Properties

Label 2496.4.a.ca.1.5
Level $2496$
Weight $4$
Character 2496.1
Self dual yes
Analytic conductor $147.269$
Analytic rank $1$
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] = [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: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 20x^{3} - 33x^{2} + 17x + 40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 1248)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.04635\) 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)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} +22.0884 q^{5} +0.570636 q^{7} +9.00000 q^{9} -14.5147 q^{11} +13.0000 q^{13} -66.2652 q^{15} +62.8354 q^{17} -106.914 q^{19} -1.71191 q^{21} -175.759 q^{23} +362.898 q^{25} -27.0000 q^{27} -66.4653 q^{29} -53.0145 q^{31} +43.5441 q^{33} +12.6045 q^{35} -61.7208 q^{37} -39.0000 q^{39} -244.872 q^{41} +64.2156 q^{43} +198.796 q^{45} +51.2733 q^{47} -342.674 q^{49} -188.506 q^{51} -495.893 q^{53} -320.606 q^{55} +320.742 q^{57} +322.773 q^{59} -499.239 q^{61} +5.13573 q^{63} +287.149 q^{65} -145.751 q^{67} +527.278 q^{69} -1111.55 q^{71} -592.186 q^{73} -1088.69 q^{75} -8.28261 q^{77} -124.450 q^{79} +81.0000 q^{81} -675.769 q^{83} +1387.93 q^{85} +199.396 q^{87} -411.589 q^{89} +7.41827 q^{91} +159.044 q^{93} -2361.56 q^{95} -1633.14 q^{97} -130.632 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 15 q^{3} + 10 q^{5} - 14 q^{7} + 45 q^{9} + 22 q^{11} + 65 q^{13} - 30 q^{15} - 34 q^{17} - 90 q^{19} + 42 q^{21} - 96 q^{23} + 107 q^{25} - 135 q^{27} + 54 q^{29} - 378 q^{31} - 66 q^{33} + 84 q^{35}+ \cdots + 198 q^{99}+O(q^{100}) \) 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
\(5\) 22.0884 1.97565 0.987824 0.155578i \(-0.0497239\pi\)
0.987824 + 0.155578i \(0.0497239\pi\)
\(6\) 0 0
\(7\) 0.570636 0.0308115 0.0154057 0.999881i \(-0.495096\pi\)
0.0154057 + 0.999881i \(0.495096\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −14.5147 −0.397849 −0.198925 0.980015i \(-0.563745\pi\)
−0.198925 + 0.980015i \(0.563745\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) −66.2652 −1.14064
\(16\) 0 0
\(17\) 62.8354 0.896460 0.448230 0.893918i \(-0.352055\pi\)
0.448230 + 0.893918i \(0.352055\pi\)
\(18\) 0 0
\(19\) −106.914 −1.29093 −0.645467 0.763788i \(-0.723337\pi\)
−0.645467 + 0.763788i \(0.723337\pi\)
\(20\) 0 0
\(21\) −1.71191 −0.0177890
\(22\) 0 0
\(23\) −175.759 −1.59341 −0.796704 0.604370i \(-0.793425\pi\)
−0.796704 + 0.604370i \(0.793425\pi\)
\(24\) 0 0
\(25\) 362.898 2.90318
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −66.4653 −0.425597 −0.212798 0.977096i \(-0.568258\pi\)
−0.212798 + 0.977096i \(0.568258\pi\)
\(30\) 0 0
\(31\) −53.0145 −0.307151 −0.153576 0.988137i \(-0.549079\pi\)
−0.153576 + 0.988137i \(0.549079\pi\)
\(32\) 0 0
\(33\) 43.5441 0.229698
\(34\) 0 0
\(35\) 12.6045 0.0608726
\(36\) 0 0
\(37\) −61.7208 −0.274239 −0.137119 0.990555i \(-0.543784\pi\)
−0.137119 + 0.990555i \(0.543784\pi\)
\(38\) 0 0
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) −244.872 −0.932747 −0.466373 0.884588i \(-0.654440\pi\)
−0.466373 + 0.884588i \(0.654440\pi\)
\(42\) 0 0
\(43\) 64.2156 0.227739 0.113870 0.993496i \(-0.463675\pi\)
0.113870 + 0.993496i \(0.463675\pi\)
\(44\) 0 0
\(45\) 198.796 0.658549
\(46\) 0 0
\(47\) 51.2733 0.159127 0.0795636 0.996830i \(-0.474647\pi\)
0.0795636 + 0.996830i \(0.474647\pi\)
\(48\) 0 0
\(49\) −342.674 −0.999051
\(50\) 0 0
\(51\) −188.506 −0.517571
\(52\) 0 0
\(53\) −495.893 −1.28521 −0.642605 0.766198i \(-0.722146\pi\)
−0.642605 + 0.766198i \(0.722146\pi\)
\(54\) 0 0
\(55\) −320.606 −0.786010
\(56\) 0 0
\(57\) 320.742 0.745321
\(58\) 0 0
\(59\) 322.773 0.712229 0.356115 0.934442i \(-0.384101\pi\)
0.356115 + 0.934442i \(0.384101\pi\)
\(60\) 0 0
\(61\) −499.239 −1.04788 −0.523942 0.851754i \(-0.675539\pi\)
−0.523942 + 0.851754i \(0.675539\pi\)
\(62\) 0 0
\(63\) 5.13573 0.0102705
\(64\) 0 0
\(65\) 287.149 0.547946
\(66\) 0 0
\(67\) −145.751 −0.265765 −0.132883 0.991132i \(-0.542423\pi\)
−0.132883 + 0.991132i \(0.542423\pi\)
\(68\) 0 0
\(69\) 527.278 0.919954
\(70\) 0 0
\(71\) −1111.55 −1.85799 −0.928994 0.370094i \(-0.879325\pi\)
−0.928994 + 0.370094i \(0.879325\pi\)
\(72\) 0 0
\(73\) −592.186 −0.949454 −0.474727 0.880133i \(-0.657453\pi\)
−0.474727 + 0.880133i \(0.657453\pi\)
\(74\) 0 0
\(75\) −1088.69 −1.67615
\(76\) 0 0
\(77\) −8.28261 −0.0122583
\(78\) 0 0
\(79\) −124.450 −0.177238 −0.0886188 0.996066i \(-0.528245\pi\)
−0.0886188 + 0.996066i \(0.528245\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −675.769 −0.893678 −0.446839 0.894614i \(-0.647450\pi\)
−0.446839 + 0.894614i \(0.647450\pi\)
\(84\) 0 0
\(85\) 1387.93 1.77109
\(86\) 0 0
\(87\) 199.396 0.245718
\(88\) 0 0
\(89\) −411.589 −0.490206 −0.245103 0.969497i \(-0.578822\pi\)
−0.245103 + 0.969497i \(0.578822\pi\)
\(90\) 0 0
\(91\) 7.41827 0.00854556
\(92\) 0 0
\(93\) 159.044 0.177334
\(94\) 0 0
\(95\) −2361.56 −2.55043
\(96\) 0 0
\(97\) −1633.14 −1.70949 −0.854744 0.519050i \(-0.826286\pi\)
−0.854744 + 0.519050i \(0.826286\pi\)
\(98\) 0 0
\(99\) −130.632 −0.132616
\(100\) 0 0
\(101\) 1695.34 1.67023 0.835114 0.550077i \(-0.185402\pi\)
0.835114 + 0.550077i \(0.185402\pi\)
\(102\) 0 0
\(103\) 1055.03 1.00927 0.504637 0.863331i \(-0.331626\pi\)
0.504637 + 0.863331i \(0.331626\pi\)
\(104\) 0 0
\(105\) −37.8134 −0.0351448
\(106\) 0 0
\(107\) 1434.41 1.29598 0.647990 0.761649i \(-0.275610\pi\)
0.647990 + 0.761649i \(0.275610\pi\)
\(108\) 0 0
\(109\) 210.166 0.184681 0.0923406 0.995727i \(-0.470565\pi\)
0.0923406 + 0.995727i \(0.470565\pi\)
\(110\) 0 0
\(111\) 185.163 0.158332
\(112\) 0 0
\(113\) 788.100 0.656090 0.328045 0.944662i \(-0.393610\pi\)
0.328045 + 0.944662i \(0.393610\pi\)
\(114\) 0 0
\(115\) −3882.25 −3.14801
\(116\) 0 0
\(117\) 117.000 0.0924500
\(118\) 0 0
\(119\) 35.8561 0.0276212
\(120\) 0 0
\(121\) −1120.32 −0.841716
\(122\) 0 0
\(123\) 734.617 0.538522
\(124\) 0 0
\(125\) 5254.78 3.76002
\(126\) 0 0
\(127\) −1668.08 −1.16550 −0.582750 0.812652i \(-0.698023\pi\)
−0.582750 + 0.812652i \(0.698023\pi\)
\(128\) 0 0
\(129\) −192.647 −0.131485
\(130\) 0 0
\(131\) 2635.33 1.75763 0.878817 0.477159i \(-0.158333\pi\)
0.878817 + 0.477159i \(0.158333\pi\)
\(132\) 0 0
\(133\) −61.0090 −0.0397756
\(134\) 0 0
\(135\) −596.387 −0.380214
\(136\) 0 0
\(137\) −1228.31 −0.765997 −0.382999 0.923749i \(-0.625109\pi\)
−0.382999 + 0.923749i \(0.625109\pi\)
\(138\) 0 0
\(139\) −1363.03 −0.831734 −0.415867 0.909426i \(-0.636522\pi\)
−0.415867 + 0.909426i \(0.636522\pi\)
\(140\) 0 0
\(141\) −153.820 −0.0918722
\(142\) 0 0
\(143\) −188.691 −0.110344
\(144\) 0 0
\(145\) −1468.11 −0.840829
\(146\) 0 0
\(147\) 1028.02 0.576802
\(148\) 0 0
\(149\) 298.498 0.164120 0.0820600 0.996627i \(-0.473850\pi\)
0.0820600 + 0.996627i \(0.473850\pi\)
\(150\) 0 0
\(151\) 3297.44 1.77710 0.888550 0.458781i \(-0.151714\pi\)
0.888550 + 0.458781i \(0.151714\pi\)
\(152\) 0 0
\(153\) 565.518 0.298820
\(154\) 0 0
\(155\) −1171.01 −0.606823
\(156\) 0 0
\(157\) −1969.65 −1.00124 −0.500620 0.865667i \(-0.666895\pi\)
−0.500620 + 0.865667i \(0.666895\pi\)
\(158\) 0 0
\(159\) 1487.68 0.742016
\(160\) 0 0
\(161\) −100.295 −0.0490952
\(162\) 0 0
\(163\) 2046.76 0.983525 0.491762 0.870729i \(-0.336353\pi\)
0.491762 + 0.870729i \(0.336353\pi\)
\(164\) 0 0
\(165\) 961.819 0.453803
\(166\) 0 0
\(167\) −2873.01 −1.33126 −0.665628 0.746283i \(-0.731836\pi\)
−0.665628 + 0.746283i \(0.731836\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) −962.226 −0.430311
\(172\) 0 0
\(173\) 534.246 0.234786 0.117393 0.993086i \(-0.462546\pi\)
0.117393 + 0.993086i \(0.462546\pi\)
\(174\) 0 0
\(175\) 207.083 0.0894513
\(176\) 0 0
\(177\) −968.320 −0.411206
\(178\) 0 0
\(179\) −3504.15 −1.46320 −0.731599 0.681735i \(-0.761226\pi\)
−0.731599 + 0.681735i \(0.761226\pi\)
\(180\) 0 0
\(181\) 4027.89 1.65409 0.827047 0.562133i \(-0.190019\pi\)
0.827047 + 0.562133i \(0.190019\pi\)
\(182\) 0 0
\(183\) 1497.72 0.604996
\(184\) 0 0
\(185\) −1363.32 −0.541800
\(186\) 0 0
\(187\) −912.036 −0.356656
\(188\) 0 0
\(189\) −15.4072 −0.00592967
\(190\) 0 0
\(191\) 1516.47 0.574493 0.287247 0.957857i \(-0.407260\pi\)
0.287247 + 0.957857i \(0.407260\pi\)
\(192\) 0 0
\(193\) 429.841 0.160314 0.0801571 0.996782i \(-0.474458\pi\)
0.0801571 + 0.996782i \(0.474458\pi\)
\(194\) 0 0
\(195\) −861.448 −0.316357
\(196\) 0 0
\(197\) −630.616 −0.228069 −0.114034 0.993477i \(-0.536377\pi\)
−0.114034 + 0.993477i \(0.536377\pi\)
\(198\) 0 0
\(199\) 4222.79 1.50425 0.752124 0.659021i \(-0.229029\pi\)
0.752124 + 0.659021i \(0.229029\pi\)
\(200\) 0 0
\(201\) 437.252 0.153440
\(202\) 0 0
\(203\) −37.9275 −0.0131133
\(204\) 0 0
\(205\) −5408.84 −1.84278
\(206\) 0 0
\(207\) −1581.84 −0.531136
\(208\) 0 0
\(209\) 1551.82 0.513597
\(210\) 0 0
\(211\) −1464.08 −0.477686 −0.238843 0.971058i \(-0.576768\pi\)
−0.238843 + 0.971058i \(0.576768\pi\)
\(212\) 0 0
\(213\) 3334.66 1.07271
\(214\) 0 0
\(215\) 1418.42 0.449933
\(216\) 0 0
\(217\) −30.2520 −0.00946379
\(218\) 0 0
\(219\) 1776.56 0.548167
\(220\) 0 0
\(221\) 816.860 0.248633
\(222\) 0 0
\(223\) −5488.41 −1.64812 −0.824061 0.566501i \(-0.808297\pi\)
−0.824061 + 0.566501i \(0.808297\pi\)
\(224\) 0 0
\(225\) 3266.08 0.967727
\(226\) 0 0
\(227\) −1538.34 −0.449793 −0.224896 0.974383i \(-0.572204\pi\)
−0.224896 + 0.974383i \(0.572204\pi\)
\(228\) 0 0
\(229\) −2544.64 −0.734300 −0.367150 0.930162i \(-0.619666\pi\)
−0.367150 + 0.930162i \(0.619666\pi\)
\(230\) 0 0
\(231\) 24.8478 0.00707735
\(232\) 0 0
\(233\) 588.474 0.165460 0.0827301 0.996572i \(-0.473636\pi\)
0.0827301 + 0.996572i \(0.473636\pi\)
\(234\) 0 0
\(235\) 1132.55 0.314379
\(236\) 0 0
\(237\) 373.351 0.102328
\(238\) 0 0
\(239\) −655.416 −0.177386 −0.0886932 0.996059i \(-0.528269\pi\)
−0.0886932 + 0.996059i \(0.528269\pi\)
\(240\) 0 0
\(241\) −603.681 −0.161355 −0.0806774 0.996740i \(-0.525708\pi\)
−0.0806774 + 0.996740i \(0.525708\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −7569.13 −1.97377
\(246\) 0 0
\(247\) −1389.88 −0.358041
\(248\) 0 0
\(249\) 2027.31 0.515965
\(250\) 0 0
\(251\) 6332.16 1.59236 0.796180 0.605060i \(-0.206851\pi\)
0.796180 + 0.605060i \(0.206851\pi\)
\(252\) 0 0
\(253\) 2551.09 0.633936
\(254\) 0 0
\(255\) −4163.80 −1.02254
\(256\) 0 0
\(257\) 2114.90 0.513323 0.256662 0.966501i \(-0.417377\pi\)
0.256662 + 0.966501i \(0.417377\pi\)
\(258\) 0 0
\(259\) −35.2202 −0.00844971
\(260\) 0 0
\(261\) −598.188 −0.141866
\(262\) 0 0
\(263\) 3186.29 0.747054 0.373527 0.927619i \(-0.378148\pi\)
0.373527 + 0.927619i \(0.378148\pi\)
\(264\) 0 0
\(265\) −10953.5 −2.53912
\(266\) 0 0
\(267\) 1234.77 0.283020
\(268\) 0 0
\(269\) 2056.40 0.466100 0.233050 0.972465i \(-0.425129\pi\)
0.233050 + 0.972465i \(0.425129\pi\)
\(270\) 0 0
\(271\) −399.938 −0.0896476 −0.0448238 0.998995i \(-0.514273\pi\)
−0.0448238 + 0.998995i \(0.514273\pi\)
\(272\) 0 0
\(273\) −22.2548 −0.00493378
\(274\) 0 0
\(275\) −5267.35 −1.15503
\(276\) 0 0
\(277\) 4470.24 0.969643 0.484821 0.874613i \(-0.338885\pi\)
0.484821 + 0.874613i \(0.338885\pi\)
\(278\) 0 0
\(279\) −477.131 −0.102384
\(280\) 0 0
\(281\) −401.796 −0.0852993 −0.0426497 0.999090i \(-0.513580\pi\)
−0.0426497 + 0.999090i \(0.513580\pi\)
\(282\) 0 0
\(283\) 3914.77 0.822292 0.411146 0.911569i \(-0.365129\pi\)
0.411146 + 0.911569i \(0.365129\pi\)
\(284\) 0 0
\(285\) 7084.68 1.47249
\(286\) 0 0
\(287\) −139.733 −0.0287393
\(288\) 0 0
\(289\) −964.717 −0.196360
\(290\) 0 0
\(291\) 4899.42 0.986973
\(292\) 0 0
\(293\) 433.936 0.0865216 0.0432608 0.999064i \(-0.486225\pi\)
0.0432608 + 0.999064i \(0.486225\pi\)
\(294\) 0 0
\(295\) 7129.55 1.40711
\(296\) 0 0
\(297\) 391.897 0.0765661
\(298\) 0 0
\(299\) −2284.87 −0.441932
\(300\) 0 0
\(301\) 36.6438 0.00701699
\(302\) 0 0
\(303\) −5086.03 −0.964306
\(304\) 0 0
\(305\) −11027.4 −2.07025
\(306\) 0 0
\(307\) −4680.64 −0.870157 −0.435078 0.900393i \(-0.643279\pi\)
−0.435078 + 0.900393i \(0.643279\pi\)
\(308\) 0 0
\(309\) −3165.09 −0.582705
\(310\) 0 0
\(311\) −670.394 −0.122233 −0.0611167 0.998131i \(-0.519466\pi\)
−0.0611167 + 0.998131i \(0.519466\pi\)
\(312\) 0 0
\(313\) 2198.73 0.397060 0.198530 0.980095i \(-0.436383\pi\)
0.198530 + 0.980095i \(0.436383\pi\)
\(314\) 0 0
\(315\) 113.440 0.0202909
\(316\) 0 0
\(317\) 7510.62 1.33072 0.665360 0.746523i \(-0.268278\pi\)
0.665360 + 0.746523i \(0.268278\pi\)
\(318\) 0 0
\(319\) 964.724 0.169323
\(320\) 0 0
\(321\) −4303.24 −0.748235
\(322\) 0 0
\(323\) −6717.98 −1.15727
\(324\) 0 0
\(325\) 4717.67 0.805198
\(326\) 0 0
\(327\) −630.498 −0.106626
\(328\) 0 0
\(329\) 29.2584 0.00490295
\(330\) 0 0
\(331\) −8426.86 −1.39934 −0.699671 0.714465i \(-0.746670\pi\)
−0.699671 + 0.714465i \(0.746670\pi\)
\(332\) 0 0
\(333\) −555.488 −0.0914130
\(334\) 0 0
\(335\) −3219.40 −0.525058
\(336\) 0 0
\(337\) −9064.12 −1.46515 −0.732573 0.680689i \(-0.761681\pi\)
−0.732573 + 0.680689i \(0.761681\pi\)
\(338\) 0 0
\(339\) −2364.30 −0.378794
\(340\) 0 0
\(341\) 769.489 0.122200
\(342\) 0 0
\(343\) −391.271 −0.0615937
\(344\) 0 0
\(345\) 11646.7 1.81751
\(346\) 0 0
\(347\) 4496.43 0.695622 0.347811 0.937565i \(-0.386925\pi\)
0.347811 + 0.937565i \(0.386925\pi\)
\(348\) 0 0
\(349\) −9397.56 −1.44137 −0.720687 0.693260i \(-0.756174\pi\)
−0.720687 + 0.693260i \(0.756174\pi\)
\(350\) 0 0
\(351\) −351.000 −0.0533761
\(352\) 0 0
\(353\) −8189.26 −1.23476 −0.617380 0.786665i \(-0.711806\pi\)
−0.617380 + 0.786665i \(0.711806\pi\)
\(354\) 0 0
\(355\) −24552.5 −3.67073
\(356\) 0 0
\(357\) −107.568 −0.0159471
\(358\) 0 0
\(359\) −4176.24 −0.613965 −0.306983 0.951715i \(-0.599319\pi\)
−0.306983 + 0.951715i \(0.599319\pi\)
\(360\) 0 0
\(361\) 4571.60 0.666511
\(362\) 0 0
\(363\) 3360.97 0.485965
\(364\) 0 0
\(365\) −13080.4 −1.87579
\(366\) 0 0
\(367\) −5272.91 −0.749983 −0.374991 0.927028i \(-0.622354\pi\)
−0.374991 + 0.927028i \(0.622354\pi\)
\(368\) 0 0
\(369\) −2203.85 −0.310916
\(370\) 0 0
\(371\) −282.975 −0.0395992
\(372\) 0 0
\(373\) −9751.86 −1.35371 −0.676853 0.736118i \(-0.736657\pi\)
−0.676853 + 0.736118i \(0.736657\pi\)
\(374\) 0 0
\(375\) −15764.4 −2.17085
\(376\) 0 0
\(377\) −864.049 −0.118039
\(378\) 0 0
\(379\) 12570.3 1.70368 0.851839 0.523803i \(-0.175487\pi\)
0.851839 + 0.523803i \(0.175487\pi\)
\(380\) 0 0
\(381\) 5004.25 0.672901
\(382\) 0 0
\(383\) −10564.7 −1.40948 −0.704741 0.709464i \(-0.748937\pi\)
−0.704741 + 0.709464i \(0.748937\pi\)
\(384\) 0 0
\(385\) −182.950 −0.0242181
\(386\) 0 0
\(387\) 577.941 0.0759131
\(388\) 0 0
\(389\) 9376.03 1.22207 0.611033 0.791605i \(-0.290754\pi\)
0.611033 + 0.791605i \(0.290754\pi\)
\(390\) 0 0
\(391\) −11043.9 −1.42843
\(392\) 0 0
\(393\) −7906.00 −1.01477
\(394\) 0 0
\(395\) −2748.91 −0.350159
\(396\) 0 0
\(397\) 3449.75 0.436116 0.218058 0.975936i \(-0.430028\pi\)
0.218058 + 0.975936i \(0.430028\pi\)
\(398\) 0 0
\(399\) 183.027 0.0229644
\(400\) 0 0
\(401\) 6580.34 0.819467 0.409734 0.912205i \(-0.365622\pi\)
0.409734 + 0.912205i \(0.365622\pi\)
\(402\) 0 0
\(403\) −689.189 −0.0851885
\(404\) 0 0
\(405\) 1789.16 0.219516
\(406\) 0 0
\(407\) 895.859 0.109106
\(408\) 0 0
\(409\) −6394.78 −0.773109 −0.386554 0.922267i \(-0.626335\pi\)
−0.386554 + 0.922267i \(0.626335\pi\)
\(410\) 0 0
\(411\) 3684.93 0.442249
\(412\) 0 0
\(413\) 184.186 0.0219448
\(414\) 0 0
\(415\) −14926.7 −1.76559
\(416\) 0 0
\(417\) 4089.10 0.480202
\(418\) 0 0
\(419\) −8951.92 −1.04375 −0.521873 0.853023i \(-0.674767\pi\)
−0.521873 + 0.853023i \(0.674767\pi\)
\(420\) 0 0
\(421\) 414.532 0.0479883 0.0239941 0.999712i \(-0.492362\pi\)
0.0239941 + 0.999712i \(0.492362\pi\)
\(422\) 0 0
\(423\) 461.460 0.0530424
\(424\) 0 0
\(425\) 22802.8 2.60259
\(426\) 0 0
\(427\) −284.884 −0.0322869
\(428\) 0 0
\(429\) 566.073 0.0637069
\(430\) 0 0
\(431\) −945.412 −0.105659 −0.0528293 0.998604i \(-0.516824\pi\)
−0.0528293 + 0.998604i \(0.516824\pi\)
\(432\) 0 0
\(433\) 6691.85 0.742702 0.371351 0.928493i \(-0.378895\pi\)
0.371351 + 0.928493i \(0.378895\pi\)
\(434\) 0 0
\(435\) 4404.34 0.485453
\(436\) 0 0
\(437\) 18791.1 2.05698
\(438\) 0 0
\(439\) 799.787 0.0869516 0.0434758 0.999054i \(-0.486157\pi\)
0.0434758 + 0.999054i \(0.486157\pi\)
\(440\) 0 0
\(441\) −3084.07 −0.333017
\(442\) 0 0
\(443\) −4856.67 −0.520874 −0.260437 0.965491i \(-0.583867\pi\)
−0.260437 + 0.965491i \(0.583867\pi\)
\(444\) 0 0
\(445\) −9091.34 −0.968474
\(446\) 0 0
\(447\) −895.493 −0.0947548
\(448\) 0 0
\(449\) −16801.4 −1.76594 −0.882969 0.469431i \(-0.844459\pi\)
−0.882969 + 0.469431i \(0.844459\pi\)
\(450\) 0 0
\(451\) 3554.24 0.371093
\(452\) 0 0
\(453\) −9892.32 −1.02601
\(454\) 0 0
\(455\) 163.858 0.0168830
\(456\) 0 0
\(457\) −3603.14 −0.368814 −0.184407 0.982850i \(-0.559036\pi\)
−0.184407 + 0.982850i \(0.559036\pi\)
\(458\) 0 0
\(459\) −1696.55 −0.172524
\(460\) 0 0
\(461\) −7442.82 −0.751946 −0.375973 0.926631i \(-0.622691\pi\)
−0.375973 + 0.926631i \(0.622691\pi\)
\(462\) 0 0
\(463\) −1873.49 −0.188053 −0.0940265 0.995570i \(-0.529974\pi\)
−0.0940265 + 0.995570i \(0.529974\pi\)
\(464\) 0 0
\(465\) 3513.02 0.350349
\(466\) 0 0
\(467\) −8925.09 −0.884377 −0.442188 0.896922i \(-0.645798\pi\)
−0.442188 + 0.896922i \(0.645798\pi\)
\(468\) 0 0
\(469\) −83.1706 −0.00818862
\(470\) 0 0
\(471\) 5908.94 0.578067
\(472\) 0 0
\(473\) −932.070 −0.0906060
\(474\) 0 0
\(475\) −38798.8 −3.74782
\(476\) 0 0
\(477\) −4463.04 −0.428403
\(478\) 0 0
\(479\) −2863.80 −0.273174 −0.136587 0.990628i \(-0.543613\pi\)
−0.136587 + 0.990628i \(0.543613\pi\)
\(480\) 0 0
\(481\) −802.371 −0.0760602
\(482\) 0 0
\(483\) 300.884 0.0283452
\(484\) 0 0
\(485\) −36073.5 −3.37734
\(486\) 0 0
\(487\) 2976.01 0.276911 0.138456 0.990369i \(-0.455786\pi\)
0.138456 + 0.990369i \(0.455786\pi\)
\(488\) 0 0
\(489\) −6140.28 −0.567838
\(490\) 0 0
\(491\) −14753.6 −1.35605 −0.678027 0.735037i \(-0.737165\pi\)
−0.678027 + 0.735037i \(0.737165\pi\)
\(492\) 0 0
\(493\) −4176.37 −0.381530
\(494\) 0 0
\(495\) −2885.46 −0.262003
\(496\) 0 0
\(497\) −634.293 −0.0572474
\(498\) 0 0
\(499\) 16239.3 1.45685 0.728426 0.685124i \(-0.240252\pi\)
0.728426 + 0.685124i \(0.240252\pi\)
\(500\) 0 0
\(501\) 8619.02 0.768601
\(502\) 0 0
\(503\) 1975.94 0.175155 0.0875774 0.996158i \(-0.472087\pi\)
0.0875774 + 0.996158i \(0.472087\pi\)
\(504\) 0 0
\(505\) 37447.4 3.29978
\(506\) 0 0
\(507\) −507.000 −0.0444116
\(508\) 0 0
\(509\) −18898.0 −1.64565 −0.822827 0.568292i \(-0.807605\pi\)
−0.822827 + 0.568292i \(0.807605\pi\)
\(510\) 0 0
\(511\) −337.923 −0.0292541
\(512\) 0 0
\(513\) 2886.68 0.248440
\(514\) 0 0
\(515\) 23304.0 1.99397
\(516\) 0 0
\(517\) −744.216 −0.0633087
\(518\) 0 0
\(519\) −1602.74 −0.135554
\(520\) 0 0
\(521\) 19031.4 1.60035 0.800175 0.599767i \(-0.204740\pi\)
0.800175 + 0.599767i \(0.204740\pi\)
\(522\) 0 0
\(523\) −20162.4 −1.68574 −0.842870 0.538117i \(-0.819136\pi\)
−0.842870 + 0.538117i \(0.819136\pi\)
\(524\) 0 0
\(525\) −621.248 −0.0516447
\(526\) 0 0
\(527\) −3331.19 −0.275349
\(528\) 0 0
\(529\) 18724.4 1.53895
\(530\) 0 0
\(531\) 2904.96 0.237410
\(532\) 0 0
\(533\) −3183.34 −0.258697
\(534\) 0 0
\(535\) 31683.9 2.56040
\(536\) 0 0
\(537\) 10512.4 0.844778
\(538\) 0 0
\(539\) 4973.81 0.397472
\(540\) 0 0
\(541\) −1067.93 −0.0848688 −0.0424344 0.999099i \(-0.513511\pi\)
−0.0424344 + 0.999099i \(0.513511\pi\)
\(542\) 0 0
\(543\) −12083.7 −0.954992
\(544\) 0 0
\(545\) 4642.23 0.364865
\(546\) 0 0
\(547\) −18158.4 −1.41938 −0.709688 0.704516i \(-0.751164\pi\)
−0.709688 + 0.704516i \(0.751164\pi\)
\(548\) 0 0
\(549\) −4493.15 −0.349295
\(550\) 0 0
\(551\) 7106.07 0.549417
\(552\) 0 0
\(553\) −71.0159 −0.00546095
\(554\) 0 0
\(555\) 4089.95 0.312808
\(556\) 0 0
\(557\) −21903.8 −1.66624 −0.833120 0.553092i \(-0.813448\pi\)
−0.833120 + 0.553092i \(0.813448\pi\)
\(558\) 0 0
\(559\) 834.803 0.0631635
\(560\) 0 0
\(561\) 2736.11 0.205915
\(562\) 0 0
\(563\) 6633.07 0.496538 0.248269 0.968691i \(-0.420138\pi\)
0.248269 + 0.968691i \(0.420138\pi\)
\(564\) 0 0
\(565\) 17407.9 1.29620
\(566\) 0 0
\(567\) 46.2216 0.00342350
\(568\) 0 0
\(569\) −14739.5 −1.08596 −0.542981 0.839745i \(-0.682704\pi\)
−0.542981 + 0.839745i \(0.682704\pi\)
\(570\) 0 0
\(571\) −22073.0 −1.61774 −0.808868 0.587990i \(-0.799920\pi\)
−0.808868 + 0.587990i \(0.799920\pi\)
\(572\) 0 0
\(573\) −4549.42 −0.331684
\(574\) 0 0
\(575\) −63782.7 −4.62595
\(576\) 0 0
\(577\) 9921.43 0.715831 0.357915 0.933754i \(-0.383488\pi\)
0.357915 + 0.933754i \(0.383488\pi\)
\(578\) 0 0
\(579\) −1289.52 −0.0925574
\(580\) 0 0
\(581\) −385.618 −0.0275355
\(582\) 0 0
\(583\) 7197.73 0.511320
\(584\) 0 0
\(585\) 2584.34 0.182649
\(586\) 0 0
\(587\) 22175.4 1.55924 0.779622 0.626250i \(-0.215411\pi\)
0.779622 + 0.626250i \(0.215411\pi\)
\(588\) 0 0
\(589\) 5668.00 0.396512
\(590\) 0 0
\(591\) 1891.85 0.131676
\(592\) 0 0
\(593\) −19410.7 −1.34419 −0.672093 0.740467i \(-0.734605\pi\)
−0.672093 + 0.740467i \(0.734605\pi\)
\(594\) 0 0
\(595\) 792.005 0.0545698
\(596\) 0 0
\(597\) −12668.4 −0.868478
\(598\) 0 0
\(599\) −23121.5 −1.57716 −0.788582 0.614930i \(-0.789184\pi\)
−0.788582 + 0.614930i \(0.789184\pi\)
\(600\) 0 0
\(601\) −11376.1 −0.772117 −0.386059 0.922474i \(-0.626164\pi\)
−0.386059 + 0.922474i \(0.626164\pi\)
\(602\) 0 0
\(603\) −1311.76 −0.0885884
\(604\) 0 0
\(605\) −24746.2 −1.66293
\(606\) 0 0
\(607\) 10687.2 0.714628 0.357314 0.933984i \(-0.383693\pi\)
0.357314 + 0.933984i \(0.383693\pi\)
\(608\) 0 0
\(609\) 113.783 0.00757094
\(610\) 0 0
\(611\) 666.553 0.0441340
\(612\) 0 0
\(613\) 23589.4 1.55427 0.777135 0.629334i \(-0.216672\pi\)
0.777135 + 0.629334i \(0.216672\pi\)
\(614\) 0 0
\(615\) 16226.5 1.06393
\(616\) 0 0
\(617\) −14195.7 −0.926249 −0.463124 0.886293i \(-0.653272\pi\)
−0.463124 + 0.886293i \(0.653272\pi\)
\(618\) 0 0
\(619\) −11206.6 −0.727678 −0.363839 0.931462i \(-0.618534\pi\)
−0.363839 + 0.931462i \(0.618534\pi\)
\(620\) 0 0
\(621\) 4745.51 0.306651
\(622\) 0 0
\(623\) −234.867 −0.0151040
\(624\) 0 0
\(625\) 70707.6 4.52529
\(626\) 0 0
\(627\) −4655.47 −0.296526
\(628\) 0 0
\(629\) −3878.25 −0.245844
\(630\) 0 0
\(631\) −2757.97 −0.173999 −0.0869993 0.996208i \(-0.527728\pi\)
−0.0869993 + 0.996208i \(0.527728\pi\)
\(632\) 0 0
\(633\) 4392.25 0.275792
\(634\) 0 0
\(635\) −36845.3 −2.30262
\(636\) 0 0
\(637\) −4454.77 −0.277087
\(638\) 0 0
\(639\) −10004.0 −0.619329
\(640\) 0 0
\(641\) −11974.7 −0.737869 −0.368934 0.929455i \(-0.620277\pi\)
−0.368934 + 0.929455i \(0.620277\pi\)
\(642\) 0 0
\(643\) 16145.5 0.990227 0.495114 0.868828i \(-0.335126\pi\)
0.495114 + 0.868828i \(0.335126\pi\)
\(644\) 0 0
\(645\) −4255.26 −0.259769
\(646\) 0 0
\(647\) 21437.9 1.30264 0.651322 0.758802i \(-0.274215\pi\)
0.651322 + 0.758802i \(0.274215\pi\)
\(648\) 0 0
\(649\) −4684.96 −0.283360
\(650\) 0 0
\(651\) 90.7561 0.00546392
\(652\) 0 0
\(653\) 17735.5 1.06286 0.531428 0.847104i \(-0.321656\pi\)
0.531428 + 0.847104i \(0.321656\pi\)
\(654\) 0 0
\(655\) 58210.3 3.47246
\(656\) 0 0
\(657\) −5329.67 −0.316485
\(658\) 0 0
\(659\) 21518.8 1.27201 0.636004 0.771686i \(-0.280586\pi\)
0.636004 + 0.771686i \(0.280586\pi\)
\(660\) 0 0
\(661\) 30544.1 1.79732 0.898660 0.438646i \(-0.144542\pi\)
0.898660 + 0.438646i \(0.144542\pi\)
\(662\) 0 0
\(663\) −2450.58 −0.143548
\(664\) 0 0
\(665\) −1347.59 −0.0785825
\(666\) 0 0
\(667\) 11681.9 0.678149
\(668\) 0 0
\(669\) 16465.2 0.951543
\(670\) 0 0
\(671\) 7246.29 0.416900
\(672\) 0 0
\(673\) 6938.35 0.397405 0.198703 0.980060i \(-0.436327\pi\)
0.198703 + 0.980060i \(0.436327\pi\)
\(674\) 0 0
\(675\) −9798.24 −0.558718
\(676\) 0 0
\(677\) −8394.43 −0.476550 −0.238275 0.971198i \(-0.576582\pi\)
−0.238275 + 0.971198i \(0.576582\pi\)
\(678\) 0 0
\(679\) −931.930 −0.0526718
\(680\) 0 0
\(681\) 4615.01 0.259688
\(682\) 0 0
\(683\) 18375.5 1.02946 0.514729 0.857353i \(-0.327892\pi\)
0.514729 + 0.857353i \(0.327892\pi\)
\(684\) 0 0
\(685\) −27131.4 −1.51334
\(686\) 0 0
\(687\) 7633.93 0.423948
\(688\) 0 0
\(689\) −6446.61 −0.356453
\(690\) 0 0
\(691\) 11058.8 0.608824 0.304412 0.952541i \(-0.401540\pi\)
0.304412 + 0.952541i \(0.401540\pi\)
\(692\) 0 0
\(693\) −74.5435 −0.00408611
\(694\) 0 0
\(695\) −30107.2 −1.64321
\(696\) 0 0
\(697\) −15386.6 −0.836170
\(698\) 0 0
\(699\) −1765.42 −0.0955284
\(700\) 0 0
\(701\) −7955.43 −0.428634 −0.214317 0.976764i \(-0.568753\pi\)
−0.214317 + 0.976764i \(0.568753\pi\)
\(702\) 0 0
\(703\) 6598.82 0.354024
\(704\) 0 0
\(705\) −3397.64 −0.181507
\(706\) 0 0
\(707\) 967.425 0.0514622
\(708\) 0 0
\(709\) 12814.7 0.678794 0.339397 0.940643i \(-0.389777\pi\)
0.339397 + 0.940643i \(0.389777\pi\)
\(710\) 0 0
\(711\) −1120.05 −0.0590792
\(712\) 0 0
\(713\) 9317.81 0.489417
\(714\) 0 0
\(715\) −4167.88 −0.218000
\(716\) 0 0
\(717\) 1966.25 0.102414
\(718\) 0 0
\(719\) −2221.14 −0.115208 −0.0576039 0.998340i \(-0.518346\pi\)
−0.0576039 + 0.998340i \(0.518346\pi\)
\(720\) 0 0
\(721\) 602.039 0.0310972
\(722\) 0 0
\(723\) 1811.04 0.0931582
\(724\) 0 0
\(725\) −24120.1 −1.23558
\(726\) 0 0
\(727\) 17320.1 0.883587 0.441794 0.897117i \(-0.354342\pi\)
0.441794 + 0.897117i \(0.354342\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 4035.01 0.204159
\(732\) 0 0
\(733\) 28448.3 1.43351 0.716755 0.697325i \(-0.245626\pi\)
0.716755 + 0.697325i \(0.245626\pi\)
\(734\) 0 0
\(735\) 22707.4 1.13956
\(736\) 0 0
\(737\) 2115.52 0.105735
\(738\) 0 0
\(739\) 19610.1 0.976141 0.488070 0.872804i \(-0.337701\pi\)
0.488070 + 0.872804i \(0.337701\pi\)
\(740\) 0 0
\(741\) 4169.65 0.206715
\(742\) 0 0
\(743\) 36346.5 1.79465 0.897325 0.441370i \(-0.145507\pi\)
0.897325 + 0.441370i \(0.145507\pi\)
\(744\) 0 0
\(745\) 6593.34 0.324243
\(746\) 0 0
\(747\) −6081.92 −0.297893
\(748\) 0 0
\(749\) 818.528 0.0399311
\(750\) 0 0
\(751\) 30350.0 1.47468 0.737342 0.675520i \(-0.236081\pi\)
0.737342 + 0.675520i \(0.236081\pi\)
\(752\) 0 0
\(753\) −18996.5 −0.919349
\(754\) 0 0
\(755\) 72835.2 3.51092
\(756\) 0 0
\(757\) 19424.8 0.932635 0.466318 0.884617i \(-0.345580\pi\)
0.466318 + 0.884617i \(0.345580\pi\)
\(758\) 0 0
\(759\) −7653.28 −0.366003
\(760\) 0 0
\(761\) −4640.01 −0.221025 −0.110513 0.993875i \(-0.535249\pi\)
−0.110513 + 0.993875i \(0.535249\pi\)
\(762\) 0 0
\(763\) 119.928 0.00569030
\(764\) 0 0
\(765\) 12491.4 0.590363
\(766\) 0 0
\(767\) 4196.06 0.197537
\(768\) 0 0
\(769\) 34544.3 1.61990 0.809949 0.586501i \(-0.199495\pi\)
0.809949 + 0.586501i \(0.199495\pi\)
\(770\) 0 0
\(771\) −6344.71 −0.296367
\(772\) 0 0
\(773\) 9123.70 0.424523 0.212262 0.977213i \(-0.431917\pi\)
0.212262 + 0.977213i \(0.431917\pi\)
\(774\) 0 0
\(775\) −19238.9 −0.891716
\(776\) 0 0
\(777\) 105.660 0.00487844
\(778\) 0 0
\(779\) 26180.3 1.20411
\(780\) 0 0
\(781\) 16133.9 0.739199
\(782\) 0 0
\(783\) 1794.56 0.0819061
\(784\) 0 0
\(785\) −43506.3 −1.97810
\(786\) 0 0
\(787\) 13224.2 0.598974 0.299487 0.954100i \(-0.403185\pi\)
0.299487 + 0.954100i \(0.403185\pi\)
\(788\) 0 0
\(789\) −9558.87 −0.431312
\(790\) 0 0
\(791\) 449.719 0.0202151
\(792\) 0 0
\(793\) −6490.10 −0.290631
\(794\) 0 0
\(795\) 32860.5 1.46596
\(796\) 0 0
\(797\) 14713.4 0.653922 0.326961 0.945038i \(-0.393975\pi\)
0.326961 + 0.945038i \(0.393975\pi\)
\(798\) 0 0
\(799\) 3221.78 0.142651
\(800\) 0 0
\(801\) −3704.30 −0.163402
\(802\) 0 0
\(803\) 8595.39 0.377740
\(804\) 0 0
\(805\) −2215.35 −0.0969949
\(806\) 0 0
\(807\) −6169.20 −0.269103
\(808\) 0 0
\(809\) 2335.80 0.101511 0.0507555 0.998711i \(-0.483837\pi\)
0.0507555 + 0.998711i \(0.483837\pi\)
\(810\) 0 0
\(811\) 26446.9 1.14510 0.572550 0.819870i \(-0.305954\pi\)
0.572550 + 0.819870i \(0.305954\pi\)
\(812\) 0 0
\(813\) 1199.81 0.0517581
\(814\) 0 0
\(815\) 45209.7 1.94310
\(816\) 0 0
\(817\) −6865.55 −0.293997
\(818\) 0 0
\(819\) 66.7645 0.00284852
\(820\) 0 0
\(821\) −30682.1 −1.30428 −0.652140 0.758098i \(-0.726129\pi\)
−0.652140 + 0.758098i \(0.726129\pi\)
\(822\) 0 0
\(823\) −10242.8 −0.433829 −0.216914 0.976191i \(-0.569599\pi\)
−0.216914 + 0.976191i \(0.569599\pi\)
\(824\) 0 0
\(825\) 15802.0 0.666856
\(826\) 0 0
\(827\) −35864.5 −1.50802 −0.754008 0.656865i \(-0.771882\pi\)
−0.754008 + 0.656865i \(0.771882\pi\)
\(828\) 0 0
\(829\) 3633.70 0.152236 0.0761179 0.997099i \(-0.475747\pi\)
0.0761179 + 0.997099i \(0.475747\pi\)
\(830\) 0 0
\(831\) −13410.7 −0.559823
\(832\) 0 0
\(833\) −21532.1 −0.895609
\(834\) 0 0
\(835\) −63460.1 −2.63009
\(836\) 0 0
\(837\) 1431.39 0.0591113
\(838\) 0 0
\(839\) −22098.2 −0.909312 −0.454656 0.890667i \(-0.650238\pi\)
−0.454656 + 0.890667i \(0.650238\pi\)
\(840\) 0 0
\(841\) −19971.4 −0.818867
\(842\) 0 0
\(843\) 1205.39 0.0492476
\(844\) 0 0
\(845\) 3732.94 0.151973
\(846\) 0 0
\(847\) −639.298 −0.0259345
\(848\) 0 0
\(849\) −11744.3 −0.474751
\(850\) 0 0
\(851\) 10848.0 0.436975
\(852\) 0 0
\(853\) −8946.55 −0.359114 −0.179557 0.983748i \(-0.557466\pi\)
−0.179557 + 0.983748i \(0.557466\pi\)
\(854\) 0 0
\(855\) −21254.0 −0.850144
\(856\) 0 0
\(857\) −22867.5 −0.911479 −0.455740 0.890113i \(-0.650625\pi\)
−0.455740 + 0.890113i \(0.650625\pi\)
\(858\) 0 0
\(859\) −31281.5 −1.24250 −0.621252 0.783611i \(-0.713375\pi\)
−0.621252 + 0.783611i \(0.713375\pi\)
\(860\) 0 0
\(861\) 419.199 0.0165926
\(862\) 0 0
\(863\) 23071.5 0.910040 0.455020 0.890481i \(-0.349632\pi\)
0.455020 + 0.890481i \(0.349632\pi\)
\(864\) 0 0
\(865\) 11800.6 0.463854
\(866\) 0 0
\(867\) 2894.15 0.113369
\(868\) 0 0
\(869\) 1806.36 0.0705138
\(870\) 0 0
\(871\) −1894.76 −0.0737100
\(872\) 0 0
\(873\) −14698.3 −0.569829
\(874\) 0 0
\(875\) 2998.57 0.115852
\(876\) 0 0
\(877\) 26100.7 1.00497 0.502485 0.864586i \(-0.332419\pi\)
0.502485 + 0.864586i \(0.332419\pi\)
\(878\) 0 0
\(879\) −1301.81 −0.0499533
\(880\) 0 0
\(881\) 48698.2 1.86230 0.931148 0.364642i \(-0.118809\pi\)
0.931148 + 0.364642i \(0.118809\pi\)
\(882\) 0 0
\(883\) −870.968 −0.0331941 −0.0165971 0.999862i \(-0.505283\pi\)
−0.0165971 + 0.999862i \(0.505283\pi\)
\(884\) 0 0
\(885\) −21388.7 −0.812398
\(886\) 0 0
\(887\) −33022.5 −1.25004 −0.625022 0.780608i \(-0.714910\pi\)
−0.625022 + 0.780608i \(0.714910\pi\)
\(888\) 0 0
\(889\) −951.869 −0.0359107
\(890\) 0 0
\(891\) −1175.69 −0.0442055
\(892\) 0 0
\(893\) −5481.83 −0.205423
\(894\) 0 0
\(895\) −77401.1 −2.89076
\(896\) 0 0
\(897\) 6854.62 0.255149
\(898\) 0 0
\(899\) 3523.63 0.130723
\(900\) 0 0
\(901\) −31159.6 −1.15214
\(902\) 0 0
\(903\) −109.931 −0.00405126
\(904\) 0 0
\(905\) 88969.8 3.26791
\(906\) 0 0
\(907\) −42835.0 −1.56815 −0.784076 0.620665i \(-0.786863\pi\)
−0.784076 + 0.620665i \(0.786863\pi\)
\(908\) 0 0
\(909\) 15258.1 0.556742
\(910\) 0 0
\(911\) −5584.98 −0.203116 −0.101558 0.994830i \(-0.532383\pi\)
−0.101558 + 0.994830i \(0.532383\pi\)
\(912\) 0 0
\(913\) 9808.57 0.355549
\(914\) 0 0
\(915\) 33082.2 1.19526
\(916\) 0 0
\(917\) 1503.82 0.0541553
\(918\) 0 0
\(919\) 50229.1 1.80294 0.901472 0.432838i \(-0.142488\pi\)
0.901472 + 0.432838i \(0.142488\pi\)
\(920\) 0 0
\(921\) 14041.9 0.502385
\(922\) 0 0
\(923\) −14450.2 −0.515313
\(924\) 0 0
\(925\) −22398.4 −0.796166
\(926\) 0 0
\(927\) 9495.28 0.336425
\(928\) 0 0
\(929\) −13286.2 −0.469222 −0.234611 0.972089i \(-0.575382\pi\)
−0.234611 + 0.972089i \(0.575382\pi\)
\(930\) 0 0
\(931\) 36636.7 1.28971
\(932\) 0 0
\(933\) 2011.18 0.0705714
\(934\) 0 0
\(935\) −20145.4 −0.704626
\(936\) 0 0
\(937\) −1627.36 −0.0567380 −0.0283690 0.999598i \(-0.509031\pi\)
−0.0283690 + 0.999598i \(0.509031\pi\)
\(938\) 0 0
\(939\) −6596.20 −0.229243
\(940\) 0 0
\(941\) 44267.2 1.53355 0.766774 0.641917i \(-0.221861\pi\)
0.766774 + 0.641917i \(0.221861\pi\)
\(942\) 0 0
\(943\) 43038.6 1.48625
\(944\) 0 0
\(945\) −340.320 −0.0117149
\(946\) 0 0
\(947\) −47433.9 −1.62766 −0.813830 0.581103i \(-0.802621\pi\)
−0.813830 + 0.581103i \(0.802621\pi\)
\(948\) 0 0
\(949\) −7698.42 −0.263331
\(950\) 0 0
\(951\) −22531.8 −0.768292
\(952\) 0 0
\(953\) −38935.8 −1.32346 −0.661728 0.749744i \(-0.730177\pi\)
−0.661728 + 0.749744i \(0.730177\pi\)
\(954\) 0 0
\(955\) 33496.5 1.13500
\(956\) 0 0
\(957\) −2894.17 −0.0977589
\(958\) 0 0
\(959\) −700.918 −0.0236015
\(960\) 0 0
\(961\) −26980.5 −0.905658
\(962\) 0 0
\(963\) 12909.7 0.431994
\(964\) 0 0
\(965\) 9494.50 0.316724
\(966\) 0 0
\(967\) −18350.7 −0.610257 −0.305129 0.952311i \(-0.598699\pi\)
−0.305129 + 0.952311i \(0.598699\pi\)
\(968\) 0 0
\(969\) 20153.9 0.668150
\(970\) 0 0
\(971\) −3679.77 −0.121616 −0.0608082 0.998149i \(-0.519368\pi\)
−0.0608082 + 0.998149i \(0.519368\pi\)
\(972\) 0 0
\(973\) −777.796 −0.0256269
\(974\) 0 0
\(975\) −14153.0 −0.464881
\(976\) 0 0
\(977\) −16462.8 −0.539092 −0.269546 0.962988i \(-0.586874\pi\)
−0.269546 + 0.962988i \(0.586874\pi\)
\(978\) 0 0
\(979\) 5974.08 0.195028
\(980\) 0 0
\(981\) 1891.49 0.0615604
\(982\) 0 0
\(983\) 38116.1 1.23674 0.618370 0.785887i \(-0.287793\pi\)
0.618370 + 0.785887i \(0.287793\pi\)
\(984\) 0 0
\(985\) −13929.3 −0.450583
\(986\) 0 0
\(987\) −87.7753 −0.00283072
\(988\) 0 0
\(989\) −11286.5 −0.362882
\(990\) 0 0
\(991\) 8909.11 0.285577 0.142789 0.989753i \(-0.454393\pi\)
0.142789 + 0.989753i \(0.454393\pi\)
\(992\) 0 0
\(993\) 25280.6 0.807910
\(994\) 0 0
\(995\) 93274.7 2.97186
\(996\) 0 0
\(997\) 36786.4 1.16854 0.584271 0.811559i \(-0.301381\pi\)
0.584271 + 0.811559i \(0.301381\pi\)
\(998\) 0 0
\(999\) 1666.46 0.0527773
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.ca.1.5 5
4.3 odd 2 2496.4.a.cf.1.5 5
8.3 odd 2 1248.4.a.g.1.1 5
8.5 even 2 1248.4.a.l.1.1 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1248.4.a.g.1.1 5 8.3 odd 2
1248.4.a.l.1.1 yes 5 8.5 even 2
2496.4.a.ca.1.5 5 1.1 even 1 trivial
2496.4.a.cf.1.5 5 4.3 odd 2