Properties

Label 6040.2.a.s.1.6
Level $6040$
Weight $2$
Character 6040.1
Self dual yes
Analytic conductor $48.230$
Analytic rank $0$
Dimension $24$
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] = [6040,2,Mod(1,6040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6040, 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("6040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6040.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: \(48.2296428209\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Character \(\chi\) \(=\) 6040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.29572 q^{3} +1.00000 q^{5} +4.99611 q^{7} +2.27034 q^{9} +O(q^{10})\) \(q-2.29572 q^{3} +1.00000 q^{5} +4.99611 q^{7} +2.27034 q^{9} -0.0910559 q^{11} +4.74101 q^{13} -2.29572 q^{15} +3.08447 q^{17} +3.48388 q^{19} -11.4697 q^{21} +5.20981 q^{23} +1.00000 q^{25} +1.67510 q^{27} +4.87156 q^{29} -4.75626 q^{31} +0.209039 q^{33} +4.99611 q^{35} +2.85526 q^{37} -10.8840 q^{39} +7.46870 q^{41} -2.78036 q^{43} +2.27034 q^{45} +8.24075 q^{47} +17.9611 q^{49} -7.08109 q^{51} -2.79674 q^{53} -0.0910559 q^{55} -7.99802 q^{57} +4.93952 q^{59} -15.5686 q^{61} +11.3429 q^{63} +4.74101 q^{65} +0.961539 q^{67} -11.9603 q^{69} -2.81571 q^{71} +16.7476 q^{73} -2.29572 q^{75} -0.454925 q^{77} -5.64257 q^{79} -10.6566 q^{81} -7.20514 q^{83} +3.08447 q^{85} -11.1838 q^{87} +2.59491 q^{89} +23.6866 q^{91} +10.9191 q^{93} +3.48388 q^{95} +13.3132 q^{97} -0.206728 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 2 q^{3} + 24 q^{5} + 3 q^{7} + 40 q^{9} + 17 q^{11} + 16 q^{13} + 2 q^{15} + 22 q^{17} + 16 q^{19} - q^{21} + 7 q^{23} + 24 q^{25} - 4 q^{27} + 25 q^{29} + 28 q^{31} + 11 q^{33} + 3 q^{35} + 26 q^{37} + 13 q^{39} + 38 q^{41} - 13 q^{43} + 40 q^{45} + 12 q^{47} + 61 q^{49} + 53 q^{53} + 17 q^{55} + 30 q^{57} + 35 q^{59} + 44 q^{61} - 9 q^{63} + 16 q^{65} - 15 q^{67} + 9 q^{69} + 22 q^{71} + 31 q^{73} + 2 q^{75} + 26 q^{77} + 20 q^{79} + 88 q^{81} - 14 q^{83} + 22 q^{85} - 18 q^{87} + 37 q^{89} - 26 q^{91} + 13 q^{93} + 16 q^{95} + 21 q^{97} - 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.29572 −1.32544 −0.662718 0.748869i \(-0.730597\pi\)
−0.662718 + 0.748869i \(0.730597\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.99611 1.88835 0.944176 0.329442i \(-0.106861\pi\)
0.944176 + 0.329442i \(0.106861\pi\)
\(8\) 0 0
\(9\) 2.27034 0.756780
\(10\) 0 0
\(11\) −0.0910559 −0.0274544 −0.0137272 0.999906i \(-0.504370\pi\)
−0.0137272 + 0.999906i \(0.504370\pi\)
\(12\) 0 0
\(13\) 4.74101 1.31492 0.657459 0.753490i \(-0.271631\pi\)
0.657459 + 0.753490i \(0.271631\pi\)
\(14\) 0 0
\(15\) −2.29572 −0.592753
\(16\) 0 0
\(17\) 3.08447 0.748094 0.374047 0.927410i \(-0.377970\pi\)
0.374047 + 0.927410i \(0.377970\pi\)
\(18\) 0 0
\(19\) 3.48388 0.799257 0.399628 0.916677i \(-0.369139\pi\)
0.399628 + 0.916677i \(0.369139\pi\)
\(20\) 0 0
\(21\) −11.4697 −2.50289
\(22\) 0 0
\(23\) 5.20981 1.08632 0.543161 0.839629i \(-0.317227\pi\)
0.543161 + 0.839629i \(0.317227\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.67510 0.322372
\(28\) 0 0
\(29\) 4.87156 0.904627 0.452313 0.891859i \(-0.350599\pi\)
0.452313 + 0.891859i \(0.350599\pi\)
\(30\) 0 0
\(31\) −4.75626 −0.854250 −0.427125 0.904193i \(-0.640474\pi\)
−0.427125 + 0.904193i \(0.640474\pi\)
\(32\) 0 0
\(33\) 0.209039 0.0363890
\(34\) 0 0
\(35\) 4.99611 0.844497
\(36\) 0 0
\(37\) 2.85526 0.469402 0.234701 0.972068i \(-0.424589\pi\)
0.234701 + 0.972068i \(0.424589\pi\)
\(38\) 0 0
\(39\) −10.8840 −1.74284
\(40\) 0 0
\(41\) 7.46870 1.16641 0.583207 0.812323i \(-0.301798\pi\)
0.583207 + 0.812323i \(0.301798\pi\)
\(42\) 0 0
\(43\) −2.78036 −0.424001 −0.212000 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(44\) 0 0
\(45\) 2.27034 0.338442
\(46\) 0 0
\(47\) 8.24075 1.20204 0.601018 0.799235i \(-0.294762\pi\)
0.601018 + 0.799235i \(0.294762\pi\)
\(48\) 0 0
\(49\) 17.9611 2.56587
\(50\) 0 0
\(51\) −7.08109 −0.991551
\(52\) 0 0
\(53\) −2.79674 −0.384162 −0.192081 0.981379i \(-0.561524\pi\)
−0.192081 + 0.981379i \(0.561524\pi\)
\(54\) 0 0
\(55\) −0.0910559 −0.0122780
\(56\) 0 0
\(57\) −7.99802 −1.05936
\(58\) 0 0
\(59\) 4.93952 0.643071 0.321535 0.946898i \(-0.395801\pi\)
0.321535 + 0.946898i \(0.395801\pi\)
\(60\) 0 0
\(61\) −15.5686 −1.99336 −0.996680 0.0814198i \(-0.974055\pi\)
−0.996680 + 0.0814198i \(0.974055\pi\)
\(62\) 0 0
\(63\) 11.3429 1.42907
\(64\) 0 0
\(65\) 4.74101 0.588049
\(66\) 0 0
\(67\) 0.961539 0.117471 0.0587353 0.998274i \(-0.481293\pi\)
0.0587353 + 0.998274i \(0.481293\pi\)
\(68\) 0 0
\(69\) −11.9603 −1.43985
\(70\) 0 0
\(71\) −2.81571 −0.334164 −0.167082 0.985943i \(-0.553434\pi\)
−0.167082 + 0.985943i \(0.553434\pi\)
\(72\) 0 0
\(73\) 16.7476 1.96016 0.980081 0.198597i \(-0.0636386\pi\)
0.980081 + 0.198597i \(0.0636386\pi\)
\(74\) 0 0
\(75\) −2.29572 −0.265087
\(76\) 0 0
\(77\) −0.454925 −0.0518435
\(78\) 0 0
\(79\) −5.64257 −0.634839 −0.317419 0.948285i \(-0.602816\pi\)
−0.317419 + 0.948285i \(0.602816\pi\)
\(80\) 0 0
\(81\) −10.6566 −1.18406
\(82\) 0 0
\(83\) −7.20514 −0.790867 −0.395433 0.918495i \(-0.629406\pi\)
−0.395433 + 0.918495i \(0.629406\pi\)
\(84\) 0 0
\(85\) 3.08447 0.334558
\(86\) 0 0
\(87\) −11.1838 −1.19902
\(88\) 0 0
\(89\) 2.59491 0.275060 0.137530 0.990498i \(-0.456084\pi\)
0.137530 + 0.990498i \(0.456084\pi\)
\(90\) 0 0
\(91\) 23.6866 2.48303
\(92\) 0 0
\(93\) 10.9191 1.13225
\(94\) 0 0
\(95\) 3.48388 0.357438
\(96\) 0 0
\(97\) 13.3132 1.35175 0.675877 0.737015i \(-0.263765\pi\)
0.675877 + 0.737015i \(0.263765\pi\)
\(98\) 0 0
\(99\) −0.206728 −0.0207769
\(100\) 0 0
\(101\) 3.41251 0.339557 0.169779 0.985482i \(-0.445695\pi\)
0.169779 + 0.985482i \(0.445695\pi\)
\(102\) 0 0
\(103\) −13.6336 −1.34335 −0.671677 0.740844i \(-0.734426\pi\)
−0.671677 + 0.740844i \(0.734426\pi\)
\(104\) 0 0
\(105\) −11.4697 −1.11933
\(106\) 0 0
\(107\) −9.22568 −0.891880 −0.445940 0.895063i \(-0.647131\pi\)
−0.445940 + 0.895063i \(0.647131\pi\)
\(108\) 0 0
\(109\) −2.17922 −0.208732 −0.104366 0.994539i \(-0.533281\pi\)
−0.104366 + 0.994539i \(0.533281\pi\)
\(110\) 0 0
\(111\) −6.55488 −0.622162
\(112\) 0 0
\(113\) −13.8473 −1.30264 −0.651322 0.758801i \(-0.725785\pi\)
−0.651322 + 0.758801i \(0.725785\pi\)
\(114\) 0 0
\(115\) 5.20981 0.485818
\(116\) 0 0
\(117\) 10.7637 0.995104
\(118\) 0 0
\(119\) 15.4104 1.41267
\(120\) 0 0
\(121\) −10.9917 −0.999246
\(122\) 0 0
\(123\) −17.1461 −1.54601
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −10.3742 −0.920558 −0.460279 0.887774i \(-0.652251\pi\)
−0.460279 + 0.887774i \(0.652251\pi\)
\(128\) 0 0
\(129\) 6.38293 0.561986
\(130\) 0 0
\(131\) −16.0953 −1.40625 −0.703127 0.711064i \(-0.748214\pi\)
−0.703127 + 0.711064i \(0.748214\pi\)
\(132\) 0 0
\(133\) 17.4058 1.50928
\(134\) 0 0
\(135\) 1.67510 0.144169
\(136\) 0 0
\(137\) −6.40018 −0.546804 −0.273402 0.961900i \(-0.588149\pi\)
−0.273402 + 0.961900i \(0.588149\pi\)
\(138\) 0 0
\(139\) −15.6231 −1.32513 −0.662567 0.749002i \(-0.730533\pi\)
−0.662567 + 0.749002i \(0.730533\pi\)
\(140\) 0 0
\(141\) −18.9185 −1.59322
\(142\) 0 0
\(143\) −0.431697 −0.0361003
\(144\) 0 0
\(145\) 4.87156 0.404561
\(146\) 0 0
\(147\) −41.2337 −3.40090
\(148\) 0 0
\(149\) 3.99341 0.327153 0.163576 0.986531i \(-0.447697\pi\)
0.163576 + 0.986531i \(0.447697\pi\)
\(150\) 0 0
\(151\) 1.00000 0.0813788
\(152\) 0 0
\(153\) 7.00280 0.566143
\(154\) 0 0
\(155\) −4.75626 −0.382032
\(156\) 0 0
\(157\) −3.26269 −0.260391 −0.130195 0.991488i \(-0.541560\pi\)
−0.130195 + 0.991488i \(0.541560\pi\)
\(158\) 0 0
\(159\) 6.42054 0.509182
\(160\) 0 0
\(161\) 26.0288 2.05136
\(162\) 0 0
\(163\) −13.1793 −1.03228 −0.516141 0.856503i \(-0.672632\pi\)
−0.516141 + 0.856503i \(0.672632\pi\)
\(164\) 0 0
\(165\) 0.209039 0.0162737
\(166\) 0 0
\(167\) 4.26874 0.330325 0.165163 0.986266i \(-0.447185\pi\)
0.165163 + 0.986266i \(0.447185\pi\)
\(168\) 0 0
\(169\) 9.47714 0.729011
\(170\) 0 0
\(171\) 7.90959 0.604862
\(172\) 0 0
\(173\) 5.56099 0.422794 0.211397 0.977400i \(-0.432199\pi\)
0.211397 + 0.977400i \(0.432199\pi\)
\(174\) 0 0
\(175\) 4.99611 0.377670
\(176\) 0 0
\(177\) −11.3398 −0.852349
\(178\) 0 0
\(179\) −5.12353 −0.382951 −0.191475 0.981497i \(-0.561327\pi\)
−0.191475 + 0.981497i \(0.561327\pi\)
\(180\) 0 0
\(181\) −0.888073 −0.0660100 −0.0330050 0.999455i \(-0.510508\pi\)
−0.0330050 + 0.999455i \(0.510508\pi\)
\(182\) 0 0
\(183\) 35.7413 2.64207
\(184\) 0 0
\(185\) 2.85526 0.209923
\(186\) 0 0
\(187\) −0.280859 −0.0205385
\(188\) 0 0
\(189\) 8.36896 0.608752
\(190\) 0 0
\(191\) −12.5361 −0.907082 −0.453541 0.891236i \(-0.649839\pi\)
−0.453541 + 0.891236i \(0.649839\pi\)
\(192\) 0 0
\(193\) −2.12120 −0.152688 −0.0763438 0.997082i \(-0.524325\pi\)
−0.0763438 + 0.997082i \(0.524325\pi\)
\(194\) 0 0
\(195\) −10.8840 −0.779422
\(196\) 0 0
\(197\) 4.92506 0.350896 0.175448 0.984489i \(-0.443863\pi\)
0.175448 + 0.984489i \(0.443863\pi\)
\(198\) 0 0
\(199\) −19.7707 −1.40151 −0.700755 0.713402i \(-0.747153\pi\)
−0.700755 + 0.713402i \(0.747153\pi\)
\(200\) 0 0
\(201\) −2.20743 −0.155700
\(202\) 0 0
\(203\) 24.3389 1.70825
\(204\) 0 0
\(205\) 7.46870 0.521636
\(206\) 0 0
\(207\) 11.8281 0.822106
\(208\) 0 0
\(209\) −0.317228 −0.0219431
\(210\) 0 0
\(211\) 17.7409 1.22134 0.610668 0.791887i \(-0.290901\pi\)
0.610668 + 0.791887i \(0.290901\pi\)
\(212\) 0 0
\(213\) 6.46410 0.442913
\(214\) 0 0
\(215\) −2.78036 −0.189619
\(216\) 0 0
\(217\) −23.7628 −1.61312
\(218\) 0 0
\(219\) −38.4479 −2.59807
\(220\) 0 0
\(221\) 14.6235 0.983683
\(222\) 0 0
\(223\) −23.0406 −1.54291 −0.771456 0.636282i \(-0.780471\pi\)
−0.771456 + 0.636282i \(0.780471\pi\)
\(224\) 0 0
\(225\) 2.27034 0.151356
\(226\) 0 0
\(227\) −16.6592 −1.10571 −0.552855 0.833278i \(-0.686461\pi\)
−0.552855 + 0.833278i \(0.686461\pi\)
\(228\) 0 0
\(229\) −21.8292 −1.44251 −0.721256 0.692668i \(-0.756435\pi\)
−0.721256 + 0.692668i \(0.756435\pi\)
\(230\) 0 0
\(231\) 1.04438 0.0687153
\(232\) 0 0
\(233\) −2.42410 −0.158808 −0.0794040 0.996843i \(-0.525302\pi\)
−0.0794040 + 0.996843i \(0.525302\pi\)
\(234\) 0 0
\(235\) 8.24075 0.537567
\(236\) 0 0
\(237\) 12.9538 0.841438
\(238\) 0 0
\(239\) 10.0709 0.651435 0.325717 0.945467i \(-0.394394\pi\)
0.325717 + 0.945467i \(0.394394\pi\)
\(240\) 0 0
\(241\) 3.33315 0.214707 0.107354 0.994221i \(-0.465762\pi\)
0.107354 + 0.994221i \(0.465762\pi\)
\(242\) 0 0
\(243\) 19.4393 1.24703
\(244\) 0 0
\(245\) 17.9611 1.14749
\(246\) 0 0
\(247\) 16.5171 1.05096
\(248\) 0 0
\(249\) 16.5410 1.04824
\(250\) 0 0
\(251\) 14.1622 0.893911 0.446955 0.894556i \(-0.352508\pi\)
0.446955 + 0.894556i \(0.352508\pi\)
\(252\) 0 0
\(253\) −0.474384 −0.0298243
\(254\) 0 0
\(255\) −7.08109 −0.443435
\(256\) 0 0
\(257\) 5.38431 0.335864 0.167932 0.985799i \(-0.446291\pi\)
0.167932 + 0.985799i \(0.446291\pi\)
\(258\) 0 0
\(259\) 14.2652 0.886395
\(260\) 0 0
\(261\) 11.0601 0.684603
\(262\) 0 0
\(263\) 15.6732 0.966449 0.483225 0.875496i \(-0.339465\pi\)
0.483225 + 0.875496i \(0.339465\pi\)
\(264\) 0 0
\(265\) −2.79674 −0.171802
\(266\) 0 0
\(267\) −5.95718 −0.364574
\(268\) 0 0
\(269\) 21.9301 1.33710 0.668551 0.743666i \(-0.266915\pi\)
0.668551 + 0.743666i \(0.266915\pi\)
\(270\) 0 0
\(271\) −1.45274 −0.0882477 −0.0441239 0.999026i \(-0.514050\pi\)
−0.0441239 + 0.999026i \(0.514050\pi\)
\(272\) 0 0
\(273\) −54.3778 −3.29110
\(274\) 0 0
\(275\) −0.0910559 −0.00549088
\(276\) 0 0
\(277\) −21.3443 −1.28245 −0.641227 0.767351i \(-0.721574\pi\)
−0.641227 + 0.767351i \(0.721574\pi\)
\(278\) 0 0
\(279\) −10.7983 −0.646479
\(280\) 0 0
\(281\) 18.7183 1.11664 0.558319 0.829626i \(-0.311446\pi\)
0.558319 + 0.829626i \(0.311446\pi\)
\(282\) 0 0
\(283\) 9.39773 0.558637 0.279319 0.960199i \(-0.409891\pi\)
0.279319 + 0.960199i \(0.409891\pi\)
\(284\) 0 0
\(285\) −7.99802 −0.473762
\(286\) 0 0
\(287\) 37.3144 2.20260
\(288\) 0 0
\(289\) −7.48603 −0.440355
\(290\) 0 0
\(291\) −30.5635 −1.79166
\(292\) 0 0
\(293\) −4.71506 −0.275456 −0.137728 0.990470i \(-0.543980\pi\)
−0.137728 + 0.990470i \(0.543980\pi\)
\(294\) 0 0
\(295\) 4.93952 0.287590
\(296\) 0 0
\(297\) −0.152527 −0.00885053
\(298\) 0 0
\(299\) 24.6998 1.42842
\(300\) 0 0
\(301\) −13.8910 −0.800663
\(302\) 0 0
\(303\) −7.83417 −0.450061
\(304\) 0 0
\(305\) −15.5686 −0.891458
\(306\) 0 0
\(307\) −32.0915 −1.83156 −0.915781 0.401679i \(-0.868427\pi\)
−0.915781 + 0.401679i \(0.868427\pi\)
\(308\) 0 0
\(309\) 31.2989 1.78053
\(310\) 0 0
\(311\) 29.5615 1.67628 0.838140 0.545456i \(-0.183643\pi\)
0.838140 + 0.545456i \(0.183643\pi\)
\(312\) 0 0
\(313\) 21.3182 1.20498 0.602489 0.798127i \(-0.294176\pi\)
0.602489 + 0.798127i \(0.294176\pi\)
\(314\) 0 0
\(315\) 11.3429 0.639098
\(316\) 0 0
\(317\) −19.7507 −1.10931 −0.554655 0.832081i \(-0.687150\pi\)
−0.554655 + 0.832081i \(0.687150\pi\)
\(318\) 0 0
\(319\) −0.443585 −0.0248360
\(320\) 0 0
\(321\) 21.1796 1.18213
\(322\) 0 0
\(323\) 10.7459 0.597919
\(324\) 0 0
\(325\) 4.74101 0.262984
\(326\) 0 0
\(327\) 5.00289 0.276660
\(328\) 0 0
\(329\) 41.1717 2.26987
\(330\) 0 0
\(331\) 23.2818 1.27968 0.639842 0.768507i \(-0.279000\pi\)
0.639842 + 0.768507i \(0.279000\pi\)
\(332\) 0 0
\(333\) 6.48241 0.355234
\(334\) 0 0
\(335\) 0.961539 0.0525345
\(336\) 0 0
\(337\) −2.86711 −0.156182 −0.0780908 0.996946i \(-0.524882\pi\)
−0.0780908 + 0.996946i \(0.524882\pi\)
\(338\) 0 0
\(339\) 31.7895 1.72657
\(340\) 0 0
\(341\) 0.433086 0.0234529
\(342\) 0 0
\(343\) 54.7629 2.95692
\(344\) 0 0
\(345\) −11.9603 −0.643920
\(346\) 0 0
\(347\) −29.8394 −1.60187 −0.800933 0.598754i \(-0.795663\pi\)
−0.800933 + 0.598754i \(0.795663\pi\)
\(348\) 0 0
\(349\) 27.7895 1.48754 0.743770 0.668436i \(-0.233036\pi\)
0.743770 + 0.668436i \(0.233036\pi\)
\(350\) 0 0
\(351\) 7.94164 0.423893
\(352\) 0 0
\(353\) 6.38821 0.340010 0.170005 0.985443i \(-0.445622\pi\)
0.170005 + 0.985443i \(0.445622\pi\)
\(354\) 0 0
\(355\) −2.81571 −0.149443
\(356\) 0 0
\(357\) −35.3779 −1.87240
\(358\) 0 0
\(359\) −11.6112 −0.612816 −0.306408 0.951900i \(-0.599127\pi\)
−0.306408 + 0.951900i \(0.599127\pi\)
\(360\) 0 0
\(361\) −6.86259 −0.361189
\(362\) 0 0
\(363\) 25.2339 1.32444
\(364\) 0 0
\(365\) 16.7476 0.876611
\(366\) 0 0
\(367\) −35.9009 −1.87401 −0.937006 0.349314i \(-0.886415\pi\)
−0.937006 + 0.349314i \(0.886415\pi\)
\(368\) 0 0
\(369\) 16.9565 0.882719
\(370\) 0 0
\(371\) −13.9728 −0.725433
\(372\) 0 0
\(373\) −5.29396 −0.274111 −0.137055 0.990563i \(-0.543764\pi\)
−0.137055 + 0.990563i \(0.543764\pi\)
\(374\) 0 0
\(375\) −2.29572 −0.118551
\(376\) 0 0
\(377\) 23.0961 1.18951
\(378\) 0 0
\(379\) −23.8767 −1.22646 −0.613232 0.789903i \(-0.710131\pi\)
−0.613232 + 0.789903i \(0.710131\pi\)
\(380\) 0 0
\(381\) 23.8162 1.22014
\(382\) 0 0
\(383\) −4.33195 −0.221352 −0.110676 0.993857i \(-0.535302\pi\)
−0.110676 + 0.993857i \(0.535302\pi\)
\(384\) 0 0
\(385\) −0.454925 −0.0231851
\(386\) 0 0
\(387\) −6.31236 −0.320876
\(388\) 0 0
\(389\) 25.6380 1.29990 0.649948 0.759979i \(-0.274791\pi\)
0.649948 + 0.759979i \(0.274791\pi\)
\(390\) 0 0
\(391\) 16.0695 0.812671
\(392\) 0 0
\(393\) 36.9504 1.86390
\(394\) 0 0
\(395\) −5.64257 −0.283908
\(396\) 0 0
\(397\) −9.67229 −0.485438 −0.242719 0.970097i \(-0.578039\pi\)
−0.242719 + 0.970097i \(0.578039\pi\)
\(398\) 0 0
\(399\) −39.9590 −2.00045
\(400\) 0 0
\(401\) 33.0926 1.65256 0.826282 0.563257i \(-0.190452\pi\)
0.826282 + 0.563257i \(0.190452\pi\)
\(402\) 0 0
\(403\) −22.5495 −1.12327
\(404\) 0 0
\(405\) −10.6566 −0.529529
\(406\) 0 0
\(407\) −0.259988 −0.0128871
\(408\) 0 0
\(409\) 22.4378 1.10948 0.554739 0.832024i \(-0.312818\pi\)
0.554739 + 0.832024i \(0.312818\pi\)
\(410\) 0 0
\(411\) 14.6930 0.724754
\(412\) 0 0
\(413\) 24.6784 1.21434
\(414\) 0 0
\(415\) −7.20514 −0.353686
\(416\) 0 0
\(417\) 35.8663 1.75638
\(418\) 0 0
\(419\) −5.37539 −0.262605 −0.131302 0.991342i \(-0.541916\pi\)
−0.131302 + 0.991342i \(0.541916\pi\)
\(420\) 0 0
\(421\) −0.0199154 −0.000970619 0 −0.000485309 1.00000i \(-0.500154\pi\)
−0.000485309 1.00000i \(0.500154\pi\)
\(422\) 0 0
\(423\) 18.7093 0.909678
\(424\) 0 0
\(425\) 3.08447 0.149619
\(426\) 0 0
\(427\) −77.7826 −3.76416
\(428\) 0 0
\(429\) 0.991055 0.0478486
\(430\) 0 0
\(431\) 13.3684 0.643933 0.321966 0.946751i \(-0.395656\pi\)
0.321966 + 0.946751i \(0.395656\pi\)
\(432\) 0 0
\(433\) 26.4711 1.27212 0.636059 0.771640i \(-0.280563\pi\)
0.636059 + 0.771640i \(0.280563\pi\)
\(434\) 0 0
\(435\) −11.1838 −0.536220
\(436\) 0 0
\(437\) 18.1504 0.868250
\(438\) 0 0
\(439\) 10.1526 0.484556 0.242278 0.970207i \(-0.422105\pi\)
0.242278 + 0.970207i \(0.422105\pi\)
\(440\) 0 0
\(441\) 40.7778 1.94180
\(442\) 0 0
\(443\) 0.671932 0.0319245 0.0159622 0.999873i \(-0.494919\pi\)
0.0159622 + 0.999873i \(0.494919\pi\)
\(444\) 0 0
\(445\) 2.59491 0.123010
\(446\) 0 0
\(447\) −9.16776 −0.433620
\(448\) 0 0
\(449\) −18.1103 −0.854680 −0.427340 0.904091i \(-0.640549\pi\)
−0.427340 + 0.904091i \(0.640549\pi\)
\(450\) 0 0
\(451\) −0.680069 −0.0320232
\(452\) 0 0
\(453\) −2.29572 −0.107862
\(454\) 0 0
\(455\) 23.6866 1.11044
\(456\) 0 0
\(457\) −8.23106 −0.385033 −0.192516 0.981294i \(-0.561665\pi\)
−0.192516 + 0.981294i \(0.561665\pi\)
\(458\) 0 0
\(459\) 5.16679 0.241165
\(460\) 0 0
\(461\) −18.8788 −0.879272 −0.439636 0.898176i \(-0.644892\pi\)
−0.439636 + 0.898176i \(0.644892\pi\)
\(462\) 0 0
\(463\) 4.68209 0.217595 0.108797 0.994064i \(-0.465300\pi\)
0.108797 + 0.994064i \(0.465300\pi\)
\(464\) 0 0
\(465\) 10.9191 0.506359
\(466\) 0 0
\(467\) −42.1173 −1.94896 −0.974479 0.224478i \(-0.927932\pi\)
−0.974479 + 0.224478i \(0.927932\pi\)
\(468\) 0 0
\(469\) 4.80395 0.221826
\(470\) 0 0
\(471\) 7.49023 0.345131
\(472\) 0 0
\(473\) 0.253168 0.0116407
\(474\) 0 0
\(475\) 3.48388 0.159851
\(476\) 0 0
\(477\) −6.34955 −0.290726
\(478\) 0 0
\(479\) 27.5051 1.25674 0.628371 0.777914i \(-0.283722\pi\)
0.628371 + 0.777914i \(0.283722\pi\)
\(480\) 0 0
\(481\) 13.5368 0.617225
\(482\) 0 0
\(483\) −59.7549 −2.71894
\(484\) 0 0
\(485\) 13.3132 0.604522
\(486\) 0 0
\(487\) 30.3512 1.37535 0.687673 0.726020i \(-0.258632\pi\)
0.687673 + 0.726020i \(0.258632\pi\)
\(488\) 0 0
\(489\) 30.2560 1.36822
\(490\) 0 0
\(491\) −4.73478 −0.213677 −0.106839 0.994276i \(-0.534073\pi\)
−0.106839 + 0.994276i \(0.534073\pi\)
\(492\) 0 0
\(493\) 15.0262 0.676746
\(494\) 0 0
\(495\) −0.206728 −0.00929173
\(496\) 0 0
\(497\) −14.0676 −0.631019
\(498\) 0 0
\(499\) −40.2043 −1.79979 −0.899896 0.436104i \(-0.856358\pi\)
−0.899896 + 0.436104i \(0.856358\pi\)
\(500\) 0 0
\(501\) −9.79985 −0.437825
\(502\) 0 0
\(503\) 34.3196 1.53024 0.765118 0.643890i \(-0.222681\pi\)
0.765118 + 0.643890i \(0.222681\pi\)
\(504\) 0 0
\(505\) 3.41251 0.151855
\(506\) 0 0
\(507\) −21.7569 −0.966257
\(508\) 0 0
\(509\) 17.4105 0.771707 0.385854 0.922560i \(-0.373907\pi\)
0.385854 + 0.922560i \(0.373907\pi\)
\(510\) 0 0
\(511\) 83.6730 3.70148
\(512\) 0 0
\(513\) 5.83583 0.257658
\(514\) 0 0
\(515\) −13.6336 −0.600767
\(516\) 0 0
\(517\) −0.750369 −0.0330012
\(518\) 0 0
\(519\) −12.7665 −0.560386
\(520\) 0 0
\(521\) 1.03291 0.0452526 0.0226263 0.999744i \(-0.492797\pi\)
0.0226263 + 0.999744i \(0.492797\pi\)
\(522\) 0 0
\(523\) 13.5864 0.594091 0.297046 0.954863i \(-0.403999\pi\)
0.297046 + 0.954863i \(0.403999\pi\)
\(524\) 0 0
\(525\) −11.4697 −0.500578
\(526\) 0 0
\(527\) −14.6706 −0.639059
\(528\) 0 0
\(529\) 4.14216 0.180094
\(530\) 0 0
\(531\) 11.2144 0.486663
\(532\) 0 0
\(533\) 35.4091 1.53374
\(534\) 0 0
\(535\) −9.22568 −0.398861
\(536\) 0 0
\(537\) 11.7622 0.507576
\(538\) 0 0
\(539\) −1.63546 −0.0704445
\(540\) 0 0
\(541\) −7.07734 −0.304279 −0.152139 0.988359i \(-0.548616\pi\)
−0.152139 + 0.988359i \(0.548616\pi\)
\(542\) 0 0
\(543\) 2.03877 0.0874920
\(544\) 0 0
\(545\) −2.17922 −0.0933476
\(546\) 0 0
\(547\) 8.10611 0.346592 0.173296 0.984870i \(-0.444558\pi\)
0.173296 + 0.984870i \(0.444558\pi\)
\(548\) 0 0
\(549\) −35.3461 −1.50854
\(550\) 0 0
\(551\) 16.9719 0.723029
\(552\) 0 0
\(553\) −28.1909 −1.19880
\(554\) 0 0
\(555\) −6.55488 −0.278239
\(556\) 0 0
\(557\) 41.4217 1.75509 0.877546 0.479492i \(-0.159179\pi\)
0.877546 + 0.479492i \(0.159179\pi\)
\(558\) 0 0
\(559\) −13.1817 −0.557527
\(560\) 0 0
\(561\) 0.644775 0.0272224
\(562\) 0 0
\(563\) −28.2371 −1.19005 −0.595025 0.803707i \(-0.702858\pi\)
−0.595025 + 0.803707i \(0.702858\pi\)
\(564\) 0 0
\(565\) −13.8473 −0.582560
\(566\) 0 0
\(567\) −53.2414 −2.23593
\(568\) 0 0
\(569\) 12.7009 0.532450 0.266225 0.963911i \(-0.414224\pi\)
0.266225 + 0.963911i \(0.414224\pi\)
\(570\) 0 0
\(571\) −4.73768 −0.198266 −0.0991330 0.995074i \(-0.531607\pi\)
−0.0991330 + 0.995074i \(0.531607\pi\)
\(572\) 0 0
\(573\) 28.7794 1.20228
\(574\) 0 0
\(575\) 5.20981 0.217264
\(576\) 0 0
\(577\) −22.4676 −0.935339 −0.467669 0.883903i \(-0.654906\pi\)
−0.467669 + 0.883903i \(0.654906\pi\)
\(578\) 0 0
\(579\) 4.86969 0.202378
\(580\) 0 0
\(581\) −35.9977 −1.49344
\(582\) 0 0
\(583\) 0.254660 0.0105469
\(584\) 0 0
\(585\) 10.7637 0.445024
\(586\) 0 0
\(587\) 1.93739 0.0799646 0.0399823 0.999200i \(-0.487270\pi\)
0.0399823 + 0.999200i \(0.487270\pi\)
\(588\) 0 0
\(589\) −16.5702 −0.682765
\(590\) 0 0
\(591\) −11.3066 −0.465090
\(592\) 0 0
\(593\) −25.4894 −1.04672 −0.523361 0.852111i \(-0.675322\pi\)
−0.523361 + 0.852111i \(0.675322\pi\)
\(594\) 0 0
\(595\) 15.4104 0.631763
\(596\) 0 0
\(597\) 45.3881 1.85761
\(598\) 0 0
\(599\) 1.44120 0.0588859 0.0294429 0.999566i \(-0.490627\pi\)
0.0294429 + 0.999566i \(0.490627\pi\)
\(600\) 0 0
\(601\) 16.5905 0.676741 0.338370 0.941013i \(-0.390124\pi\)
0.338370 + 0.941013i \(0.390124\pi\)
\(602\) 0 0
\(603\) 2.18302 0.0888995
\(604\) 0 0
\(605\) −10.9917 −0.446877
\(606\) 0 0
\(607\) 23.8372 0.967521 0.483761 0.875200i \(-0.339271\pi\)
0.483761 + 0.875200i \(0.339271\pi\)
\(608\) 0 0
\(609\) −55.8753 −2.26418
\(610\) 0 0
\(611\) 39.0694 1.58058
\(612\) 0 0
\(613\) −23.9049 −0.965510 −0.482755 0.875756i \(-0.660364\pi\)
−0.482755 + 0.875756i \(0.660364\pi\)
\(614\) 0 0
\(615\) −17.1461 −0.691396
\(616\) 0 0
\(617\) 23.8968 0.962050 0.481025 0.876707i \(-0.340265\pi\)
0.481025 + 0.876707i \(0.340265\pi\)
\(618\) 0 0
\(619\) 31.3190 1.25882 0.629409 0.777074i \(-0.283297\pi\)
0.629409 + 0.777074i \(0.283297\pi\)
\(620\) 0 0
\(621\) 8.72694 0.350200
\(622\) 0 0
\(623\) 12.9644 0.519409
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0.728267 0.0290842
\(628\) 0 0
\(629\) 8.80696 0.351157
\(630\) 0 0
\(631\) 32.3148 1.28643 0.643215 0.765685i \(-0.277600\pi\)
0.643215 + 0.765685i \(0.277600\pi\)
\(632\) 0 0
\(633\) −40.7282 −1.61880
\(634\) 0 0
\(635\) −10.3742 −0.411686
\(636\) 0 0
\(637\) 85.1537 3.37391
\(638\) 0 0
\(639\) −6.39263 −0.252889
\(640\) 0 0
\(641\) −35.8511 −1.41603 −0.708016 0.706197i \(-0.750409\pi\)
−0.708016 + 0.706197i \(0.750409\pi\)
\(642\) 0 0
\(643\) 28.3341 1.11739 0.558695 0.829373i \(-0.311302\pi\)
0.558695 + 0.829373i \(0.311302\pi\)
\(644\) 0 0
\(645\) 6.38293 0.251328
\(646\) 0 0
\(647\) −41.4329 −1.62890 −0.814448 0.580236i \(-0.802960\pi\)
−0.814448 + 0.580236i \(0.802960\pi\)
\(648\) 0 0
\(649\) −0.449772 −0.0176551
\(650\) 0 0
\(651\) 54.5528 2.13809
\(652\) 0 0
\(653\) 26.7251 1.04583 0.522916 0.852384i \(-0.324844\pi\)
0.522916 + 0.852384i \(0.324844\pi\)
\(654\) 0 0
\(655\) −16.0953 −0.628896
\(656\) 0 0
\(657\) 38.0228 1.48341
\(658\) 0 0
\(659\) 38.2412 1.48967 0.744833 0.667251i \(-0.232529\pi\)
0.744833 + 0.667251i \(0.232529\pi\)
\(660\) 0 0
\(661\) −25.4182 −0.988653 −0.494327 0.869276i \(-0.664585\pi\)
−0.494327 + 0.869276i \(0.664585\pi\)
\(662\) 0 0
\(663\) −33.5715 −1.30381
\(664\) 0 0
\(665\) 17.4058 0.674970
\(666\) 0 0
\(667\) 25.3799 0.982715
\(668\) 0 0
\(669\) 52.8948 2.04503
\(670\) 0 0
\(671\) 1.41762 0.0547265
\(672\) 0 0
\(673\) −16.3957 −0.632009 −0.316004 0.948758i \(-0.602341\pi\)
−0.316004 + 0.948758i \(0.602341\pi\)
\(674\) 0 0
\(675\) 1.67510 0.0644745
\(676\) 0 0
\(677\) −22.4707 −0.863621 −0.431810 0.901964i \(-0.642125\pi\)
−0.431810 + 0.901964i \(0.642125\pi\)
\(678\) 0 0
\(679\) 66.5143 2.55259
\(680\) 0 0
\(681\) 38.2449 1.46555
\(682\) 0 0
\(683\) −15.9550 −0.610499 −0.305250 0.952272i \(-0.598740\pi\)
−0.305250 + 0.952272i \(0.598740\pi\)
\(684\) 0 0
\(685\) −6.40018 −0.244538
\(686\) 0 0
\(687\) 50.1137 1.91196
\(688\) 0 0
\(689\) −13.2594 −0.505141
\(690\) 0 0
\(691\) −39.6298 −1.50759 −0.753795 0.657110i \(-0.771779\pi\)
−0.753795 + 0.657110i \(0.771779\pi\)
\(692\) 0 0
\(693\) −1.03284 −0.0392342
\(694\) 0 0
\(695\) −15.6231 −0.592618
\(696\) 0 0
\(697\) 23.0370 0.872588
\(698\) 0 0
\(699\) 5.56506 0.210490
\(700\) 0 0
\(701\) 13.6715 0.516367 0.258184 0.966096i \(-0.416876\pi\)
0.258184 + 0.966096i \(0.416876\pi\)
\(702\) 0 0
\(703\) 9.94738 0.375172
\(704\) 0 0
\(705\) −18.9185 −0.712511
\(706\) 0 0
\(707\) 17.0493 0.641203
\(708\) 0 0
\(709\) −16.4047 −0.616093 −0.308046 0.951371i \(-0.599675\pi\)
−0.308046 + 0.951371i \(0.599675\pi\)
\(710\) 0 0
\(711\) −12.8106 −0.480433
\(712\) 0 0
\(713\) −24.7792 −0.927990
\(714\) 0 0
\(715\) −0.431697 −0.0161445
\(716\) 0 0
\(717\) −23.1201 −0.863435
\(718\) 0 0
\(719\) −40.6607 −1.51639 −0.758195 0.652028i \(-0.773918\pi\)
−0.758195 + 0.652028i \(0.773918\pi\)
\(720\) 0 0
\(721\) −68.1148 −2.53673
\(722\) 0 0
\(723\) −7.65199 −0.284580
\(724\) 0 0
\(725\) 4.87156 0.180925
\(726\) 0 0
\(727\) −13.5670 −0.503172 −0.251586 0.967835i \(-0.580952\pi\)
−0.251586 + 0.967835i \(0.580952\pi\)
\(728\) 0 0
\(729\) −12.6574 −0.468792
\(730\) 0 0
\(731\) −8.57594 −0.317193
\(732\) 0 0
\(733\) 5.19748 0.191973 0.0959866 0.995383i \(-0.469399\pi\)
0.0959866 + 0.995383i \(0.469399\pi\)
\(734\) 0 0
\(735\) −41.2337 −1.52093
\(736\) 0 0
\(737\) −0.0875538 −0.00322508
\(738\) 0 0
\(739\) −48.8323 −1.79633 −0.898163 0.439663i \(-0.855098\pi\)
−0.898163 + 0.439663i \(0.855098\pi\)
\(740\) 0 0
\(741\) −37.9187 −1.39298
\(742\) 0 0
\(743\) 34.4597 1.26420 0.632101 0.774886i \(-0.282193\pi\)
0.632101 + 0.774886i \(0.282193\pi\)
\(744\) 0 0
\(745\) 3.99341 0.146307
\(746\) 0 0
\(747\) −16.3581 −0.598512
\(748\) 0 0
\(749\) −46.0925 −1.68418
\(750\) 0 0
\(751\) −27.6104 −1.00752 −0.503758 0.863845i \(-0.668050\pi\)
−0.503758 + 0.863845i \(0.668050\pi\)
\(752\) 0 0
\(753\) −32.5125 −1.18482
\(754\) 0 0
\(755\) 1.00000 0.0363937
\(756\) 0 0
\(757\) −7.67066 −0.278795 −0.139397 0.990237i \(-0.544517\pi\)
−0.139397 + 0.990237i \(0.544517\pi\)
\(758\) 0 0
\(759\) 1.08905 0.0395302
\(760\) 0 0
\(761\) −24.0784 −0.872840 −0.436420 0.899743i \(-0.643754\pi\)
−0.436420 + 0.899743i \(0.643754\pi\)
\(762\) 0 0
\(763\) −10.8876 −0.394159
\(764\) 0 0
\(765\) 7.00280 0.253187
\(766\) 0 0
\(767\) 23.4183 0.845585
\(768\) 0 0
\(769\) −21.3074 −0.768363 −0.384182 0.923258i \(-0.625516\pi\)
−0.384182 + 0.923258i \(0.625516\pi\)
\(770\) 0 0
\(771\) −12.3609 −0.445166
\(772\) 0 0
\(773\) −22.3680 −0.804522 −0.402261 0.915525i \(-0.631776\pi\)
−0.402261 + 0.915525i \(0.631776\pi\)
\(774\) 0 0
\(775\) −4.75626 −0.170850
\(776\) 0 0
\(777\) −32.7489 −1.17486
\(778\) 0 0
\(779\) 26.0200 0.932265
\(780\) 0 0
\(781\) 0.256387 0.00917426
\(782\) 0 0
\(783\) 8.16033 0.291627
\(784\) 0 0
\(785\) −3.26269 −0.116450
\(786\) 0 0
\(787\) −11.8576 −0.422678 −0.211339 0.977413i \(-0.567782\pi\)
−0.211339 + 0.977413i \(0.567782\pi\)
\(788\) 0 0
\(789\) −35.9812 −1.28097
\(790\) 0 0
\(791\) −69.1826 −2.45985
\(792\) 0 0
\(793\) −73.8110 −2.62111
\(794\) 0 0
\(795\) 6.42054 0.227713
\(796\) 0 0
\(797\) −12.8401 −0.454821 −0.227410 0.973799i \(-0.573026\pi\)
−0.227410 + 0.973799i \(0.573026\pi\)
\(798\) 0 0
\(799\) 25.4184 0.899237
\(800\) 0 0
\(801\) 5.89132 0.208160
\(802\) 0 0
\(803\) −1.52497 −0.0538150
\(804\) 0 0
\(805\) 26.0288 0.917395
\(806\) 0 0
\(807\) −50.3455 −1.77224
\(808\) 0 0
\(809\) 53.9336 1.89621 0.948103 0.317964i \(-0.102999\pi\)
0.948103 + 0.317964i \(0.102999\pi\)
\(810\) 0 0
\(811\) 44.9370 1.57795 0.788975 0.614425i \(-0.210612\pi\)
0.788975 + 0.614425i \(0.210612\pi\)
\(812\) 0 0
\(813\) 3.33509 0.116967
\(814\) 0 0
\(815\) −13.1793 −0.461651
\(816\) 0 0
\(817\) −9.68644 −0.338886
\(818\) 0 0
\(819\) 53.7766 1.87911
\(820\) 0 0
\(821\) −48.7495 −1.70137 −0.850685 0.525675i \(-0.823813\pi\)
−0.850685 + 0.525675i \(0.823813\pi\)
\(822\) 0 0
\(823\) −35.3637 −1.23270 −0.616350 0.787472i \(-0.711389\pi\)
−0.616350 + 0.787472i \(0.711389\pi\)
\(824\) 0 0
\(825\) 0.209039 0.00727781
\(826\) 0 0
\(827\) −39.7094 −1.38083 −0.690416 0.723412i \(-0.742573\pi\)
−0.690416 + 0.723412i \(0.742573\pi\)
\(828\) 0 0
\(829\) 17.8533 0.620072 0.310036 0.950725i \(-0.399659\pi\)
0.310036 + 0.950725i \(0.399659\pi\)
\(830\) 0 0
\(831\) 49.0005 1.69981
\(832\) 0 0
\(833\) 55.4005 1.91951
\(834\) 0 0
\(835\) 4.26874 0.147726
\(836\) 0 0
\(837\) −7.96719 −0.275386
\(838\) 0 0
\(839\) 15.2648 0.527000 0.263500 0.964659i \(-0.415123\pi\)
0.263500 + 0.964659i \(0.415123\pi\)
\(840\) 0 0
\(841\) −5.26788 −0.181651
\(842\) 0 0
\(843\) −42.9719 −1.48003
\(844\) 0 0
\(845\) 9.47714 0.326024
\(846\) 0 0
\(847\) −54.9158 −1.88693
\(848\) 0 0
\(849\) −21.5746 −0.740438
\(850\) 0 0
\(851\) 14.8754 0.509921
\(852\) 0 0
\(853\) −52.3822 −1.79353 −0.896766 0.442506i \(-0.854090\pi\)
−0.896766 + 0.442506i \(0.854090\pi\)
\(854\) 0 0
\(855\) 7.90959 0.270502
\(856\) 0 0
\(857\) 5.09873 0.174169 0.0870847 0.996201i \(-0.472245\pi\)
0.0870847 + 0.996201i \(0.472245\pi\)
\(858\) 0 0
\(859\) 23.9449 0.816988 0.408494 0.912761i \(-0.366054\pi\)
0.408494 + 0.912761i \(0.366054\pi\)
\(860\) 0 0
\(861\) −85.6636 −2.91941
\(862\) 0 0
\(863\) −37.6116 −1.28031 −0.640157 0.768244i \(-0.721131\pi\)
−0.640157 + 0.768244i \(0.721131\pi\)
\(864\) 0 0
\(865\) 5.56099 0.189079
\(866\) 0 0
\(867\) 17.1859 0.583662
\(868\) 0 0
\(869\) 0.513789 0.0174291
\(870\) 0 0
\(871\) 4.55866 0.154464
\(872\) 0 0
\(873\) 30.2256 1.02298
\(874\) 0 0
\(875\) 4.99611 0.168899
\(876\) 0 0
\(877\) −20.6925 −0.698737 −0.349369 0.936985i \(-0.613604\pi\)
−0.349369 + 0.936985i \(0.613604\pi\)
\(878\) 0 0
\(879\) 10.8245 0.365100
\(880\) 0 0
\(881\) 17.2843 0.582324 0.291162 0.956674i \(-0.405958\pi\)
0.291162 + 0.956674i \(0.405958\pi\)
\(882\) 0 0
\(883\) −21.1431 −0.711523 −0.355761 0.934577i \(-0.615778\pi\)
−0.355761 + 0.934577i \(0.615778\pi\)
\(884\) 0 0
\(885\) −11.3398 −0.381182
\(886\) 0 0
\(887\) 21.0783 0.707739 0.353869 0.935295i \(-0.384866\pi\)
0.353869 + 0.935295i \(0.384866\pi\)
\(888\) 0 0
\(889\) −51.8304 −1.73834
\(890\) 0 0
\(891\) 0.970344 0.0325077
\(892\) 0 0
\(893\) 28.7098 0.960736
\(894\) 0 0
\(895\) −5.12353 −0.171261
\(896\) 0 0
\(897\) −56.7038 −1.89328
\(898\) 0 0
\(899\) −23.1704 −0.772777
\(900\) 0 0
\(901\) −8.62647 −0.287389
\(902\) 0 0
\(903\) 31.8898 1.06123
\(904\) 0 0
\(905\) −0.888073 −0.0295206
\(906\) 0 0
\(907\) 19.3399 0.642172 0.321086 0.947050i \(-0.395952\pi\)
0.321086 + 0.947050i \(0.395952\pi\)
\(908\) 0 0
\(909\) 7.74755 0.256970
\(910\) 0 0
\(911\) 34.0223 1.12721 0.563604 0.826045i \(-0.309414\pi\)
0.563604 + 0.826045i \(0.309414\pi\)
\(912\) 0 0
\(913\) 0.656071 0.0217128
\(914\) 0 0
\(915\) 35.7413 1.18157
\(916\) 0 0
\(917\) −80.4140 −2.65550
\(918\) 0 0
\(919\) 38.2942 1.26321 0.631604 0.775291i \(-0.282397\pi\)
0.631604 + 0.775291i \(0.282397\pi\)
\(920\) 0 0
\(921\) 73.6733 2.42762
\(922\) 0 0
\(923\) −13.3493 −0.439398
\(924\) 0 0
\(925\) 2.85526 0.0938803
\(926\) 0 0
\(927\) −30.9528 −1.01662
\(928\) 0 0
\(929\) −30.5559 −1.00251 −0.501253 0.865301i \(-0.667127\pi\)
−0.501253 + 0.865301i \(0.667127\pi\)
\(930\) 0 0
\(931\) 62.5743 2.05079
\(932\) 0 0
\(933\) −67.8650 −2.22180
\(934\) 0 0
\(935\) −0.280859 −0.00918508
\(936\) 0 0
\(937\) 30.1420 0.984696 0.492348 0.870398i \(-0.336139\pi\)
0.492348 + 0.870398i \(0.336139\pi\)
\(938\) 0 0
\(939\) −48.9407 −1.59712
\(940\) 0 0
\(941\) 18.4882 0.602697 0.301348 0.953514i \(-0.402563\pi\)
0.301348 + 0.953514i \(0.402563\pi\)
\(942\) 0 0
\(943\) 38.9105 1.26710
\(944\) 0 0
\(945\) 8.36896 0.272242
\(946\) 0 0
\(947\) 29.5051 0.958788 0.479394 0.877600i \(-0.340856\pi\)
0.479394 + 0.877600i \(0.340856\pi\)
\(948\) 0 0
\(949\) 79.4006 2.57745
\(950\) 0 0
\(951\) 45.3421 1.47032
\(952\) 0 0
\(953\) −30.9527 −1.00266 −0.501329 0.865257i \(-0.667155\pi\)
−0.501329 + 0.865257i \(0.667155\pi\)
\(954\) 0 0
\(955\) −12.5361 −0.405659
\(956\) 0 0
\(957\) 1.01835 0.0329185
\(958\) 0 0
\(959\) −31.9760 −1.03256
\(960\) 0 0
\(961\) −8.37798 −0.270257
\(962\) 0 0
\(963\) −20.9454 −0.674957
\(964\) 0 0
\(965\) −2.12120 −0.0682840
\(966\) 0 0
\(967\) 52.8683 1.70013 0.850065 0.526678i \(-0.176563\pi\)
0.850065 + 0.526678i \(0.176563\pi\)
\(968\) 0 0
\(969\) −24.6697 −0.792504
\(970\) 0 0
\(971\) 43.8004 1.40562 0.702811 0.711376i \(-0.251928\pi\)
0.702811 + 0.711376i \(0.251928\pi\)
\(972\) 0 0
\(973\) −78.0548 −2.50232
\(974\) 0 0
\(975\) −10.8840 −0.348568
\(976\) 0 0
\(977\) 10.0987 0.323085 0.161542 0.986866i \(-0.448353\pi\)
0.161542 + 0.986866i \(0.448353\pi\)
\(978\) 0 0
\(979\) −0.236282 −0.00755159
\(980\) 0 0
\(981\) −4.94757 −0.157964
\(982\) 0 0
\(983\) 0.469222 0.0149658 0.00748292 0.999972i \(-0.497618\pi\)
0.00748292 + 0.999972i \(0.497618\pi\)
\(984\) 0 0
\(985\) 4.92506 0.156925
\(986\) 0 0
\(987\) −94.5188 −3.00857
\(988\) 0 0
\(989\) −14.4852 −0.460601
\(990\) 0 0
\(991\) −38.2793 −1.21598 −0.607992 0.793943i \(-0.708025\pi\)
−0.607992 + 0.793943i \(0.708025\pi\)
\(992\) 0 0
\(993\) −53.4485 −1.69614
\(994\) 0 0
\(995\) −19.7707 −0.626774
\(996\) 0 0
\(997\) −29.4094 −0.931405 −0.465703 0.884941i \(-0.654198\pi\)
−0.465703 + 0.884941i \(0.654198\pi\)
\(998\) 0 0
\(999\) 4.78283 0.151322
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 6040.2.a.s.1.6 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6040.2.a.s.1.6 24 1.1 even 1 trivial