(479\) −35.2838 −1.61216 −0.806080 0.591807i \(-0.798415\pi\)
−0.806080 + 0.591807i \(0.798415\pi\)
\(480\) 0.632619 0.0288750
\(481\) −45.9370 −2.09455
\(482\) 7.64883 0.348395
\(483\) 0 0
\(484\) −6.38780 −0.290355
\(485\) −0.0739133 −0.00335623
\(486\) −21.8055 −0.989119
\(487\) 4.73153 0.214406 0.107203 0.994237i \(-0.465811\pi\)
0.107203 + 0.994237i \(0.465811\pi\)
\(488\) 9.83960 0.445418
\(489\) −16.3837 −0.740898
\(490\) 0 0
\(491\) −18.5337 −0.836413 −0.418206 0.908352i \(-0.637341\pi\)
−0.418206 + 0.908352i \(0.637341\pi\)
\(492\) −26.0345 −1.17373
\(493\) −2.22705 −0.100301
\(494\) −53.4983 −2.40700
\(495\) 1.61652 0.0726572
\(496\) −3.04515 −0.136731
\(497\) 0 0
\(498\) 4.64365 0.208087
\(499\) −8.49139 −0.380127 −0.190063 0.981772i \(-0.560869\pi\)
−0.190063 + 0.981772i \(0.560869\pi\)
\(500\) −2.58963 −0.115812
\(501\) −17.5282 −0.783101
\(502\) −10.3042 −0.459899
\(503\) 25.6049 1.14167 0.570833 0.821066i \(-0.306620\pi\)
0.570833 + 0.821066i \(0.306620\pi\)
\(504\) 0 0
\(505\) −2.84522 −0.126610
\(506\) −9.41794 −0.418679
\(507\) −65.9953 −2.93095
\(508\) 4.82334 0.214001
\(509\) −2.16096 −0.0957829 −0.0478914 0.998853i \(-0.515250\pi\)
−0.0478914 + 0.998853i \(0.515250\pi\)
\(510\) 1.40887 0.0623859
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 2.31610 0.102258
\(514\) 4.43144 0.195462
\(515\) −3.01003 −0.132638
\(516\) −0.473244 −0.0208334
\(517\) 26.2533 1.15462
\(518\) 0 0
\(519\) −20.8859 −0.916790
\(520\) −1.65316 −0.0724957
\(521\) 38.3906 1.68192 0.840961 0.541096i \(-0.181991\pi\)
0.840961 + 0.541096i \(0.181991\pi\)
\(522\) 2.88687 0.126355
\(523\) −36.1101 −1.57898 −0.789492 0.613761i \(-0.789656\pi\)
−0.789492 + 0.613761i \(0.789656\pi\)
\(524\) 14.0889 0.615475
\(525\) 0 0
\(526\) −8.75489 −0.381731
\(527\) −6.78169 −0.295415
\(528\) −5.21070 −0.226767
\(529\) −3.76889 −0.163865
\(530\) 0.101789 0.00442142
\(531\) −42.5919 −1.84833
\(532\) 0 0
\(533\) 68.0332 2.94685
\(534\) 28.6808 1.24114
\(535\) 2.59862 0.112348
\(536\) −9.53790 −0.411975
\(537\) −51.1454 −2.20709
\(538\) −21.7855 −0.939238
\(539\) 0 0
\(540\) 0.0715699 0.00307988
\(541\) −6.89449 −0.296417 −0.148209 0.988956i \(-0.547351\pi\)
−0.148209 + 0.988956i \(0.547351\pi\)
\(542\) 16.9776 0.729251
\(543\) 20.8477 0.894662
\(544\) −2.22705 −0.0954838
\(545\) −2.56167 −0.109730
\(546\) 0 0
\(547\) −39.4686 −1.68756 −0.843779 0.536691i \(-0.819674\pi\)
−0.843779 + 0.536691i \(0.819674\pi\)
\(548\) −13.0249 −0.556396
\(549\) −28.4056 −1.21232
\(550\) 10.5920 0.451645
\(551\) −8.43775 −0.359460
\(552\) 10.6401 0.452871
\(553\) 0 0
\(554\) 0.608653 0.0258592
\(555\) 4.58344 0.194556
\(556\) −16.8987 −0.716664
\(557\) 21.9613 0.930530 0.465265 0.885171i \(-0.345959\pi\)
0.465265 + 0.885171i \(0.345959\pi\)
\(558\) 8.79094 0.372150
\(559\) 1.23668 0.0523059
\(560\) 0 0
\(561\) −11.6045 −0.489941
\(562\) 25.5555 1.07799
\(563\) −17.4707 −0.736303 −0.368151 0.929766i \(-0.620009\pi\)
−0.368151 + 0.929766i \(0.620009\pi\)
\(564\) −29.6600 −1.24891
\(565\) −0.314005 −0.0132103
\(566\) −24.7141 −1.03881
\(567\) 0 0
\(568\) −2.18856 −0.0918300
\(569\) 11.0179 0.461896 0.230948 0.972966i \(-0.425817\pi\)
0.230948 + 0.972966i \(0.425817\pi\)
\(570\) 5.33788 0.223579
\(571\) 4.82287 0.201831 0.100915 0.994895i \(-0.467823\pi\)
0.100915 + 0.994895i \(0.467823\pi\)
\(572\) 13.6166 0.569337
\(573\) −20.6760 −0.863751
\(574\) 0 0
\(575\) −21.6285 −0.901971
\(576\) 2.88687 0.120286
\(577\) −11.9775 −0.498629 −0.249314 0.968423i \(-0.580205\pi\)
−0.249314 + 0.968423i \(0.580205\pi\)
\(578\) 12.0403 0.500809
\(579\) −31.9760 −1.32888
\(580\) −0.260736 −0.0108265
\(581\) 0 0
\(582\) −0.687804 −0.0285104
\(583\) −0.838404 −0.0347231
\(584\) −4.18350 −0.173114
\(585\) 4.77244 0.197316
\(586\) −20.9908 −0.867122
\(587\) 28.0949 1.15960 0.579800 0.814759i \(-0.303131\pi\)
0.579800 + 0.814759i \(0.303131\pi\)
\(588\) 0 0
\(589\) −25.6942 −1.05871
\(590\) 3.84681 0.158371
\(591\) 14.4827 0.595738
\(592\) −7.24517 −0.297775
\(593\) 25.6900 1.05496 0.527481 0.849567i \(-0.323137\pi\)
0.527481 + 0.849567i \(0.323137\pi\)
\(594\) −0.589500 −0.0241875
\(595\) 0 0
\(596\) −0.491974 −0.0201520
\(597\) −32.3459 −1.32383
\(598\) −27.8045 −1.13701
\(599\) −8.23688 −0.336550 −0.168275 0.985740i \(-0.553820\pi\)
−0.168275 + 0.985740i \(0.553820\pi\)
\(600\) −11.9665 −0.488530
\(601\) 12.0901 0.493166 0.246583 0.969122i \(-0.420692\pi\)
0.246583 + 0.969122i \(0.420692\pi\)
\(602\) 0 0
\(603\) 27.5347 1.12130
\(604\) 2.10185 0.0855230
\(605\) −1.66553 −0.0677133
\(606\) −26.4763 −1.07553
\(607\) 25.0685 1.01750 0.508749 0.860915i \(-0.330108\pi\)
0.508749 + 0.860915i \(0.330108\pi\)
\(608\) −8.43775 −0.342196
\(609\) 0 0
\(610\) 2.56553 0.103875
\(611\) 77.5074 3.13561
\(612\) 6.42919 0.259885
\(613\) −33.6592 −1.35948 −0.679741 0.733453i \(-0.737908\pi\)
−0.679741 + 0.733453i \(0.737908\pi\)
\(614\) −24.9164 −1.00554
\(615\) −6.78813 −0.273724
\(616\) 0 0
\(617\) −29.3861 −1.18304 −0.591519 0.806291i \(-0.701472\pi\)
−0.591519 + 0.806291i \(0.701472\pi\)
\(618\) −28.0100 −1.12673
\(619\) 41.3752 1.66301 0.831504 0.555518i \(-0.187480\pi\)
0.831504 + 0.555518i \(0.187480\pi\)
\(620\) −0.793979 −0.0318870
\(621\) 1.20374 0.0483044
\(622\) −3.38252 −0.135627
\(623\) 0 0
\(624\) −15.3835 −0.615834
\(625\) 23.9849 0.959395
\(626\) −27.7834 −1.11045
\(627\) −43.9666 −1.75586
\(628\) 9.92354 0.395993
\(629\) −16.1353 −0.643358
\(630\) 0 0
\(631\) −3.03083 −0.120656 −0.0603278 0.998179i \(-0.519215\pi\)
−0.0603278 + 0.998179i \(0.519215\pi\)
\(632\) 4.48597 0.178442
\(633\) −24.0653 −0.956510
\(634\) 13.6437 0.541859
\(635\) 1.25762 0.0499070
\(636\) 0.947199 0.0375589
\(637\) 0 0
\(638\) 2.14760 0.0850244
\(639\) 6.31809 0.249940
\(640\) −0.260736 −0.0103065
\(641\) 25.6543 1.01328 0.506642 0.862157i \(-0.330887\pi\)
0.506642 + 0.862157i \(0.330887\pi\)
\(642\) 24.1816 0.954370
\(643\) −11.3614 −0.448048 −0.224024 0.974584i \(-0.571919\pi\)
−0.224024 + 0.974584i \(0.571919\pi\)
\(644\) 0 0
\(645\) −0.123392 −0.00485854
\(646\) −18.7913 −0.739332
\(647\) −19.0817 −0.750181 −0.375090 0.926988i \(-0.622388\pi\)
−0.375090 + 0.926988i \(0.622388\pi\)
\(648\) 9.32660 0.366383
\(649\) −31.6850 −1.24375
\(650\) 31.2707 1.22654
\(651\) 0 0
\(652\) 6.75259 0.264452
\(653\) 48.3668 1.89274 0.946370 0.323086i \(-0.104720\pi\)
0.946370 + 0.323086i \(0.104720\pi\)
\(654\) −23.8377 −0.932128
\(655\) 3.67347 0.143534
\(656\) 10.7302 0.418943
\(657\) 12.0772 0.471177
\(658\) 0 0
\(659\) −7.57465 −0.295067 −0.147533 0.989057i \(-0.547133\pi\)
−0.147533 + 0.989057i \(0.547133\pi\)
\(660\) −1.35862 −0.0528840
\(661\) −44.4433 −1.72864 −0.864322 0.502940i \(-0.832252\pi\)
−0.864322 + 0.502940i \(0.832252\pi\)
\(662\) 12.1576 0.472520
\(663\) −34.2598 −1.33054
\(664\) −1.91389 −0.0742734
\(665\) 0 0
\(666\) 20.9158 0.810473
\(667\) −4.38533 −0.169801
\(668\) 7.22428 0.279516
\(669\) 47.5274 1.83751
\(670\) −2.48687 −0.0960762
\(671\) −21.1316 −0.815774
\(672\) 0 0
\(673\) −22.7141 −0.875564 −0.437782 0.899081i \(-0.644236\pi\)
−0.437782 + 0.899081i \(0.644236\pi\)
\(674\) −12.7616 −0.491558
\(675\) −1.35380 −0.0521078
\(676\) 27.2001 1.04616
\(677\) −1.67725 −0.0644621 −0.0322310 0.999480i \(-0.510261\pi\)
−0.0322310 + 0.999480i \(0.510261\pi\)
\(678\) −2.92198 −0.112218
\(679\) 0 0
\(680\) −0.580670 −0.0222677
\(681\) −38.0729 −1.45895
\(682\) 6.53977 0.250421
\(683\) 49.9200 1.91014 0.955069 0.296385i \(-0.0957811\pi\)
0.955069 + 0.296385i \(0.0957811\pi\)
\(684\) 24.3587 0.931377
\(685\) −3.39606 −0.129757
\(686\) 0 0
\(687\) 34.9863 1.33481
\(688\) 0.195049 0.00743617
\(689\) −2.47521 −0.0942982
\(690\) 2.77424 0.105614
\(691\) 25.5835 0.973240 0.486620 0.873614i \(-0.338230\pi\)
0.486620 + 0.873614i \(0.338230\pi\)
\(692\) 8.60818 0.327234
\(693\) 0 0
\(694\) 3.20642 0.121714
\(695\) −4.40609 −0.167132
\(696\) −2.42629 −0.0919682
\(697\) 23.8966 0.905149
\(698\) −3.95168 −0.149573
\(699\) 24.3335 0.920378
\(700\) 0 0
\(701\) 29.5973 1.11787 0.558936 0.829211i \(-0.311210\pi\)
0.558936 + 0.829211i \(0.311210\pi\)
\(702\) −1.74038 −0.0656864
\(703\) −61.1329 −2.30567
\(704\) 2.14760 0.0809408
\(705\) −7.73343 −0.291258
\(706\) 15.0305 0.565682
\(707\) 0 0
\(708\) 35.7966 1.34532
\(709\) 0.307107 0.0115337 0.00576683 0.999983i \(-0.498164\pi\)
0.00576683 + 0.999983i \(0.498164\pi\)
\(710\) −0.570636 −0.0214156
\(711\) −12.9504 −0.485678
\(712\) −11.8209 −0.443006
\(713\) −13.3540 −0.500111
\(714\) 0 0
\(715\) 3.55032 0.132775
\(716\) 21.0797 0.787786
\(717\) 2.23311 0.0833969
\(718\) 12.5777 0.469394
\(719\) 22.2426 0.829511 0.414755 0.909933i \(-0.363867\pi\)
0.414755 + 0.909933i \(0.363867\pi\)
\(720\) 0.752709 0.0280518
\(721\) 0 0
\(722\) −52.1956 −1.94252
\(723\) 18.5583 0.690189
\(724\) −8.59244 −0.319336
\(725\) 4.93202 0.183171
\(726\) −15.4986 −0.575208
\(727\) −23.8821 −0.885739 −0.442869 0.896586i \(-0.646039\pi\)
−0.442869 + 0.896586i \(0.646039\pi\)
\(728\) 0 0
\(729\) −24.9267 −0.923210
\(730\) −1.09079 −0.0403719
\(731\) 0.434383 0.0160662
\(732\) 23.8737 0.882397
\(733\) −2.31726 −0.0855899 −0.0427949 0.999084i \(-0.513626\pi\)
−0.0427949 + 0.999084i \(0.513626\pi\)
\(734\) 15.2524 0.562975
\(735\) 0 0
\(736\) −4.38533 −0.161645
\(737\) 20.4836 0.754524
\(738\) −30.9766 −1.14027
\(739\) −20.4310 −0.751566 −0.375783 0.926708i \(-0.622626\pi\)
−0.375783 + 0.926708i \(0.622626\pi\)
\(740\) −1.88907 −0.0694438
\(741\) −129.802 −4.76841
\(742\) 0 0
\(743\) 16.7514 0.614550 0.307275 0.951621i \(-0.400583\pi\)
0.307275 + 0.951621i \(0.400583\pi\)
\(744\) −7.38841 −0.270872
\(745\) −0.128275 −0.00469963
\(746\) 14.9777 0.548373
\(747\) 5.52515 0.202155
\(748\) 4.78281 0.174877
\(749\) 0 0
\(750\) −6.28319 −0.229430
\(751\) 27.2245 0.993438 0.496719 0.867911i \(-0.334538\pi\)
0.496719 + 0.867911i \(0.334538\pi\)
\(752\) 12.2245 0.445780
\(753\) −25.0009 −0.911084
\(754\) 6.34036 0.230902
\(755\) 0.548027 0.0199447
\(756\) 0 0
\(757\) 7.31697 0.265940 0.132970 0.991120i \(-0.457549\pi\)
0.132970 + 0.991120i \(0.457549\pi\)
\(758\) −0.578081 −0.0209969
\(759\) −22.8506 −0.829426
\(760\) −2.20002 −0.0798032
\(761\) −24.8920 −0.902334 −0.451167 0.892440i \(-0.648992\pi\)
−0.451167 + 0.892440i \(0.648992\pi\)
\(762\) 11.7028 0.423948
\(763\) 0 0
\(764\) 8.52165 0.308302
\(765\) 1.67632 0.0606074
\(766\) 28.9218 1.04499
\(767\) −93.5435 −3.37766
\(768\) −2.42629 −0.0875511
\(769\) −47.5147 −1.71342 −0.856712 0.515795i \(-0.827497\pi\)
−0.856712 + 0.515795i \(0.827497\pi\)
\(770\) 0 0
\(771\) 10.7519 0.387221
\(772\) 13.1790 0.474322
\(773\) 35.8534 1.28956 0.644778 0.764370i \(-0.276950\pi\)
0.644778 + 0.764370i \(0.276950\pi\)
\(774\) −0.563080 −0.0202395
\(775\) 15.0187 0.539489
\(776\) 0.283480 0.0101763
\(777\) 0 0
\(778\) 9.91719 0.355548
\(779\) 90.5386 3.24388
\(780\) −4.01103 −0.143618
\(781\) 4.70016 0.168185
\(782\) −9.76633 −0.349243
\(783\) −0.274492 −0.00980955
\(784\) 0 0
\(785\) 2.58742 0.0923490
\(786\) 34.1836 1.21929
\(787\) −4.53121 −0.161520 −0.0807601 0.996734i \(-0.525735\pi\)
−0.0807601 + 0.996734i \(0.525735\pi\)
\(788\) −5.96908 −0.212640
\(789\) −21.2419 −0.756231
\(790\) 1.16965 0.0416144
\(791\) 0 0
\(792\) −6.19984 −0.220302
\(793\) −62.3865 −2.21541
\(794\) −10.3356 −0.366797
\(795\) 0.246969 0.00875908
\(796\) 13.3315 0.472521
\(797\) −1.22332 −0.0433324 −0.0216662 0.999765i \(-0.506897\pi\)
−0.0216662 + 0.999765i \(0.506897\pi\)
\(798\) 0 0
\(799\) 27.2244 0.963131
\(800\) 4.93202 0.174373
\(801\) 34.1253 1.20576
\(802\) 20.8127 0.734921
\(803\) 8.98450 0.317056
\(804\) −23.1417 −0.816145
\(805\) 0 0
\(806\) 19.3073 0.680072
\(807\) −52.8578 −1.86068
\(808\) 10.9123 0.383892
\(809\) −46.2305 −1.62538 −0.812689 0.582697i \(-0.801997\pi\)
−0.812689 + 0.582697i \(0.801997\pi\)
\(810\) 2.43178 0.0854439
\(811\) 20.0839 0.705241 0.352621 0.935766i \(-0.385291\pi\)
0.352621 + 0.935766i \(0.385291\pi\)
\(812\) 0 0
\(813\) 41.1925 1.44469
\(814\) 15.5598 0.545369
\(815\) 1.76064 0.0616726
\(816\) −5.40345 −0.189159
\(817\) 1.64577 0.0575783
\(818\) −9.80067 −0.342672
\(819\) 0 0
\(820\) 2.79774 0.0977014
\(821\) 27.8598 0.972314 0.486157 0.873871i \(-0.338398\pi\)
0.486157 + 0.873871i \(0.338398\pi\)
\(822\) −31.6022 −1.10225
\(823\) 54.1598 1.88789 0.943946 0.330099i \(-0.107082\pi\)
0.943946 + 0.330099i \(0.107082\pi\)
\(824\) 11.5444 0.402168
\(825\) 25.6993 0.894734
\(826\) 0 0
\(827\) −16.8852 −0.587157 −0.293579 0.955935i \(-0.594846\pi\)
−0.293579 + 0.955935i \(0.594846\pi\)
\(828\) 12.6599 0.439961
\(829\) −46.3467 −1.60969 −0.804844 0.593486i \(-0.797751\pi\)
−0.804844 + 0.593486i \(0.797751\pi\)
\(830\) −0.499020 −0.0173212
\(831\) 1.47677 0.0512285
\(832\) 6.34036 0.219812
\(833\) 0 0
\(834\) −41.0010 −1.41975
\(835\) 1.88363 0.0651856
\(836\) 18.1209 0.626726
\(837\) −0.835870 −0.0288919
\(838\) −0.564842 −0.0195121
\(839\) −24.9065 −0.859866 −0.429933 0.902861i \(-0.641463\pi\)
−0.429933 + 0.902861i \(0.641463\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −28.1593 −0.970434
\(843\) 62.0049 2.13556
\(844\) 9.91857 0.341411
\(845\) 7.09204 0.243974
\(846\) −35.2904 −1.21331
\(847\) 0 0
\(848\) −0.390391 −0.0134061
\(849\) −59.9636 −2.05794
\(850\) 10.9838 0.376742
\(851\) −31.7725 −1.08915
\(852\) −5.31008 −0.181920
\(853\) −13.4612 −0.460903 −0.230451 0.973084i \(-0.574020\pi\)
−0.230451 + 0.973084i \(0.574020\pi\)
\(854\) 0 0
\(855\) 6.35117 0.217205
\(856\) −9.96649 −0.340648
\(857\) 28.3203 0.967403 0.483702 0.875233i \(-0.339292\pi\)
0.483702 + 0.875233i \(0.339292\pi\)
\(858\) 33.0377 1.12789
\(859\) 21.8641 0.745995 0.372997 0.927832i \(-0.378330\pi\)
0.372997 + 0.927832i \(0.378330\pi\)
\(860\) 0.0508562 0.00173418
\(861\) 0 0
\(862\) −28.1658 −0.959331
\(863\) 32.4502 1.10462 0.552308 0.833640i \(-0.313747\pi\)
0.552308 + 0.833640i \(0.313747\pi\)
\(864\) −0.274492 −0.00933842
\(865\) 2.24446 0.0763139
\(866\) −17.6927 −0.601224
\(867\) 29.2131 0.992130
\(868\) 0 0
\(869\) −9.63409 −0.326814
\(870\) −0.632619 −0.0214478
\(871\) 60.4737 2.04907
\(872\) 9.82477 0.332709
\(873\) −0.818369 −0.0276976
\(874\) −37.0023 −1.25162
\(875\) 0 0
\(876\) −10.1504 −0.342949
\(877\) −4.92965 −0.166462 −0.0832312 0.996530i \(-0.526524\pi\)
−0.0832312 + 0.996530i \(0.526524\pi\)
\(878\) 28.1245 0.949155
\(879\) −50.9297 −1.71782
\(880\) 0.559957 0.0188761
\(881\) −13.9409 −0.469682 −0.234841 0.972034i \(-0.575457\pi\)
−0.234841 + 0.972034i \(0.575457\pi\)
\(882\) 0 0
\(883\) 44.4387 1.49548 0.747740 0.663992i \(-0.231139\pi\)
0.747740 + 0.663992i \(0.231139\pi\)
\(884\) 14.1203 0.474916
\(885\) 9.33346 0.313741
\(886\) 6.21048 0.208645
\(887\) −3.19391 −0.107241 −0.0536204 0.998561i \(-0.517076\pi\)
−0.0536204 + 0.998561i \(0.517076\pi\)
\(888\) −17.5789 −0.589908
\(889\) 0 0
\(890\) −3.08212 −0.103313
\(891\) −20.0298 −0.671025
\(892\) −19.5885 −0.655872
\(893\) 103.147 3.45168
\(894\) −1.19367 −0.0399223
\(895\) 5.49623 0.183719
\(896\) 0 0
\(897\) −67.4618 −2.25248
\(898\) 5.87960 0.196205
\(899\) 3.04515 0.101561
\(900\) −14.2381 −0.474603
\(901\) −0.869418 −0.0289645
\(902\) −23.0442 −0.767287
\(903\) 0 0
\(904\) 1.20430 0.0400545
\(905\) −2.24036 −0.0744720
\(906\) 5.09969 0.169426
\(907\) 20.0688 0.666372 0.333186 0.942861i \(-0.391876\pi\)
0.333186 + 0.942861i \(0.391876\pi\)
\(908\) 15.6918 0.520751
\(909\) −31.5022 −1.04486
\(910\) 0 0
\(911\) −5.18802 −0.171887 −0.0859434 0.996300i \(-0.527390\pi\)
−0.0859434 + 0.996300i \(0.527390\pi\)
\(912\) −20.4724 −0.677909
\(913\) 4.11028 0.136030
\(914\) 34.2340 1.13236
\(915\) 6.22472 0.205783
\(916\) −14.4197 −0.476440
\(917\) 0 0
\(918\) −0.611307 −0.0201761
\(919\) 10.2725 0.338859 0.169429 0.985542i \(-0.445808\pi\)
0.169429 + 0.985542i \(0.445808\pi\)
\(920\) −1.14341 −0.0376972
\(921\) −60.4543 −1.99204
\(922\) 7.77623 0.256097
\(923\) 13.8763 0.456743
\(924\) 0 0
\(925\) 35.7333 1.17490
\(926\) 6.50388 0.213731
\(927\) −33.3271 −1.09461
\(928\) 1.00000 0.0328266
\(929\) −4.27777 −0.140349 −0.0701745 0.997535i \(-0.522356\pi\)
−0.0701745 + 0.997535i \(0.522356\pi\)
\(930\) −1.92642 −0.0631698
\(931\) 0 0
\(932\) −10.0291 −0.328515
\(933\) −8.20696 −0.268684
\(934\) 5.00733 0.163845
\(935\) 1.24705 0.0407829
\(936\) −18.3038 −0.598277
\(937\) −59.1093 −1.93102 −0.965508 0.260374i \(-0.916154\pi\)
−0.965508 + 0.260374i \(0.916154\pi\)
\(938\) 0 0
\(939\) −67.4106 −2.19986
\(940\) 3.18735 0.103960
\(941\) 35.6173 1.16109 0.580545 0.814228i \(-0.302839\pi\)
0.580545 + 0.814228i \(0.302839\pi\)
\(942\) 24.0774 0.784483
\(943\) 47.0554 1.53233
\(944\) −14.7537 −0.480191
\(945\) 0 0
\(946\) −0.418887 −0.0136192
\(947\) −56.0906 −1.82270 −0.911349 0.411634i \(-0.864958\pi\)
−0.911349 + 0.411634i \(0.864958\pi\)
\(948\) 10.8843 0.353504
\(949\) 26.5249 0.861034
\(950\) 41.6151 1.35017
\(951\) 33.1034 1.07345
\(952\) 0 0
\(953\) 30.7780 0.996998 0.498499 0.866890i \(-0.333885\pi\)
0.498499 + 0.866890i \(0.333885\pi\)
\(954\) 1.12701 0.0364882
\(955\) 2.22190 0.0718989
\(956\) −0.920380 −0.0297672
\(957\) 5.21070 0.168438
\(958\) 35.2838 1.13997
\(959\) 0 0
\(960\) −0.632619 −0.0204177
\(961\) −21.7271 −0.700873
\(962\) 45.9370 1.48107
\(963\) 28.7719 0.927162
\(964\) −7.64883 −0.246352
\(965\) 3.43623 0.110616
\(966\) 0 0
\(967\) 18.2916 0.588217 0.294108 0.955772i \(-0.404977\pi\)
0.294108 + 0.955772i \(0.404977\pi\)
\(968\) 6.38780 0.205312
\(969\) −45.5930 −1.46466
\(970\) 0.0739133 0.00237321
\(971\) 19.2108 0.616503 0.308251 0.951305i \(-0.400256\pi\)
0.308251 + 0.951305i \(0.400256\pi\)
\(972\) 21.8055 0.699413
\(973\) 0 0
\(974\) −4.73153 −0.151608
\(975\) 75.8718 2.42984
\(976\) −9.83960 −0.314958
\(977\) 11.6484 0.372665 0.186332 0.982487i \(-0.440340\pi\)
0.186332 + 0.982487i \(0.440340\pi\)
\(978\) 16.3837 0.523894
\(979\) 25.3865 0.811357
\(980\) 0 0
\(981\) −28.3628 −0.905555
\(982\) 18.5337 0.591433
\(983\) −1.00446 −0.0320372 −0.0160186 0.999872i \(-0.505099\pi\)
−0.0160186 + 0.999872i \(0.505099\pi\)
\(984\) 26.0345 0.829950
\(985\) −1.55635 −0.0495895
\(986\) 2.22705 0.0709236
\(987\) 0 0
\(988\) 53.4983 1.70201
\(989\) 0.855353 0.0271986
\(990\) −1.61652 −0.0513764
\(991\) −47.5781 −1.51137 −0.755684 0.654936i \(-0.772695\pi\)
−0.755684 + 0.654936i \(0.772695\pi\)
\(992\) 3.04515 0.0966836
\(993\) 29.4979 0.936088
\(994\) 0 0
\(995\) 3.47599 0.110196
\(996\) −4.64365 −0.147140
\(997\) −13.6822 −0.433320 −0.216660 0.976247i \(-0.569516\pi\)
−0.216660 + 0.976247i \(0.569516\pi\)
\(998\) 8.49139 0.268790
\(999\) −1.98874 −0.0629211
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 2842.2.a.u.1.1 5
7.3 odd 6 406.2.e.d.233.1 10
7.5 odd 6 406.2.e.d.291.1 yes 10
7.6 odd 2 2842.2.a.t.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
406.2.e.d.233.1 10 7.3 odd 6
406.2.e.d.291.1 yes 10 7.5 odd 6
2842.2.a.t.1.5 5 7.6 odd 2
2842.2.a.u.1.1 5 1.1 even 1 trivial