Properties

Label 2496.4.a.s.1.2
Level $2496$
Weight $4$
Character 2496.1
Self dual yes
Analytic conductor $147.269$
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] = [2496,4,Mod(1,2496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2496, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2496.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2496.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: \(147.268767374\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.74166\) of defining polynomial
Character \(\chi\) \(=\) 2496.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -4.51669 q^{5} +7.48331 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -4.51669 q^{5} +7.48331 q^{7} +9.00000 q^{9} -66.8999 q^{11} +13.0000 q^{13} +13.5501 q^{15} +96.9666 q^{17} +31.4833 q^{19} -22.4499 q^{21} -183.600 q^{23} -104.600 q^{25} -27.0000 q^{27} -112.200 q^{29} +77.2831 q^{31} +200.700 q^{33} -33.7998 q^{35} -54.7664 q^{37} -39.0000 q^{39} +451.716 q^{41} -113.434 q^{43} -40.6502 q^{45} +42.2670 q^{47} -287.000 q^{49} -290.900 q^{51} +530.999 q^{53} +302.166 q^{55} -94.4499 q^{57} +219.666 q^{59} -822.865 q^{61} +67.3498 q^{63} -58.7169 q^{65} -872.082 q^{67} +550.799 q^{69} +100.299 q^{71} -165.634 q^{73} +313.799 q^{75} -500.633 q^{77} +545.266 q^{79} +81.0000 q^{81} -454.534 q^{83} -437.968 q^{85} +336.601 q^{87} -230.915 q^{89} +97.2831 q^{91} -231.849 q^{93} -142.200 q^{95} -1089.16 q^{97} -602.099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 24 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 24 q^{5} + 18 q^{9} - 44 q^{11} + 26 q^{13} + 72 q^{15} + 164 q^{17} + 48 q^{19} - 8 q^{23} + 150 q^{25} - 54 q^{27} - 404 q^{29} - 40 q^{31} + 132 q^{33} + 112 q^{35} + 100 q^{37} - 78 q^{39} + 200 q^{41} - 616 q^{43} - 216 q^{45} + 324 q^{47} - 574 q^{49} - 492 q^{51} + 164 q^{53} - 144 q^{55} - 144 q^{57} + 140 q^{59} - 628 q^{61} - 312 q^{65} - 472 q^{67} + 24 q^{69} - 428 q^{71} - 900 q^{73} - 450 q^{75} - 672 q^{77} + 432 q^{79} + 162 q^{81} - 1388 q^{83} - 1744 q^{85} + 1212 q^{87} + 960 q^{89} + 120 q^{93} - 464 q^{95} - 532 q^{97} - 396 q^{99}+O(q^{100}) \) 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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −4.51669 −0.403985 −0.201992 0.979387i \(-0.564742\pi\)
−0.201992 + 0.979387i \(0.564742\pi\)
\(6\) 0 0
\(7\) 7.48331 0.404061 0.202031 0.979379i \(-0.435246\pi\)
0.202031 + 0.979379i \(0.435246\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −66.8999 −1.83373 −0.916867 0.399193i \(-0.869290\pi\)
−0.916867 + 0.399193i \(0.869290\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) 13.5501 0.233241
\(16\) 0 0
\(17\) 96.9666 1.38340 0.691702 0.722183i \(-0.256861\pi\)
0.691702 + 0.722183i \(0.256861\pi\)
\(18\) 0 0
\(19\) 31.4833 0.380146 0.190073 0.981770i \(-0.439128\pi\)
0.190073 + 0.981770i \(0.439128\pi\)
\(20\) 0 0
\(21\) −22.4499 −0.233285
\(22\) 0 0
\(23\) −183.600 −1.66448 −0.832242 0.554412i \(-0.812943\pi\)
−0.832242 + 0.554412i \(0.812943\pi\)
\(24\) 0 0
\(25\) −104.600 −0.836796
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −112.200 −0.718450 −0.359225 0.933251i \(-0.616959\pi\)
−0.359225 + 0.933251i \(0.616959\pi\)
\(30\) 0 0
\(31\) 77.2831 0.447757 0.223878 0.974617i \(-0.428128\pi\)
0.223878 + 0.974617i \(0.428128\pi\)
\(32\) 0 0
\(33\) 200.700 1.05871
\(34\) 0 0
\(35\) −33.7998 −0.163234
\(36\) 0 0
\(37\) −54.7664 −0.243339 −0.121669 0.992571i \(-0.538825\pi\)
−0.121669 + 0.992571i \(0.538825\pi\)
\(38\) 0 0
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) 451.716 1.72064 0.860319 0.509756i \(-0.170264\pi\)
0.860319 + 0.509756i \(0.170264\pi\)
\(42\) 0 0
\(43\) −113.434 −0.402291 −0.201145 0.979561i \(-0.564466\pi\)
−0.201145 + 0.979561i \(0.564466\pi\)
\(44\) 0 0
\(45\) −40.6502 −0.134662
\(46\) 0 0
\(47\) 42.2670 0.131176 0.0655880 0.997847i \(-0.479108\pi\)
0.0655880 + 0.997847i \(0.479108\pi\)
\(48\) 0 0
\(49\) −287.000 −0.836735
\(50\) 0 0
\(51\) −290.900 −0.798708
\(52\) 0 0
\(53\) 530.999 1.37619 0.688097 0.725618i \(-0.258446\pi\)
0.688097 + 0.725618i \(0.258446\pi\)
\(54\) 0 0
\(55\) 302.166 0.740800
\(56\) 0 0
\(57\) −94.4499 −0.219477
\(58\) 0 0
\(59\) 219.666 0.484714 0.242357 0.970187i \(-0.422080\pi\)
0.242357 + 0.970187i \(0.422080\pi\)
\(60\) 0 0
\(61\) −822.865 −1.72717 −0.863583 0.504207i \(-0.831785\pi\)
−0.863583 + 0.504207i \(0.831785\pi\)
\(62\) 0 0
\(63\) 67.3498 0.134687
\(64\) 0 0
\(65\) −58.7169 −0.112045
\(66\) 0 0
\(67\) −872.082 −1.59018 −0.795088 0.606495i \(-0.792575\pi\)
−0.795088 + 0.606495i \(0.792575\pi\)
\(68\) 0 0
\(69\) 550.799 0.960991
\(70\) 0 0
\(71\) 100.299 0.167653 0.0838263 0.996480i \(-0.473286\pi\)
0.0838263 + 0.996480i \(0.473286\pi\)
\(72\) 0 0
\(73\) −165.634 −0.265562 −0.132781 0.991145i \(-0.542391\pi\)
−0.132781 + 0.991145i \(0.542391\pi\)
\(74\) 0 0
\(75\) 313.799 0.483125
\(76\) 0 0
\(77\) −500.633 −0.740940
\(78\) 0 0
\(79\) 545.266 0.776547 0.388273 0.921544i \(-0.373072\pi\)
0.388273 + 0.921544i \(0.373072\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −454.534 −0.601103 −0.300552 0.953766i \(-0.597171\pi\)
−0.300552 + 0.953766i \(0.597171\pi\)
\(84\) 0 0
\(85\) −437.968 −0.558874
\(86\) 0 0
\(87\) 336.601 0.414797
\(88\) 0 0
\(89\) −230.915 −0.275022 −0.137511 0.990500i \(-0.543910\pi\)
−0.137511 + 0.990500i \(0.543910\pi\)
\(90\) 0 0
\(91\) 97.2831 0.112066
\(92\) 0 0
\(93\) −231.849 −0.258512
\(94\) 0 0
\(95\) −142.200 −0.153573
\(96\) 0 0
\(97\) −1089.16 −1.14008 −0.570041 0.821616i \(-0.693073\pi\)
−0.570041 + 0.821616i \(0.693073\pi\)
\(98\) 0 0
\(99\) −602.099 −0.611245
\(100\) 0 0
\(101\) 77.2336 0.0760894 0.0380447 0.999276i \(-0.487887\pi\)
0.0380447 + 0.999276i \(0.487887\pi\)
\(102\) 0 0
\(103\) −1351.36 −1.29275 −0.646377 0.763018i \(-0.723717\pi\)
−0.646377 + 0.763018i \(0.723717\pi\)
\(104\) 0 0
\(105\) 101.399 0.0942434
\(106\) 0 0
\(107\) 1133.67 1.02426 0.512129 0.858908i \(-0.328857\pi\)
0.512129 + 0.858908i \(0.328857\pi\)
\(108\) 0 0
\(109\) −1017.66 −0.894262 −0.447131 0.894469i \(-0.647554\pi\)
−0.447131 + 0.894469i \(0.647554\pi\)
\(110\) 0 0
\(111\) 164.299 0.140492
\(112\) 0 0
\(113\) 1570.06 1.30707 0.653536 0.756895i \(-0.273285\pi\)
0.653536 + 0.756895i \(0.273285\pi\)
\(114\) 0 0
\(115\) 829.261 0.672426
\(116\) 0 0
\(117\) 117.000 0.0924500
\(118\) 0 0
\(119\) 725.632 0.558979
\(120\) 0 0
\(121\) 3144.60 2.36258
\(122\) 0 0
\(123\) −1355.15 −0.993411
\(124\) 0 0
\(125\) 1037.03 0.742037
\(126\) 0 0
\(127\) 1248.16 0.872099 0.436050 0.899923i \(-0.356377\pi\)
0.436050 + 0.899923i \(0.356377\pi\)
\(128\) 0 0
\(129\) 340.301 0.232263
\(130\) 0 0
\(131\) −1274.80 −0.850227 −0.425113 0.905140i \(-0.639766\pi\)
−0.425113 + 0.905140i \(0.639766\pi\)
\(132\) 0 0
\(133\) 235.600 0.153602
\(134\) 0 0
\(135\) 121.951 0.0777469
\(136\) 0 0
\(137\) 874.915 0.545613 0.272807 0.962069i \(-0.412048\pi\)
0.272807 + 0.962069i \(0.412048\pi\)
\(138\) 0 0
\(139\) 310.334 0.189368 0.0946840 0.995507i \(-0.469816\pi\)
0.0946840 + 0.995507i \(0.469816\pi\)
\(140\) 0 0
\(141\) −126.801 −0.0757345
\(142\) 0 0
\(143\) −869.699 −0.508586
\(144\) 0 0
\(145\) 506.773 0.290243
\(146\) 0 0
\(147\) 861.000 0.483089
\(148\) 0 0
\(149\) −5.08064 −0.00279344 −0.00139672 0.999999i \(-0.500445\pi\)
−0.00139672 + 0.999999i \(0.500445\pi\)
\(150\) 0 0
\(151\) −6.54894 −0.00352944 −0.00176472 0.999998i \(-0.500562\pi\)
−0.00176472 + 0.999998i \(0.500562\pi\)
\(152\) 0 0
\(153\) 872.700 0.461135
\(154\) 0 0
\(155\) −349.063 −0.180887
\(156\) 0 0
\(157\) 2297.60 1.16795 0.583975 0.811772i \(-0.301497\pi\)
0.583975 + 0.811772i \(0.301497\pi\)
\(158\) 0 0
\(159\) −1593.00 −0.794546
\(160\) 0 0
\(161\) −1373.93 −0.672553
\(162\) 0 0
\(163\) −1085.49 −0.521606 −0.260803 0.965392i \(-0.583987\pi\)
−0.260803 + 0.965392i \(0.583987\pi\)
\(164\) 0 0
\(165\) −906.497 −0.427701
\(166\) 0 0
\(167\) 109.066 0.0505374 0.0252687 0.999681i \(-0.491956\pi\)
0.0252687 + 0.999681i \(0.491956\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) 283.350 0.126715
\(172\) 0 0
\(173\) 1889.17 0.830236 0.415118 0.909768i \(-0.363740\pi\)
0.415118 + 0.909768i \(0.363740\pi\)
\(174\) 0 0
\(175\) −782.751 −0.338117
\(176\) 0 0
\(177\) −658.999 −0.279850
\(178\) 0 0
\(179\) 3427.86 1.43134 0.715672 0.698437i \(-0.246121\pi\)
0.715672 + 0.698437i \(0.246121\pi\)
\(180\) 0 0
\(181\) −208.403 −0.0855826 −0.0427913 0.999084i \(-0.513625\pi\)
−0.0427913 + 0.999084i \(0.513625\pi\)
\(182\) 0 0
\(183\) 2468.60 0.997180
\(184\) 0 0
\(185\) 247.363 0.0983052
\(186\) 0 0
\(187\) −6487.06 −2.53679
\(188\) 0 0
\(189\) −202.049 −0.0777616
\(190\) 0 0
\(191\) −957.735 −0.362824 −0.181412 0.983407i \(-0.558067\pi\)
−0.181412 + 0.983407i \(0.558067\pi\)
\(192\) 0 0
\(193\) −512.730 −0.191228 −0.0956142 0.995418i \(-0.530482\pi\)
−0.0956142 + 0.995418i \(0.530482\pi\)
\(194\) 0 0
\(195\) 176.151 0.0646893
\(196\) 0 0
\(197\) 3870.35 1.39975 0.699876 0.714265i \(-0.253239\pi\)
0.699876 + 0.714265i \(0.253239\pi\)
\(198\) 0 0
\(199\) −2305.83 −0.821388 −0.410694 0.911773i \(-0.634714\pi\)
−0.410694 + 0.911773i \(0.634714\pi\)
\(200\) 0 0
\(201\) 2616.25 0.918088
\(202\) 0 0
\(203\) −839.630 −0.290298
\(204\) 0 0
\(205\) −2040.26 −0.695111
\(206\) 0 0
\(207\) −1652.40 −0.554828
\(208\) 0 0
\(209\) −2106.23 −0.697086
\(210\) 0 0
\(211\) −3672.40 −1.19819 −0.599096 0.800677i \(-0.704473\pi\)
−0.599096 + 0.800677i \(0.704473\pi\)
\(212\) 0 0
\(213\) −300.898 −0.0967942
\(214\) 0 0
\(215\) 512.345 0.162519
\(216\) 0 0
\(217\) 578.334 0.180921
\(218\) 0 0
\(219\) 496.902 0.153322
\(220\) 0 0
\(221\) 1260.57 0.383687
\(222\) 0 0
\(223\) −5087.05 −1.52760 −0.763798 0.645455i \(-0.776668\pi\)
−0.763798 + 0.645455i \(0.776668\pi\)
\(224\) 0 0
\(225\) −941.396 −0.278932
\(226\) 0 0
\(227\) −2625.83 −0.767763 −0.383882 0.923382i \(-0.625413\pi\)
−0.383882 + 0.923382i \(0.625413\pi\)
\(228\) 0 0
\(229\) −1678.73 −0.484425 −0.242213 0.970223i \(-0.577873\pi\)
−0.242213 + 0.970223i \(0.577873\pi\)
\(230\) 0 0
\(231\) 1501.90 0.427782
\(232\) 0 0
\(233\) 648.506 0.182339 0.0911696 0.995835i \(-0.470939\pi\)
0.0911696 + 0.995835i \(0.470939\pi\)
\(234\) 0 0
\(235\) −190.907 −0.0529931
\(236\) 0 0
\(237\) −1635.80 −0.448340
\(238\) 0 0
\(239\) −5219.69 −1.41269 −0.706347 0.707866i \(-0.749658\pi\)
−0.706347 + 0.707866i \(0.749658\pi\)
\(240\) 0 0
\(241\) 6103.56 1.63139 0.815695 0.578483i \(-0.196355\pi\)
0.815695 + 0.578483i \(0.196355\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 1296.29 0.338028
\(246\) 0 0
\(247\) 409.283 0.105433
\(248\) 0 0
\(249\) 1363.60 0.347047
\(250\) 0 0
\(251\) 6423.40 1.61530 0.807652 0.589660i \(-0.200738\pi\)
0.807652 + 0.589660i \(0.200738\pi\)
\(252\) 0 0
\(253\) 12282.8 3.05222
\(254\) 0 0
\(255\) 1313.90 0.322666
\(256\) 0 0
\(257\) −1230.23 −0.298597 −0.149299 0.988792i \(-0.547702\pi\)
−0.149299 + 0.988792i \(0.547702\pi\)
\(258\) 0 0
\(259\) −409.834 −0.0983238
\(260\) 0 0
\(261\) −1009.80 −0.239483
\(262\) 0 0
\(263\) 514.992 0.120744 0.0603722 0.998176i \(-0.480771\pi\)
0.0603722 + 0.998176i \(0.480771\pi\)
\(264\) 0 0
\(265\) −2398.35 −0.555961
\(266\) 0 0
\(267\) 692.745 0.158784
\(268\) 0 0
\(269\) 5132.60 1.16335 0.581673 0.813423i \(-0.302398\pi\)
0.581673 + 0.813423i \(0.302398\pi\)
\(270\) 0 0
\(271\) 4300.00 0.963862 0.481931 0.876209i \(-0.339936\pi\)
0.481931 + 0.876209i \(0.339936\pi\)
\(272\) 0 0
\(273\) −291.849 −0.0647015
\(274\) 0 0
\(275\) 6997.70 1.53446
\(276\) 0 0
\(277\) 1812.80 0.393215 0.196607 0.980482i \(-0.437008\pi\)
0.196607 + 0.980482i \(0.437008\pi\)
\(278\) 0 0
\(279\) 695.548 0.149252
\(280\) 0 0
\(281\) −4073.08 −0.864696 −0.432348 0.901707i \(-0.642315\pi\)
−0.432348 + 0.901707i \(0.642315\pi\)
\(282\) 0 0
\(283\) 6346.29 1.33303 0.666516 0.745491i \(-0.267785\pi\)
0.666516 + 0.745491i \(0.267785\pi\)
\(284\) 0 0
\(285\) 426.601 0.0886654
\(286\) 0 0
\(287\) 3380.33 0.695243
\(288\) 0 0
\(289\) 4489.53 0.913806
\(290\) 0 0
\(291\) 3267.49 0.658226
\(292\) 0 0
\(293\) 8390.97 1.67306 0.836529 0.547923i \(-0.184581\pi\)
0.836529 + 0.547923i \(0.184581\pi\)
\(294\) 0 0
\(295\) −992.164 −0.195817
\(296\) 0 0
\(297\) 1806.30 0.352902
\(298\) 0 0
\(299\) −2386.79 −0.461645
\(300\) 0 0
\(301\) −848.861 −0.162550
\(302\) 0 0
\(303\) −231.701 −0.0439302
\(304\) 0 0
\(305\) 3716.62 0.697748
\(306\) 0 0
\(307\) 4005.27 0.744603 0.372301 0.928112i \(-0.378569\pi\)
0.372301 + 0.928112i \(0.378569\pi\)
\(308\) 0 0
\(309\) 4054.09 0.746372
\(310\) 0 0
\(311\) −5836.53 −1.06418 −0.532088 0.846689i \(-0.678593\pi\)
−0.532088 + 0.846689i \(0.678593\pi\)
\(312\) 0 0
\(313\) 1763.19 0.318407 0.159204 0.987246i \(-0.449107\pi\)
0.159204 + 0.987246i \(0.449107\pi\)
\(314\) 0 0
\(315\) −304.198 −0.0544115
\(316\) 0 0
\(317\) 6106.35 1.08191 0.540957 0.841050i \(-0.318062\pi\)
0.540957 + 0.841050i \(0.318062\pi\)
\(318\) 0 0
\(319\) 7506.18 1.31745
\(320\) 0 0
\(321\) −3401.00 −0.591356
\(322\) 0 0
\(323\) 3052.83 0.525895
\(324\) 0 0
\(325\) −1359.79 −0.232086
\(326\) 0 0
\(327\) 3052.99 0.516302
\(328\) 0 0
\(329\) 316.297 0.0530031
\(330\) 0 0
\(331\) 7490.38 1.24383 0.621916 0.783084i \(-0.286354\pi\)
0.621916 + 0.783084i \(0.286354\pi\)
\(332\) 0 0
\(333\) −492.898 −0.0811130
\(334\) 0 0
\(335\) 3938.92 0.642406
\(336\) 0 0
\(337\) 9462.46 1.52953 0.764767 0.644307i \(-0.222854\pi\)
0.764767 + 0.644307i \(0.222854\pi\)
\(338\) 0 0
\(339\) −4710.19 −0.754639
\(340\) 0 0
\(341\) −5170.23 −0.821066
\(342\) 0 0
\(343\) −4714.49 −0.742153
\(344\) 0 0
\(345\) −2487.78 −0.388226
\(346\) 0 0
\(347\) 11460.3 1.77297 0.886487 0.462753i \(-0.153138\pi\)
0.886487 + 0.462753i \(0.153138\pi\)
\(348\) 0 0
\(349\) 3673.96 0.563503 0.281751 0.959487i \(-0.409085\pi\)
0.281751 + 0.959487i \(0.409085\pi\)
\(350\) 0 0
\(351\) −351.000 −0.0533761
\(352\) 0 0
\(353\) 3388.65 0.510935 0.255467 0.966818i \(-0.417771\pi\)
0.255467 + 0.966818i \(0.417771\pi\)
\(354\) 0 0
\(355\) −453.020 −0.0677290
\(356\) 0 0
\(357\) −2176.90 −0.322727
\(358\) 0 0
\(359\) 9673.98 1.42221 0.711105 0.703086i \(-0.248195\pi\)
0.711105 + 0.703086i \(0.248195\pi\)
\(360\) 0 0
\(361\) −5867.80 −0.855489
\(362\) 0 0
\(363\) −9433.79 −1.36404
\(364\) 0 0
\(365\) 748.117 0.107283
\(366\) 0 0
\(367\) 8715.98 1.23970 0.619851 0.784720i \(-0.287193\pi\)
0.619851 + 0.784720i \(0.287193\pi\)
\(368\) 0 0
\(369\) 4065.44 0.573546
\(370\) 0 0
\(371\) 3973.63 0.556067
\(372\) 0 0
\(373\) −4667.99 −0.647987 −0.323994 0.946059i \(-0.605026\pi\)
−0.323994 + 0.946059i \(0.605026\pi\)
\(374\) 0 0
\(375\) −3111.09 −0.428416
\(376\) 0 0
\(377\) −1458.60 −0.199262
\(378\) 0 0
\(379\) 10862.7 1.47225 0.736123 0.676848i \(-0.236655\pi\)
0.736123 + 0.676848i \(0.236655\pi\)
\(380\) 0 0
\(381\) −3744.49 −0.503507
\(382\) 0 0
\(383\) −10054.2 −1.34137 −0.670686 0.741742i \(-0.734000\pi\)
−0.670686 + 0.741742i \(0.734000\pi\)
\(384\) 0 0
\(385\) 2261.20 0.299329
\(386\) 0 0
\(387\) −1020.90 −0.134097
\(388\) 0 0
\(389\) −6418.50 −0.836584 −0.418292 0.908313i \(-0.637371\pi\)
−0.418292 + 0.908313i \(0.637371\pi\)
\(390\) 0 0
\(391\) −17803.0 −2.30265
\(392\) 0 0
\(393\) 3824.40 0.490879
\(394\) 0 0
\(395\) −2462.79 −0.313713
\(396\) 0 0
\(397\) 12019.9 1.51955 0.759775 0.650186i \(-0.225309\pi\)
0.759775 + 0.650186i \(0.225309\pi\)
\(398\) 0 0
\(399\) −706.799 −0.0886822
\(400\) 0 0
\(401\) 3599.80 0.448293 0.224147 0.974555i \(-0.428041\pi\)
0.224147 + 0.974555i \(0.428041\pi\)
\(402\) 0 0
\(403\) 1004.68 0.124185
\(404\) 0 0
\(405\) −365.852 −0.0448872
\(406\) 0 0
\(407\) 3663.87 0.446219
\(408\) 0 0
\(409\) −48.0968 −0.00581475 −0.00290737 0.999996i \(-0.500925\pi\)
−0.00290737 + 0.999996i \(0.500925\pi\)
\(410\) 0 0
\(411\) −2624.74 −0.315010
\(412\) 0 0
\(413\) 1643.83 0.195854
\(414\) 0 0
\(415\) 2052.99 0.242837
\(416\) 0 0
\(417\) −931.001 −0.109332
\(418\) 0 0
\(419\) 723.462 0.0843518 0.0421759 0.999110i \(-0.486571\pi\)
0.0421759 + 0.999110i \(0.486571\pi\)
\(420\) 0 0
\(421\) 14845.5 1.71859 0.859295 0.511481i \(-0.170903\pi\)
0.859295 + 0.511481i \(0.170903\pi\)
\(422\) 0 0
\(423\) 380.403 0.0437253
\(424\) 0 0
\(425\) −10142.7 −1.15763
\(426\) 0 0
\(427\) −6157.76 −0.697880
\(428\) 0 0
\(429\) 2609.10 0.293632
\(430\) 0 0
\(431\) −1103.11 −0.123283 −0.0616417 0.998098i \(-0.519634\pi\)
−0.0616417 + 0.998098i \(0.519634\pi\)
\(432\) 0 0
\(433\) 8893.53 0.987057 0.493528 0.869730i \(-0.335707\pi\)
0.493528 + 0.869730i \(0.335707\pi\)
\(434\) 0 0
\(435\) −1520.32 −0.167572
\(436\) 0 0
\(437\) −5780.32 −0.632747
\(438\) 0 0
\(439\) 10901.7 1.18521 0.592607 0.805492i \(-0.298099\pi\)
0.592607 + 0.805492i \(0.298099\pi\)
\(440\) 0 0
\(441\) −2583.00 −0.278912
\(442\) 0 0
\(443\) −3781.37 −0.405550 −0.202775 0.979225i \(-0.564996\pi\)
−0.202775 + 0.979225i \(0.564996\pi\)
\(444\) 0 0
\(445\) 1042.97 0.111105
\(446\) 0 0
\(447\) 15.2419 0.00161279
\(448\) 0 0
\(449\) 106.834 0.0112289 0.00561447 0.999984i \(-0.498213\pi\)
0.00561447 + 0.999984i \(0.498213\pi\)
\(450\) 0 0
\(451\) −30219.7 −3.15519
\(452\) 0 0
\(453\) 19.6468 0.00203772
\(454\) 0 0
\(455\) −439.397 −0.0452731
\(456\) 0 0
\(457\) −1237.64 −0.126684 −0.0633419 0.997992i \(-0.520176\pi\)
−0.0633419 + 0.997992i \(0.520176\pi\)
\(458\) 0 0
\(459\) −2618.10 −0.266236
\(460\) 0 0
\(461\) −8790.90 −0.888141 −0.444071 0.895992i \(-0.646466\pi\)
−0.444071 + 0.895992i \(0.646466\pi\)
\(462\) 0 0
\(463\) 3861.55 0.387606 0.193803 0.981040i \(-0.437918\pi\)
0.193803 + 0.981040i \(0.437918\pi\)
\(464\) 0 0
\(465\) 1047.19 0.104435
\(466\) 0 0
\(467\) −8991.38 −0.890945 −0.445473 0.895296i \(-0.646964\pi\)
−0.445473 + 0.895296i \(0.646964\pi\)
\(468\) 0 0
\(469\) −6526.06 −0.642528
\(470\) 0 0
\(471\) −6892.79 −0.674316
\(472\) 0 0
\(473\) 7588.71 0.737694
\(474\) 0 0
\(475\) −3293.14 −0.318105
\(476\) 0 0
\(477\) 4778.99 0.458731
\(478\) 0 0
\(479\) −4179.82 −0.398707 −0.199354 0.979928i \(-0.563884\pi\)
−0.199354 + 0.979928i \(0.563884\pi\)
\(480\) 0 0
\(481\) −711.963 −0.0674901
\(482\) 0 0
\(483\) 4121.80 0.388299
\(484\) 0 0
\(485\) 4919.41 0.460575
\(486\) 0 0
\(487\) 18443.8 1.71616 0.858078 0.513519i \(-0.171658\pi\)
0.858078 + 0.513519i \(0.171658\pi\)
\(488\) 0 0
\(489\) 3256.46 0.301149
\(490\) 0 0
\(491\) 8093.26 0.743877 0.371939 0.928257i \(-0.378693\pi\)
0.371939 + 0.928257i \(0.378693\pi\)
\(492\) 0 0
\(493\) −10879.7 −0.993907
\(494\) 0 0
\(495\) 2719.49 0.246933
\(496\) 0 0
\(497\) 750.571 0.0677418
\(498\) 0 0
\(499\) −10941.6 −0.981591 −0.490796 0.871275i \(-0.663294\pi\)
−0.490796 + 0.871275i \(0.663294\pi\)
\(500\) 0 0
\(501\) −327.197 −0.0291778
\(502\) 0 0
\(503\) 9260.11 0.820851 0.410425 0.911894i \(-0.365380\pi\)
0.410425 + 0.911894i \(0.365380\pi\)
\(504\) 0 0
\(505\) −348.840 −0.0307389
\(506\) 0 0
\(507\) −507.000 −0.0444116
\(508\) 0 0
\(509\) 9996.40 0.870497 0.435248 0.900310i \(-0.356661\pi\)
0.435248 + 0.900310i \(0.356661\pi\)
\(510\) 0 0
\(511\) −1239.49 −0.107303
\(512\) 0 0
\(513\) −850.049 −0.0731591
\(514\) 0 0
\(515\) 6103.68 0.522253
\(516\) 0 0
\(517\) −2827.66 −0.240542
\(518\) 0 0
\(519\) −5667.51 −0.479337
\(520\) 0 0
\(521\) 11427.9 0.960973 0.480486 0.877002i \(-0.340460\pi\)
0.480486 + 0.877002i \(0.340460\pi\)
\(522\) 0 0
\(523\) −4810.47 −0.402193 −0.201097 0.979571i \(-0.564451\pi\)
−0.201097 + 0.979571i \(0.564451\pi\)
\(524\) 0 0
\(525\) 2348.25 0.195212
\(526\) 0 0
\(527\) 7493.88 0.619428
\(528\) 0 0
\(529\) 21541.8 1.77051
\(530\) 0 0
\(531\) 1977.00 0.161571
\(532\) 0 0
\(533\) 5872.31 0.477219
\(534\) 0 0
\(535\) −5120.41 −0.413785
\(536\) 0 0
\(537\) −10283.6 −0.826386
\(538\) 0 0
\(539\) 19200.3 1.53435
\(540\) 0 0
\(541\) −2411.99 −0.191681 −0.0958406 0.995397i \(-0.530554\pi\)
−0.0958406 + 0.995397i \(0.530554\pi\)
\(542\) 0 0
\(543\) 625.208 0.0494111
\(544\) 0 0
\(545\) 4596.47 0.361268
\(546\) 0 0
\(547\) −4396.34 −0.343646 −0.171823 0.985128i \(-0.554966\pi\)
−0.171823 + 0.985128i \(0.554966\pi\)
\(548\) 0 0
\(549\) −7405.79 −0.575722
\(550\) 0 0
\(551\) −3532.43 −0.273116
\(552\) 0 0
\(553\) 4080.40 0.313772
\(554\) 0 0
\(555\) −742.088 −0.0567565
\(556\) 0 0
\(557\) 17488.0 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(558\) 0 0
\(559\) −1474.64 −0.111575
\(560\) 0 0
\(561\) 19461.2 1.46462
\(562\) 0 0
\(563\) −6881.77 −0.515154 −0.257577 0.966258i \(-0.582924\pi\)
−0.257577 + 0.966258i \(0.582924\pi\)
\(564\) 0 0
\(565\) −7091.49 −0.528037
\(566\) 0 0
\(567\) 606.148 0.0448957
\(568\) 0 0
\(569\) 14733.5 1.08552 0.542758 0.839889i \(-0.317380\pi\)
0.542758 + 0.839889i \(0.317380\pi\)
\(570\) 0 0
\(571\) 4488.51 0.328964 0.164482 0.986380i \(-0.447405\pi\)
0.164482 + 0.986380i \(0.447405\pi\)
\(572\) 0 0
\(573\) 2873.21 0.209476
\(574\) 0 0
\(575\) 19204.4 1.39284
\(576\) 0 0
\(577\) −10552.2 −0.761338 −0.380669 0.924711i \(-0.624306\pi\)
−0.380669 + 0.924711i \(0.624306\pi\)
\(578\) 0 0
\(579\) 1538.19 0.110406
\(580\) 0 0
\(581\) −3401.42 −0.242882
\(582\) 0 0
\(583\) −35523.8 −2.52357
\(584\) 0 0
\(585\) −528.452 −0.0373484
\(586\) 0 0
\(587\) −1637.20 −0.115118 −0.0575591 0.998342i \(-0.518332\pi\)
−0.0575591 + 0.998342i \(0.518332\pi\)
\(588\) 0 0
\(589\) 2433.13 0.170213
\(590\) 0 0
\(591\) −11611.1 −0.808147
\(592\) 0 0
\(593\) 14024.0 0.971155 0.485577 0.874194i \(-0.338609\pi\)
0.485577 + 0.874194i \(0.338609\pi\)
\(594\) 0 0
\(595\) −3277.45 −0.225819
\(596\) 0 0
\(597\) 6917.50 0.474229
\(598\) 0 0
\(599\) −365.860 −0.0249560 −0.0124780 0.999922i \(-0.503972\pi\)
−0.0124780 + 0.999922i \(0.503972\pi\)
\(600\) 0 0
\(601\) −10128.9 −0.687465 −0.343733 0.939068i \(-0.611691\pi\)
−0.343733 + 0.939068i \(0.611691\pi\)
\(602\) 0 0
\(603\) −7848.74 −0.530058
\(604\) 0 0
\(605\) −14203.1 −0.954446
\(606\) 0 0
\(607\) −2324.97 −0.155465 −0.0777327 0.996974i \(-0.524768\pi\)
−0.0777327 + 0.996974i \(0.524768\pi\)
\(608\) 0 0
\(609\) 2518.89 0.167603
\(610\) 0 0
\(611\) 549.471 0.0363817
\(612\) 0 0
\(613\) −633.133 −0.0417162 −0.0208581 0.999782i \(-0.506640\pi\)
−0.0208581 + 0.999782i \(0.506640\pi\)
\(614\) 0 0
\(615\) 6120.77 0.401323
\(616\) 0 0
\(617\) −2981.85 −0.194562 −0.0972810 0.995257i \(-0.531015\pi\)
−0.0972810 + 0.995257i \(0.531015\pi\)
\(618\) 0 0
\(619\) 15158.6 0.984292 0.492146 0.870513i \(-0.336213\pi\)
0.492146 + 0.870513i \(0.336213\pi\)
\(620\) 0 0
\(621\) 4957.19 0.320330
\(622\) 0 0
\(623\) −1728.01 −0.111126
\(624\) 0 0
\(625\) 8391.01 0.537025
\(626\) 0 0
\(627\) 6318.69 0.402463
\(628\) 0 0
\(629\) −5310.51 −0.336636
\(630\) 0 0
\(631\) −5562.30 −0.350922 −0.175461 0.984486i \(-0.556142\pi\)
−0.175461 + 0.984486i \(0.556142\pi\)
\(632\) 0 0
\(633\) 11017.2 0.691776
\(634\) 0 0
\(635\) −5637.56 −0.352315
\(636\) 0 0
\(637\) −3731.00 −0.232068
\(638\) 0 0
\(639\) 902.693 0.0558842
\(640\) 0 0
\(641\) −24140.7 −1.48752 −0.743761 0.668446i \(-0.766960\pi\)
−0.743761 + 0.668446i \(0.766960\pi\)
\(642\) 0 0
\(643\) −1749.69 −0.107311 −0.0536555 0.998560i \(-0.517087\pi\)
−0.0536555 + 0.998560i \(0.517087\pi\)
\(644\) 0 0
\(645\) −1537.03 −0.0938305
\(646\) 0 0
\(647\) −1489.51 −0.0905083 −0.0452542 0.998976i \(-0.514410\pi\)
−0.0452542 + 0.998976i \(0.514410\pi\)
\(648\) 0 0
\(649\) −14695.7 −0.888836
\(650\) 0 0
\(651\) −1735.00 −0.104455
\(652\) 0 0
\(653\) −10668.2 −0.639327 −0.319663 0.947531i \(-0.603570\pi\)
−0.319663 + 0.947531i \(0.603570\pi\)
\(654\) 0 0
\(655\) 5757.86 0.343478
\(656\) 0 0
\(657\) −1490.71 −0.0885205
\(658\) 0 0
\(659\) −13933.1 −0.823608 −0.411804 0.911272i \(-0.635101\pi\)
−0.411804 + 0.911272i \(0.635101\pi\)
\(660\) 0 0
\(661\) 2349.69 0.138264 0.0691320 0.997608i \(-0.477977\pi\)
0.0691320 + 0.997608i \(0.477977\pi\)
\(662\) 0 0
\(663\) −3781.70 −0.221522
\(664\) 0 0
\(665\) −1064.13 −0.0620529
\(666\) 0 0
\(667\) 20599.9 1.19585
\(668\) 0 0
\(669\) 15261.1 0.881958
\(670\) 0 0
\(671\) 55049.6 3.16716
\(672\) 0 0
\(673\) −32421.0 −1.85697 −0.928483 0.371374i \(-0.878887\pi\)
−0.928483 + 0.371374i \(0.878887\pi\)
\(674\) 0 0
\(675\) 2824.19 0.161042
\(676\) 0 0
\(677\) 1071.74 0.0608421 0.0304211 0.999537i \(-0.490315\pi\)
0.0304211 + 0.999537i \(0.490315\pi\)
\(678\) 0 0
\(679\) −8150.56 −0.460663
\(680\) 0 0
\(681\) 7877.48 0.443268
\(682\) 0 0
\(683\) −305.487 −0.0171144 −0.00855721 0.999963i \(-0.502724\pi\)
−0.00855721 + 0.999963i \(0.502724\pi\)
\(684\) 0 0
\(685\) −3951.72 −0.220419
\(686\) 0 0
\(687\) 5036.18 0.279683
\(688\) 0 0
\(689\) 6902.99 0.381688
\(690\) 0 0
\(691\) 2180.81 0.120061 0.0600303 0.998197i \(-0.480880\pi\)
0.0600303 + 0.998197i \(0.480880\pi\)
\(692\) 0 0
\(693\) −4505.70 −0.246980
\(694\) 0 0
\(695\) −1401.68 −0.0765018
\(696\) 0 0
\(697\) 43801.4 2.38034
\(698\) 0 0
\(699\) −1945.52 −0.105274
\(700\) 0 0
\(701\) 15168.3 0.817259 0.408629 0.912700i \(-0.366007\pi\)
0.408629 + 0.912700i \(0.366007\pi\)
\(702\) 0 0
\(703\) −1724.23 −0.0925043
\(704\) 0 0
\(705\) 572.720 0.0305956
\(706\) 0 0
\(707\) 577.963 0.0307448
\(708\) 0 0
\(709\) 8988.17 0.476104 0.238052 0.971252i \(-0.423491\pi\)
0.238052 + 0.971252i \(0.423491\pi\)
\(710\) 0 0
\(711\) 4907.39 0.258849
\(712\) 0 0
\(713\) −14189.1 −0.745284
\(714\) 0 0
\(715\) 3928.15 0.205461
\(716\) 0 0
\(717\) 15659.1 0.815619
\(718\) 0 0
\(719\) −8448.18 −0.438198 −0.219099 0.975703i \(-0.570312\pi\)
−0.219099 + 0.975703i \(0.570312\pi\)
\(720\) 0 0
\(721\) −10112.7 −0.522352
\(722\) 0 0
\(723\) −18310.7 −0.941883
\(724\) 0 0
\(725\) 11736.1 0.601197
\(726\) 0 0
\(727\) 8624.18 0.439963 0.219982 0.975504i \(-0.429400\pi\)
0.219982 + 0.975504i \(0.429400\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −10999.3 −0.556530
\(732\) 0 0
\(733\) −31124.2 −1.56835 −0.784174 0.620541i \(-0.786913\pi\)
−0.784174 + 0.620541i \(0.786913\pi\)
\(734\) 0 0
\(735\) −3888.87 −0.195161
\(736\) 0 0
\(737\) 58342.2 2.91596
\(738\) 0 0
\(739\) −17671.1 −0.879626 −0.439813 0.898089i \(-0.644955\pi\)
−0.439813 + 0.898089i \(0.644955\pi\)
\(740\) 0 0
\(741\) −1227.85 −0.0608720
\(742\) 0 0
\(743\) 21331.1 1.05325 0.526623 0.850099i \(-0.323458\pi\)
0.526623 + 0.850099i \(0.323458\pi\)
\(744\) 0 0
\(745\) 22.9477 0.00112851
\(746\) 0 0
\(747\) −4090.81 −0.200368
\(748\) 0 0
\(749\) 8483.58 0.413863
\(750\) 0 0
\(751\) −11712.9 −0.569122 −0.284561 0.958658i \(-0.591848\pi\)
−0.284561 + 0.958658i \(0.591848\pi\)
\(752\) 0 0
\(753\) −19270.2 −0.932596
\(754\) 0 0
\(755\) 29.5795 0.00142584
\(756\) 0 0
\(757\) 16610.9 0.797537 0.398768 0.917052i \(-0.369438\pi\)
0.398768 + 0.917052i \(0.369438\pi\)
\(758\) 0 0
\(759\) −36848.4 −1.76220
\(760\) 0 0
\(761\) −29365.5 −1.39882 −0.699408 0.714723i \(-0.746553\pi\)
−0.699408 + 0.714723i \(0.746553\pi\)
\(762\) 0 0
\(763\) −7615.50 −0.361336
\(764\) 0 0
\(765\) −3941.71 −0.186291
\(766\) 0 0
\(767\) 2855.66 0.134435
\(768\) 0 0
\(769\) −28599.9 −1.34114 −0.670572 0.741844i \(-0.733951\pi\)
−0.670572 + 0.741844i \(0.733951\pi\)
\(770\) 0 0
\(771\) 3690.68 0.172395
\(772\) 0 0
\(773\) −13491.8 −0.627772 −0.313886 0.949461i \(-0.601631\pi\)
−0.313886 + 0.949461i \(0.601631\pi\)
\(774\) 0 0
\(775\) −8083.78 −0.374681
\(776\) 0 0
\(777\) 1229.50 0.0567673
\(778\) 0 0
\(779\) 14221.5 0.654093
\(780\) 0 0
\(781\) −6710.01 −0.307430
\(782\) 0 0
\(783\) 3029.41 0.138266
\(784\) 0 0
\(785\) −10377.5 −0.471834
\(786\) 0 0
\(787\) 25876.7 1.17205 0.586025 0.810293i \(-0.300692\pi\)
0.586025 + 0.810293i \(0.300692\pi\)
\(788\) 0 0
\(789\) −1544.98 −0.0697118
\(790\) 0 0
\(791\) 11749.3 0.528137
\(792\) 0 0
\(793\) −10697.3 −0.479030
\(794\) 0 0
\(795\) 7195.06 0.320984
\(796\) 0 0
\(797\) −21936.4 −0.974938 −0.487469 0.873140i \(-0.662080\pi\)
−0.487469 + 0.873140i \(0.662080\pi\)
\(798\) 0 0
\(799\) 4098.49 0.181469
\(800\) 0 0
\(801\) −2078.23 −0.0916739
\(802\) 0 0
\(803\) 11080.9 0.486969
\(804\) 0 0
\(805\) 6205.62 0.271701
\(806\) 0 0
\(807\) −15397.8 −0.671658
\(808\) 0 0
\(809\) 5583.23 0.242640 0.121320 0.992613i \(-0.461287\pi\)
0.121320 + 0.992613i \(0.461287\pi\)
\(810\) 0 0
\(811\) 12925.4 0.559647 0.279823 0.960052i \(-0.409724\pi\)
0.279823 + 0.960052i \(0.409724\pi\)
\(812\) 0 0
\(813\) −12900.0 −0.556486
\(814\) 0 0
\(815\) 4902.80 0.210721
\(816\) 0 0
\(817\) −3571.27 −0.152929
\(818\) 0 0
\(819\) 875.548 0.0373555
\(820\) 0 0
\(821\) 10153.2 0.431608 0.215804 0.976437i \(-0.430763\pi\)
0.215804 + 0.976437i \(0.430763\pi\)
\(822\) 0 0
\(823\) −3282.93 −0.139047 −0.0695235 0.997580i \(-0.522148\pi\)
−0.0695235 + 0.997580i \(0.522148\pi\)
\(824\) 0 0
\(825\) −20993.1 −0.885922
\(826\) 0 0
\(827\) −17689.3 −0.743795 −0.371897 0.928274i \(-0.621293\pi\)
−0.371897 + 0.928274i \(0.621293\pi\)
\(828\) 0 0
\(829\) −38181.5 −1.59964 −0.799818 0.600243i \(-0.795070\pi\)
−0.799818 + 0.600243i \(0.795070\pi\)
\(830\) 0 0
\(831\) −5438.40 −0.227023
\(832\) 0 0
\(833\) −27829.4 −1.15754
\(834\) 0 0
\(835\) −492.615 −0.0204163
\(836\) 0 0
\(837\) −2086.64 −0.0861708
\(838\) 0 0
\(839\) −43895.2 −1.80623 −0.903117 0.429395i \(-0.858727\pi\)
−0.903117 + 0.429395i \(0.858727\pi\)
\(840\) 0 0
\(841\) −11800.1 −0.483829
\(842\) 0 0
\(843\) 12219.2 0.499233
\(844\) 0 0
\(845\) −763.320 −0.0310757
\(846\) 0 0
\(847\) 23532.0 0.954627
\(848\) 0 0
\(849\) −19038.9 −0.769626
\(850\) 0 0
\(851\) 10055.1 0.405034
\(852\) 0 0
\(853\) 19955.2 0.800998 0.400499 0.916297i \(-0.368837\pi\)
0.400499 + 0.916297i \(0.368837\pi\)
\(854\) 0 0
\(855\) −1279.80 −0.0511910
\(856\) 0 0
\(857\) 26030.4 1.03755 0.518776 0.854910i \(-0.326388\pi\)
0.518776 + 0.854910i \(0.326388\pi\)
\(858\) 0 0
\(859\) −45617.4 −1.81193 −0.905964 0.423354i \(-0.860853\pi\)
−0.905964 + 0.423354i \(0.860853\pi\)
\(860\) 0 0
\(861\) −10141.0 −0.401399
\(862\) 0 0
\(863\) 2010.93 0.0793195 0.0396597 0.999213i \(-0.487373\pi\)
0.0396597 + 0.999213i \(0.487373\pi\)
\(864\) 0 0
\(865\) −8532.78 −0.335403
\(866\) 0 0
\(867\) −13468.6 −0.527586
\(868\) 0 0
\(869\) −36478.2 −1.42398
\(870\) 0 0
\(871\) −11337.1 −0.441035
\(872\) 0 0
\(873\) −9802.48 −0.380027
\(874\) 0 0
\(875\) 7760.41 0.299828
\(876\) 0 0
\(877\) 36767.3 1.41567 0.707836 0.706376i \(-0.249671\pi\)
0.707836 + 0.706376i \(0.249671\pi\)
\(878\) 0 0
\(879\) −25172.9 −0.965940
\(880\) 0 0
\(881\) 35401.2 1.35380 0.676899 0.736076i \(-0.263323\pi\)
0.676899 + 0.736076i \(0.263323\pi\)
\(882\) 0 0
\(883\) −11928.0 −0.454596 −0.227298 0.973825i \(-0.572989\pi\)
−0.227298 + 0.973825i \(0.572989\pi\)
\(884\) 0 0
\(885\) 2976.49 0.113055
\(886\) 0 0
\(887\) 32939.3 1.24689 0.623447 0.781866i \(-0.285732\pi\)
0.623447 + 0.781866i \(0.285732\pi\)
\(888\) 0 0
\(889\) 9340.40 0.352381
\(890\) 0 0
\(891\) −5418.89 −0.203748
\(892\) 0 0
\(893\) 1330.70 0.0498660
\(894\) 0 0
\(895\) −15482.6 −0.578241
\(896\) 0 0
\(897\) 7160.38 0.266531
\(898\) 0 0
\(899\) −8671.18 −0.321691
\(900\) 0 0
\(901\) 51489.2 1.90383
\(902\) 0 0
\(903\) 2546.58 0.0938483
\(904\) 0 0
\(905\) 941.289 0.0345740
\(906\) 0 0
\(907\) 1500.89 0.0549461 0.0274731 0.999623i \(-0.491254\pi\)
0.0274731 + 0.999623i \(0.491254\pi\)
\(908\) 0 0
\(909\) 695.102 0.0253631
\(910\) 0 0
\(911\) −19728.2 −0.717481 −0.358740 0.933437i \(-0.616794\pi\)
−0.358740 + 0.933437i \(0.616794\pi\)
\(912\) 0 0
\(913\) 30408.3 1.10226
\(914\) 0 0
\(915\) −11149.9 −0.402845
\(916\) 0 0
\(917\) −9539.72 −0.343543
\(918\) 0 0
\(919\) −19992.0 −0.717602 −0.358801 0.933414i \(-0.616814\pi\)
−0.358801 + 0.933414i \(0.616814\pi\)
\(920\) 0 0
\(921\) −12015.8 −0.429897
\(922\) 0 0
\(923\) 1303.89 0.0464984
\(924\) 0 0
\(925\) 5728.54 0.203625
\(926\) 0 0
\(927\) −12162.3 −0.430918
\(928\) 0 0
\(929\) −1923.62 −0.0679354 −0.0339677 0.999423i \(-0.510814\pi\)
−0.0339677 + 0.999423i \(0.510814\pi\)
\(930\) 0 0
\(931\) −9035.71 −0.318081
\(932\) 0 0
\(933\) 17509.6 0.614403
\(934\) 0 0
\(935\) 29300.0 1.02483
\(936\) 0 0
\(937\) −10252.9 −0.357468 −0.178734 0.983897i \(-0.557200\pi\)
−0.178734 + 0.983897i \(0.557200\pi\)
\(938\) 0 0
\(939\) −5289.58 −0.183833
\(940\) 0 0
\(941\) −11043.0 −0.382563 −0.191282 0.981535i \(-0.561264\pi\)
−0.191282 + 0.981535i \(0.561264\pi\)
\(942\) 0 0
\(943\) −82934.8 −2.86398
\(944\) 0 0
\(945\) 912.594 0.0314145
\(946\) 0 0
\(947\) 32105.2 1.10167 0.550833 0.834615i \(-0.314310\pi\)
0.550833 + 0.834615i \(0.314310\pi\)
\(948\) 0 0
\(949\) −2153.24 −0.0736535
\(950\) 0 0
\(951\) −18319.0 −0.624643
\(952\) 0 0
\(953\) −9473.37 −0.322007 −0.161003 0.986954i \(-0.551473\pi\)
−0.161003 + 0.986954i \(0.551473\pi\)
\(954\) 0 0
\(955\) 4325.79 0.146575
\(956\) 0 0
\(957\) −22518.5 −0.760628
\(958\) 0 0
\(959\) 6547.26 0.220461
\(960\) 0 0
\(961\) −23818.3 −0.799514
\(962\) 0 0
\(963\) 10203.0 0.341419
\(964\) 0 0
\(965\) 2315.84 0.0772534
\(966\) 0 0
\(967\) 5310.75 0.176610 0.0883052 0.996093i \(-0.471855\pi\)
0.0883052 + 0.996093i \(0.471855\pi\)
\(968\) 0 0
\(969\) −9158.49 −0.303626
\(970\) 0 0
\(971\) 24271.2 0.802164 0.401082 0.916042i \(-0.368634\pi\)
0.401082 + 0.916042i \(0.368634\pi\)
\(972\) 0 0
\(973\) 2322.32 0.0765163
\(974\) 0 0
\(975\) 4079.38 0.133995
\(976\) 0 0
\(977\) −49602.5 −1.62428 −0.812142 0.583460i \(-0.801699\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(978\) 0 0
\(979\) 15448.2 0.504317
\(980\) 0 0
\(981\) −9158.98 −0.298087
\(982\) 0 0
\(983\) −47385.7 −1.53751 −0.768753 0.639545i \(-0.779123\pi\)
−0.768753 + 0.639545i \(0.779123\pi\)
\(984\) 0 0
\(985\) −17481.2 −0.565478
\(986\) 0 0
\(987\) −948.891 −0.0306014
\(988\) 0 0
\(989\) 20826.4 0.669607
\(990\) 0 0
\(991\) 8947.33 0.286802 0.143401 0.989665i \(-0.454196\pi\)
0.143401 + 0.989665i \(0.454196\pi\)
\(992\) 0 0
\(993\) −22471.1 −0.718127
\(994\) 0 0
\(995\) 10414.7 0.331828
\(996\) 0 0
\(997\) 14908.6 0.473582 0.236791 0.971561i \(-0.423904\pi\)
0.236791 + 0.971561i \(0.423904\pi\)
\(998\) 0 0
\(999\) 1478.69 0.0468306
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 2496.4.a.s.1.2 2
4.3 odd 2 2496.4.a.bc.1.2 2
8.3 odd 2 39.4.a.b.1.2 2
8.5 even 2 624.4.a.r.1.1 2
24.5 odd 2 1872.4.a.t.1.2 2
24.11 even 2 117.4.a.c.1.1 2
40.19 odd 2 975.4.a.j.1.1 2
56.27 even 2 1911.4.a.h.1.2 2
104.51 odd 2 507.4.a.f.1.1 2
104.83 even 4 507.4.b.f.337.1 4
104.99 even 4 507.4.b.f.337.4 4
312.155 even 2 1521.4.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.b.1.2 2 8.3 odd 2
117.4.a.c.1.1 2 24.11 even 2
507.4.a.f.1.1 2 104.51 odd 2
507.4.b.f.337.1 4 104.83 even 4
507.4.b.f.337.4 4 104.99 even 4
624.4.a.r.1.1 2 8.5 even 2
975.4.a.j.1.1 2 40.19 odd 2
1521.4.a.s.1.2 2 312.155 even 2
1872.4.a.t.1.2 2 24.5 odd 2
1911.4.a.h.1.2 2 56.27 even 2
2496.4.a.s.1.2 2 1.1 even 1 trivial
2496.4.a.bc.1.2 2 4.3 odd 2