Properties

Label 480.4.a.m.1.1
Level $480$
Weight $4$
Character 480.1
Self dual yes
Analytic conductor $28.321$
Analytic rank $1$
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] = [480,4,Mod(1,480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(480, 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("480.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 480.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: \(28.3209168028\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.70156\) of defining polynomial
Character \(\chi\) \(=\) 480.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -5.00000 q^{5} -18.8062 q^{7} +9.00000 q^{9} +63.2250 q^{11} +1.58125 q^{13} +15.0000 q^{15} +135.256 q^{17} -97.2562 q^{19} +56.4187 q^{21} -41.1938 q^{23} +25.0000 q^{25} -27.0000 q^{27} -207.675 q^{29} -193.969 q^{31} -189.675 q^{33} +94.0312 q^{35} +339.256 q^{37} -4.74376 q^{39} -490.187 q^{41} -74.3250 q^{43} -45.0000 q^{45} -544.481 q^{47} +10.6750 q^{49} -405.769 q^{51} +663.862 q^{53} -316.125 q^{55} +291.769 q^{57} -344.775 q^{59} +5.16251 q^{61} -169.256 q^{63} -7.90627 q^{65} +671.475 q^{67} +123.581 q^{69} -425.550 q^{71} -94.8375 q^{73} -75.0000 q^{75} -1189.02 q^{77} -770.681 q^{79} +81.0000 q^{81} -589.800 q^{83} -676.281 q^{85} +623.025 q^{87} -409.813 q^{89} -29.7375 q^{91} +581.906 q^{93} +486.281 q^{95} -152.050 q^{97} +569.025 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 10 q^{5} - 12 q^{7} + 18 q^{9} + 24 q^{11} + 80 q^{13} + 30 q^{15} + 40 q^{17} + 36 q^{19} + 36 q^{21} - 108 q^{23} + 50 q^{25} - 54 q^{27} - 108 q^{29} - 516 q^{31} - 72 q^{33} + 60 q^{35}+ \cdots + 216 q^{99}+O(q^{100}) \) 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
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −18.8062 −1.01544 −0.507721 0.861522i \(-0.669512\pi\)
−0.507721 + 0.861522i \(0.669512\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 63.2250 1.73300 0.866502 0.499173i \(-0.166363\pi\)
0.866502 + 0.499173i \(0.166363\pi\)
\(12\) 0 0
\(13\) 1.58125 0.0337355 0.0168677 0.999858i \(-0.494631\pi\)
0.0168677 + 0.999858i \(0.494631\pi\)
\(14\) 0 0
\(15\) 15.0000 0.258199
\(16\) 0 0
\(17\) 135.256 1.92967 0.964837 0.262849i \(-0.0846622\pi\)
0.964837 + 0.262849i \(0.0846622\pi\)
\(18\) 0 0
\(19\) −97.2562 −1.17432 −0.587161 0.809470i \(-0.699754\pi\)
−0.587161 + 0.809470i \(0.699754\pi\)
\(20\) 0 0
\(21\) 56.4187 0.586266
\(22\) 0 0
\(23\) −41.1938 −0.373456 −0.186728 0.982412i \(-0.559788\pi\)
−0.186728 + 0.982412i \(0.559788\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −207.675 −1.32980 −0.664901 0.746931i \(-0.731526\pi\)
−0.664901 + 0.746931i \(0.731526\pi\)
\(30\) 0 0
\(31\) −193.969 −1.12380 −0.561900 0.827205i \(-0.689930\pi\)
−0.561900 + 0.827205i \(0.689930\pi\)
\(32\) 0 0
\(33\) −189.675 −1.00055
\(34\) 0 0
\(35\) 94.0312 0.454119
\(36\) 0 0
\(37\) 339.256 1.50739 0.753694 0.657225i \(-0.228270\pi\)
0.753694 + 0.657225i \(0.228270\pi\)
\(38\) 0 0
\(39\) −4.74376 −0.0194772
\(40\) 0 0
\(41\) −490.187 −1.86718 −0.933590 0.358342i \(-0.883342\pi\)
−0.933590 + 0.358342i \(0.883342\pi\)
\(42\) 0 0
\(43\) −74.3250 −0.263592 −0.131796 0.991277i \(-0.542074\pi\)
−0.131796 + 0.991277i \(0.542074\pi\)
\(44\) 0 0
\(45\) −45.0000 −0.149071
\(46\) 0 0
\(47\) −544.481 −1.68980 −0.844902 0.534922i \(-0.820341\pi\)
−0.844902 + 0.534922i \(0.820341\pi\)
\(48\) 0 0
\(49\) 10.6750 0.0311224
\(50\) 0 0
\(51\) −405.769 −1.11410
\(52\) 0 0
\(53\) 663.862 1.72054 0.860269 0.509840i \(-0.170296\pi\)
0.860269 + 0.509840i \(0.170296\pi\)
\(54\) 0 0
\(55\) −316.125 −0.775023
\(56\) 0 0
\(57\) 291.769 0.677995
\(58\) 0 0
\(59\) −344.775 −0.760778 −0.380389 0.924827i \(-0.624210\pi\)
−0.380389 + 0.924827i \(0.624210\pi\)
\(60\) 0 0
\(61\) 5.16251 0.0108359 0.00541796 0.999985i \(-0.498275\pi\)
0.00541796 + 0.999985i \(0.498275\pi\)
\(62\) 0 0
\(63\) −169.256 −0.338481
\(64\) 0 0
\(65\) −7.90627 −0.0150870
\(66\) 0 0
\(67\) 671.475 1.22438 0.612192 0.790709i \(-0.290288\pi\)
0.612192 + 0.790709i \(0.290288\pi\)
\(68\) 0 0
\(69\) 123.581 0.215615
\(70\) 0 0
\(71\) −425.550 −0.711317 −0.355658 0.934616i \(-0.615743\pi\)
−0.355658 + 0.934616i \(0.615743\pi\)
\(72\) 0 0
\(73\) −94.8375 −0.152053 −0.0760266 0.997106i \(-0.524223\pi\)
−0.0760266 + 0.997106i \(0.524223\pi\)
\(74\) 0 0
\(75\) −75.0000 −0.115470
\(76\) 0 0
\(77\) −1189.02 −1.75977
\(78\) 0 0
\(79\) −770.681 −1.09757 −0.548787 0.835962i \(-0.684910\pi\)
−0.548787 + 0.835962i \(0.684910\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −589.800 −0.779987 −0.389994 0.920818i \(-0.627523\pi\)
−0.389994 + 0.920818i \(0.627523\pi\)
\(84\) 0 0
\(85\) −676.281 −0.862976
\(86\) 0 0
\(87\) 623.025 0.767762
\(88\) 0 0
\(89\) −409.813 −0.488090 −0.244045 0.969764i \(-0.578475\pi\)
−0.244045 + 0.969764i \(0.578475\pi\)
\(90\) 0 0
\(91\) −29.7375 −0.0342564
\(92\) 0 0
\(93\) 581.906 0.648827
\(94\) 0 0
\(95\) 486.281 0.525173
\(96\) 0 0
\(97\) −152.050 −0.159158 −0.0795790 0.996829i \(-0.525358\pi\)
−0.0795790 + 0.996829i \(0.525358\pi\)
\(98\) 0 0
\(99\) 569.025 0.577668
\(100\) 0 0
\(101\) −484.325 −0.477150 −0.238575 0.971124i \(-0.576680\pi\)
−0.238575 + 0.971124i \(0.576680\pi\)
\(102\) 0 0
\(103\) −1617.51 −1.54736 −0.773678 0.633579i \(-0.781585\pi\)
−0.773678 + 0.633579i \(0.781585\pi\)
\(104\) 0 0
\(105\) −282.094 −0.262186
\(106\) 0 0
\(107\) −1010.36 −0.912854 −0.456427 0.889761i \(-0.650871\pi\)
−0.456427 + 0.889761i \(0.650871\pi\)
\(108\) 0 0
\(109\) 1.72487 0.00151571 0.000757857 1.00000i \(-0.499759\pi\)
0.000757857 1.00000i \(0.499759\pi\)
\(110\) 0 0
\(111\) −1017.77 −0.870291
\(112\) 0 0
\(113\) −729.119 −0.606989 −0.303494 0.952833i \(-0.598153\pi\)
−0.303494 + 0.952833i \(0.598153\pi\)
\(114\) 0 0
\(115\) 205.969 0.167015
\(116\) 0 0
\(117\) 14.2313 0.0112452
\(118\) 0 0
\(119\) −2543.66 −1.95947
\(120\) 0 0
\(121\) 2666.40 2.00331
\(122\) 0 0
\(123\) 1470.56 1.07802
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −388.481 −0.271434 −0.135717 0.990748i \(-0.543334\pi\)
−0.135717 + 0.990748i \(0.543334\pi\)
\(128\) 0 0
\(129\) 222.975 0.152185
\(130\) 0 0
\(131\) 2777.06 1.85216 0.926080 0.377326i \(-0.123156\pi\)
0.926080 + 0.377326i \(0.123156\pi\)
\(132\) 0 0
\(133\) 1829.02 1.19246
\(134\) 0 0
\(135\) 135.000 0.0860663
\(136\) 0 0
\(137\) 173.119 0.107960 0.0539800 0.998542i \(-0.482809\pi\)
0.0539800 + 0.998542i \(0.482809\pi\)
\(138\) 0 0
\(139\) 450.806 0.275086 0.137543 0.990496i \(-0.456080\pi\)
0.137543 + 0.990496i \(0.456080\pi\)
\(140\) 0 0
\(141\) 1633.44 0.975608
\(142\) 0 0
\(143\) 99.9748 0.0584637
\(144\) 0 0
\(145\) 1038.37 0.594706
\(146\) 0 0
\(147\) −32.0249 −0.0179685
\(148\) 0 0
\(149\) −1059.02 −0.582273 −0.291137 0.956681i \(-0.594033\pi\)
−0.291137 + 0.956681i \(0.594033\pi\)
\(150\) 0 0
\(151\) 1273.63 0.686402 0.343201 0.939262i \(-0.388489\pi\)
0.343201 + 0.939262i \(0.388489\pi\)
\(152\) 0 0
\(153\) 1217.31 0.643225
\(154\) 0 0
\(155\) 969.844 0.502579
\(156\) 0 0
\(157\) −1442.61 −0.733328 −0.366664 0.930353i \(-0.619500\pi\)
−0.366664 + 0.930353i \(0.619500\pi\)
\(158\) 0 0
\(159\) −1991.59 −0.993353
\(160\) 0 0
\(161\) 774.700 0.379223
\(162\) 0 0
\(163\) −2440.35 −1.17266 −0.586328 0.810074i \(-0.699427\pi\)
−0.586328 + 0.810074i \(0.699427\pi\)
\(164\) 0 0
\(165\) 948.375 0.447460
\(166\) 0 0
\(167\) 3655.18 1.69369 0.846846 0.531839i \(-0.178499\pi\)
0.846846 + 0.531839i \(0.178499\pi\)
\(168\) 0 0
\(169\) −2194.50 −0.998862
\(170\) 0 0
\(171\) −875.306 −0.391441
\(172\) 0 0
\(173\) −3014.75 −1.32490 −0.662449 0.749107i \(-0.730483\pi\)
−0.662449 + 0.749107i \(0.730483\pi\)
\(174\) 0 0
\(175\) −470.156 −0.203088
\(176\) 0 0
\(177\) 1034.33 0.439235
\(178\) 0 0
\(179\) 1683.04 0.702772 0.351386 0.936231i \(-0.385711\pi\)
0.351386 + 0.936231i \(0.385711\pi\)
\(180\) 0 0
\(181\) −2507.12 −1.02958 −0.514788 0.857318i \(-0.672129\pi\)
−0.514788 + 0.857318i \(0.672129\pi\)
\(182\) 0 0
\(183\) −15.4875 −0.00625613
\(184\) 0 0
\(185\) −1696.28 −0.674125
\(186\) 0 0
\(187\) 8551.57 3.34413
\(188\) 0 0
\(189\) 507.769 0.195422
\(190\) 0 0
\(191\) −3251.17 −1.23166 −0.615829 0.787880i \(-0.711179\pi\)
−0.615829 + 0.787880i \(0.711179\pi\)
\(192\) 0 0
\(193\) 2468.89 0.920800 0.460400 0.887712i \(-0.347706\pi\)
0.460400 + 0.887712i \(0.347706\pi\)
\(194\) 0 0
\(195\) 23.7188 0.00871046
\(196\) 0 0
\(197\) 155.212 0.0561341 0.0280671 0.999606i \(-0.491065\pi\)
0.0280671 + 0.999606i \(0.491065\pi\)
\(198\) 0 0
\(199\) 2117.98 0.754471 0.377235 0.926117i \(-0.376875\pi\)
0.377235 + 0.926117i \(0.376875\pi\)
\(200\) 0 0
\(201\) −2014.42 −0.706898
\(202\) 0 0
\(203\) 3905.59 1.35034
\(204\) 0 0
\(205\) 2450.94 0.835029
\(206\) 0 0
\(207\) −370.744 −0.124485
\(208\) 0 0
\(209\) −6149.02 −2.03511
\(210\) 0 0
\(211\) −613.444 −0.200148 −0.100074 0.994980i \(-0.531908\pi\)
−0.100074 + 0.994980i \(0.531908\pi\)
\(212\) 0 0
\(213\) 1276.65 0.410679
\(214\) 0 0
\(215\) 371.625 0.117882
\(216\) 0 0
\(217\) 3647.82 1.14115
\(218\) 0 0
\(219\) 284.512 0.0877880
\(220\) 0 0
\(221\) 213.875 0.0650985
\(222\) 0 0
\(223\) 1153.26 0.346313 0.173156 0.984894i \(-0.444603\pi\)
0.173156 + 0.984894i \(0.444603\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) 4873.65 1.42500 0.712501 0.701671i \(-0.247562\pi\)
0.712501 + 0.701671i \(0.247562\pi\)
\(228\) 0 0
\(229\) 597.163 0.172321 0.0861607 0.996281i \(-0.472540\pi\)
0.0861607 + 0.996281i \(0.472540\pi\)
\(230\) 0 0
\(231\) 3567.07 1.01600
\(232\) 0 0
\(233\) −5302.61 −1.49092 −0.745462 0.666548i \(-0.767771\pi\)
−0.745462 + 0.666548i \(0.767771\pi\)
\(234\) 0 0
\(235\) 2722.41 0.755703
\(236\) 0 0
\(237\) 2312.04 0.633685
\(238\) 0 0
\(239\) −3571.31 −0.966565 −0.483282 0.875464i \(-0.660556\pi\)
−0.483282 + 0.875464i \(0.660556\pi\)
\(240\) 0 0
\(241\) 4796.98 1.28216 0.641080 0.767474i \(-0.278487\pi\)
0.641080 + 0.767474i \(0.278487\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −53.3749 −0.0139184
\(246\) 0 0
\(247\) −153.787 −0.0396163
\(248\) 0 0
\(249\) 1769.40 0.450326
\(250\) 0 0
\(251\) −2441.44 −0.613953 −0.306976 0.951717i \(-0.599317\pi\)
−0.306976 + 0.951717i \(0.599317\pi\)
\(252\) 0 0
\(253\) −2604.47 −0.647201
\(254\) 0 0
\(255\) 2028.84 0.498240
\(256\) 0 0
\(257\) −4163.17 −1.01047 −0.505236 0.862981i \(-0.668595\pi\)
−0.505236 + 0.862981i \(0.668595\pi\)
\(258\) 0 0
\(259\) −6380.14 −1.53067
\(260\) 0 0
\(261\) −1869.07 −0.443268
\(262\) 0 0
\(263\) 543.694 0.127474 0.0637369 0.997967i \(-0.479698\pi\)
0.0637369 + 0.997967i \(0.479698\pi\)
\(264\) 0 0
\(265\) −3319.31 −0.769448
\(266\) 0 0
\(267\) 1229.44 0.281799
\(268\) 0 0
\(269\) 992.325 0.224919 0.112459 0.993656i \(-0.464127\pi\)
0.112459 + 0.993656i \(0.464127\pi\)
\(270\) 0 0
\(271\) 5302.11 1.18849 0.594244 0.804285i \(-0.297452\pi\)
0.594244 + 0.804285i \(0.297452\pi\)
\(272\) 0 0
\(273\) 89.2124 0.0197779
\(274\) 0 0
\(275\) 1580.62 0.346601
\(276\) 0 0
\(277\) −929.107 −0.201533 −0.100766 0.994910i \(-0.532129\pi\)
−0.100766 + 0.994910i \(0.532129\pi\)
\(278\) 0 0
\(279\) −1745.72 −0.374600
\(280\) 0 0
\(281\) −7372.24 −1.56509 −0.782546 0.622593i \(-0.786079\pi\)
−0.782546 + 0.622593i \(0.786079\pi\)
\(282\) 0 0
\(283\) −4510.05 −0.947331 −0.473665 0.880705i \(-0.657069\pi\)
−0.473665 + 0.880705i \(0.657069\pi\)
\(284\) 0 0
\(285\) −1458.84 −0.303209
\(286\) 0 0
\(287\) 9218.59 1.89601
\(288\) 0 0
\(289\) 13381.2 2.72364
\(290\) 0 0
\(291\) 456.150 0.0918899
\(292\) 0 0
\(293\) 4358.94 0.869119 0.434559 0.900643i \(-0.356904\pi\)
0.434559 + 0.900643i \(0.356904\pi\)
\(294\) 0 0
\(295\) 1723.88 0.340230
\(296\) 0 0
\(297\) −1707.07 −0.333517
\(298\) 0 0
\(299\) −65.1378 −0.0125987
\(300\) 0 0
\(301\) 1397.77 0.267662
\(302\) 0 0
\(303\) 1452.98 0.275483
\(304\) 0 0
\(305\) −25.8125 −0.00484597
\(306\) 0 0
\(307\) 788.962 0.146673 0.0733363 0.997307i \(-0.476635\pi\)
0.0733363 + 0.997307i \(0.476635\pi\)
\(308\) 0 0
\(309\) 4852.52 0.893366
\(310\) 0 0
\(311\) −956.062 −0.174319 −0.0871597 0.996194i \(-0.527779\pi\)
−0.0871597 + 0.996194i \(0.527779\pi\)
\(312\) 0 0
\(313\) −728.887 −0.131627 −0.0658133 0.997832i \(-0.520964\pi\)
−0.0658133 + 0.997832i \(0.520964\pi\)
\(314\) 0 0
\(315\) 846.281 0.151373
\(316\) 0 0
\(317\) 816.788 0.144717 0.0723586 0.997379i \(-0.476947\pi\)
0.0723586 + 0.997379i \(0.476947\pi\)
\(318\) 0 0
\(319\) −13130.2 −2.30455
\(320\) 0 0
\(321\) 3031.09 0.527037
\(322\) 0 0
\(323\) −13154.5 −2.26606
\(324\) 0 0
\(325\) 39.5314 0.00674709
\(326\) 0 0
\(327\) −5.17462 −0.000875098 0
\(328\) 0 0
\(329\) 10239.6 1.71590
\(330\) 0 0
\(331\) 3787.26 0.628902 0.314451 0.949274i \(-0.398180\pi\)
0.314451 + 0.949274i \(0.398180\pi\)
\(332\) 0 0
\(333\) 3053.31 0.502463
\(334\) 0 0
\(335\) −3357.37 −0.547561
\(336\) 0 0
\(337\) −4959.79 −0.801712 −0.400856 0.916141i \(-0.631287\pi\)
−0.400856 + 0.916141i \(0.631287\pi\)
\(338\) 0 0
\(339\) 2187.36 0.350445
\(340\) 0 0
\(341\) −12263.7 −1.94755
\(342\) 0 0
\(343\) 6249.79 0.983839
\(344\) 0 0
\(345\) −617.906 −0.0964260
\(346\) 0 0
\(347\) −1001.51 −0.154939 −0.0774697 0.996995i \(-0.524684\pi\)
−0.0774697 + 0.996995i \(0.524684\pi\)
\(348\) 0 0
\(349\) −11048.8 −1.69464 −0.847320 0.531083i \(-0.821785\pi\)
−0.847320 + 0.531083i \(0.821785\pi\)
\(350\) 0 0
\(351\) −42.6939 −0.00649239
\(352\) 0 0
\(353\) −1216.63 −0.183441 −0.0917207 0.995785i \(-0.529237\pi\)
−0.0917207 + 0.995785i \(0.529237\pi\)
\(354\) 0 0
\(355\) 2127.75 0.318111
\(356\) 0 0
\(357\) 7630.99 1.13130
\(358\) 0 0
\(359\) −967.125 −0.142181 −0.0710904 0.997470i \(-0.522648\pi\)
−0.0710904 + 0.997470i \(0.522648\pi\)
\(360\) 0 0
\(361\) 2599.78 0.379031
\(362\) 0 0
\(363\) −7999.20 −1.15661
\(364\) 0 0
\(365\) 474.187 0.0680003
\(366\) 0 0
\(367\) 4356.06 0.619576 0.309788 0.950806i \(-0.399742\pi\)
0.309788 + 0.950806i \(0.399742\pi\)
\(368\) 0 0
\(369\) −4411.69 −0.622394
\(370\) 0 0
\(371\) −12484.8 −1.74711
\(372\) 0 0
\(373\) −319.982 −0.0444183 −0.0222092 0.999753i \(-0.507070\pi\)
−0.0222092 + 0.999753i \(0.507070\pi\)
\(374\) 0 0
\(375\) 375.000 0.0516398
\(376\) 0 0
\(377\) −328.387 −0.0448615
\(378\) 0 0
\(379\) 1188.39 0.161065 0.0805326 0.996752i \(-0.474338\pi\)
0.0805326 + 0.996752i \(0.474338\pi\)
\(380\) 0 0
\(381\) 1165.44 0.156713
\(382\) 0 0
\(383\) −3150.51 −0.420322 −0.210161 0.977667i \(-0.567399\pi\)
−0.210161 + 0.977667i \(0.567399\pi\)
\(384\) 0 0
\(385\) 5945.12 0.786991
\(386\) 0 0
\(387\) −668.925 −0.0878640
\(388\) 0 0
\(389\) 6201.45 0.808293 0.404147 0.914694i \(-0.367569\pi\)
0.404147 + 0.914694i \(0.367569\pi\)
\(390\) 0 0
\(391\) −5571.71 −0.720649
\(392\) 0 0
\(393\) −8331.19 −1.06935
\(394\) 0 0
\(395\) 3853.41 0.490850
\(396\) 0 0
\(397\) −4225.39 −0.534172 −0.267086 0.963673i \(-0.586061\pi\)
−0.267086 + 0.963673i \(0.586061\pi\)
\(398\) 0 0
\(399\) −5487.07 −0.688464
\(400\) 0 0
\(401\) −5691.02 −0.708719 −0.354359 0.935109i \(-0.615301\pi\)
−0.354359 + 0.935109i \(0.615301\pi\)
\(402\) 0 0
\(403\) −306.714 −0.0379119
\(404\) 0 0
\(405\) −405.000 −0.0496904
\(406\) 0 0
\(407\) 21449.5 2.61231
\(408\) 0 0
\(409\) 2831.60 0.342331 0.171166 0.985242i \(-0.445247\pi\)
0.171166 + 0.985242i \(0.445247\pi\)
\(410\) 0 0
\(411\) −519.356 −0.0623308
\(412\) 0 0
\(413\) 6483.92 0.772526
\(414\) 0 0
\(415\) 2949.00 0.348821
\(416\) 0 0
\(417\) −1352.42 −0.158821
\(418\) 0 0
\(419\) −13660.9 −1.59279 −0.796395 0.604776i \(-0.793262\pi\)
−0.796395 + 0.604776i \(0.793262\pi\)
\(420\) 0 0
\(421\) 9252.46 1.07111 0.535555 0.844500i \(-0.320102\pi\)
0.535555 + 0.844500i \(0.320102\pi\)
\(422\) 0 0
\(423\) −4900.33 −0.563268
\(424\) 0 0
\(425\) 3381.41 0.385935
\(426\) 0 0
\(427\) −97.0874 −0.0110033
\(428\) 0 0
\(429\) −299.924 −0.0337541
\(430\) 0 0
\(431\) 397.275 0.0443992 0.0221996 0.999754i \(-0.492933\pi\)
0.0221996 + 0.999754i \(0.492933\pi\)
\(432\) 0 0
\(433\) −1122.39 −0.124569 −0.0622846 0.998058i \(-0.519839\pi\)
−0.0622846 + 0.998058i \(0.519839\pi\)
\(434\) 0 0
\(435\) −3115.12 −0.343354
\(436\) 0 0
\(437\) 4006.35 0.438558
\(438\) 0 0
\(439\) −17424.7 −1.89439 −0.947193 0.320665i \(-0.896094\pi\)
−0.947193 + 0.320665i \(0.896094\pi\)
\(440\) 0 0
\(441\) 96.0748 0.0103741
\(442\) 0 0
\(443\) −6448.20 −0.691565 −0.345782 0.938315i \(-0.612386\pi\)
−0.345782 + 0.938315i \(0.612386\pi\)
\(444\) 0 0
\(445\) 2049.06 0.218281
\(446\) 0 0
\(447\) 3177.07 0.336176
\(448\) 0 0
\(449\) −9622.97 −1.01144 −0.505720 0.862698i \(-0.668773\pi\)
−0.505720 + 0.862698i \(0.668773\pi\)
\(450\) 0 0
\(451\) −30992.1 −3.23583
\(452\) 0 0
\(453\) −3820.89 −0.396294
\(454\) 0 0
\(455\) 148.687 0.0153199
\(456\) 0 0
\(457\) 6424.55 0.657610 0.328805 0.944398i \(-0.393354\pi\)
0.328805 + 0.944398i \(0.393354\pi\)
\(458\) 0 0
\(459\) −3651.92 −0.371366
\(460\) 0 0
\(461\) 7773.75 0.785379 0.392689 0.919671i \(-0.371545\pi\)
0.392689 + 0.919671i \(0.371545\pi\)
\(462\) 0 0
\(463\) −7122.96 −0.714972 −0.357486 0.933919i \(-0.616366\pi\)
−0.357486 + 0.933919i \(0.616366\pi\)
\(464\) 0 0
\(465\) −2909.53 −0.290164
\(466\) 0 0
\(467\) −4292.21 −0.425310 −0.212655 0.977127i \(-0.568211\pi\)
−0.212655 + 0.977127i \(0.568211\pi\)
\(468\) 0 0
\(469\) −12627.9 −1.24329
\(470\) 0 0
\(471\) 4327.82 0.423387
\(472\) 0 0
\(473\) −4699.20 −0.456806
\(474\) 0 0
\(475\) −2431.41 −0.234864
\(476\) 0 0
\(477\) 5974.76 0.573513
\(478\) 0 0
\(479\) −12607.8 −1.20264 −0.601321 0.799008i \(-0.705359\pi\)
−0.601321 + 0.799008i \(0.705359\pi\)
\(480\) 0 0
\(481\) 536.450 0.0508525
\(482\) 0 0
\(483\) −2324.10 −0.218945
\(484\) 0 0
\(485\) 760.249 0.0711776
\(486\) 0 0
\(487\) 14242.9 1.32527 0.662634 0.748943i \(-0.269439\pi\)
0.662634 + 0.748943i \(0.269439\pi\)
\(488\) 0 0
\(489\) 7321.05 0.677033
\(490\) 0 0
\(491\) 19491.8 1.79156 0.895778 0.444502i \(-0.146619\pi\)
0.895778 + 0.444502i \(0.146619\pi\)
\(492\) 0 0
\(493\) −28089.3 −2.56609
\(494\) 0 0
\(495\) −2845.12 −0.258341
\(496\) 0 0
\(497\) 8003.00 0.722301
\(498\) 0 0
\(499\) 8728.93 0.783087 0.391544 0.920160i \(-0.371941\pi\)
0.391544 + 0.920160i \(0.371941\pi\)
\(500\) 0 0
\(501\) −10965.5 −0.977853
\(502\) 0 0
\(503\) 10470.8 0.928175 0.464087 0.885789i \(-0.346382\pi\)
0.464087 + 0.885789i \(0.346382\pi\)
\(504\) 0 0
\(505\) 2421.63 0.213388
\(506\) 0 0
\(507\) 6583.50 0.576693
\(508\) 0 0
\(509\) −4779.80 −0.416230 −0.208115 0.978104i \(-0.566733\pi\)
−0.208115 + 0.978104i \(0.566733\pi\)
\(510\) 0 0
\(511\) 1783.54 0.154401
\(512\) 0 0
\(513\) 2625.92 0.225998
\(514\) 0 0
\(515\) 8087.53 0.691998
\(516\) 0 0
\(517\) −34424.8 −2.92844
\(518\) 0 0
\(519\) 9044.25 0.764930
\(520\) 0 0
\(521\) 20647.1 1.73621 0.868104 0.496382i \(-0.165338\pi\)
0.868104 + 0.496382i \(0.165338\pi\)
\(522\) 0 0
\(523\) −18853.5 −1.57630 −0.788149 0.615484i \(-0.788961\pi\)
−0.788149 + 0.615484i \(0.788961\pi\)
\(524\) 0 0
\(525\) 1410.47 0.117253
\(526\) 0 0
\(527\) −26235.5 −2.16857
\(528\) 0 0
\(529\) −10470.1 −0.860531
\(530\) 0 0
\(531\) −3102.98 −0.253593
\(532\) 0 0
\(533\) −775.111 −0.0629902
\(534\) 0 0
\(535\) 5051.81 0.408241
\(536\) 0 0
\(537\) −5049.11 −0.405745
\(538\) 0 0
\(539\) 674.926 0.0539353
\(540\) 0 0
\(541\) 8575.48 0.681494 0.340747 0.940155i \(-0.389320\pi\)
0.340747 + 0.940155i \(0.389320\pi\)
\(542\) 0 0
\(543\) 7521.37 0.594426
\(544\) 0 0
\(545\) −8.62436 −0.000677848 0
\(546\) 0 0
\(547\) −14884.0 −1.16343 −0.581715 0.813393i \(-0.697618\pi\)
−0.581715 + 0.813393i \(0.697618\pi\)
\(548\) 0 0
\(549\) 46.4626 0.00361198
\(550\) 0 0
\(551\) 20197.7 1.56162
\(552\) 0 0
\(553\) 14493.6 1.11452
\(554\) 0 0
\(555\) 5088.84 0.389206
\(556\) 0 0
\(557\) 9780.58 0.744015 0.372007 0.928230i \(-0.378670\pi\)
0.372007 + 0.928230i \(0.378670\pi\)
\(558\) 0 0
\(559\) −117.527 −0.00889240
\(560\) 0 0
\(561\) −25654.7 −1.93074
\(562\) 0 0
\(563\) −25413.3 −1.90239 −0.951194 0.308594i \(-0.900142\pi\)
−0.951194 + 0.308594i \(0.900142\pi\)
\(564\) 0 0
\(565\) 3645.59 0.271454
\(566\) 0 0
\(567\) −1523.31 −0.112827
\(568\) 0 0
\(569\) −766.399 −0.0564659 −0.0282330 0.999601i \(-0.508988\pi\)
−0.0282330 + 0.999601i \(0.508988\pi\)
\(570\) 0 0
\(571\) 2066.98 0.151489 0.0757447 0.997127i \(-0.475867\pi\)
0.0757447 + 0.997127i \(0.475867\pi\)
\(572\) 0 0
\(573\) 9753.52 0.711098
\(574\) 0 0
\(575\) −1029.84 −0.0746912
\(576\) 0 0
\(577\) 2235.48 0.161290 0.0806448 0.996743i \(-0.474302\pi\)
0.0806448 + 0.996743i \(0.474302\pi\)
\(578\) 0 0
\(579\) −7406.66 −0.531624
\(580\) 0 0
\(581\) 11091.9 0.792032
\(582\) 0 0
\(583\) 41972.7 2.98170
\(584\) 0 0
\(585\) −71.1565 −0.00502899
\(586\) 0 0
\(587\) 14951.7 1.05131 0.525657 0.850697i \(-0.323820\pi\)
0.525657 + 0.850697i \(0.323820\pi\)
\(588\) 0 0
\(589\) 18864.7 1.31970
\(590\) 0 0
\(591\) −465.637 −0.0324091
\(592\) 0 0
\(593\) 14265.4 0.987872 0.493936 0.869498i \(-0.335558\pi\)
0.493936 + 0.869498i \(0.335558\pi\)
\(594\) 0 0
\(595\) 12718.3 0.876302
\(596\) 0 0
\(597\) −6353.94 −0.435594
\(598\) 0 0
\(599\) −16069.6 −1.09614 −0.548068 0.836434i \(-0.684636\pi\)
−0.548068 + 0.836434i \(0.684636\pi\)
\(600\) 0 0
\(601\) 19227.5 1.30500 0.652501 0.757788i \(-0.273720\pi\)
0.652501 + 0.757788i \(0.273720\pi\)
\(602\) 0 0
\(603\) 6043.27 0.408128
\(604\) 0 0
\(605\) −13332.0 −0.895906
\(606\) 0 0
\(607\) 22962.2 1.53543 0.767716 0.640790i \(-0.221393\pi\)
0.767716 + 0.640790i \(0.221393\pi\)
\(608\) 0 0
\(609\) −11716.8 −0.779618
\(610\) 0 0
\(611\) −860.963 −0.0570063
\(612\) 0 0
\(613\) 23035.5 1.51777 0.758887 0.651223i \(-0.225744\pi\)
0.758887 + 0.651223i \(0.225744\pi\)
\(614\) 0 0
\(615\) −7352.81 −0.482104
\(616\) 0 0
\(617\) 17460.2 1.13926 0.569628 0.821903i \(-0.307087\pi\)
0.569628 + 0.821903i \(0.307087\pi\)
\(618\) 0 0
\(619\) 14611.4 0.948759 0.474380 0.880320i \(-0.342672\pi\)
0.474380 + 0.880320i \(0.342672\pi\)
\(620\) 0 0
\(621\) 1112.23 0.0718717
\(622\) 0 0
\(623\) 7707.04 0.495627
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 18447.1 1.17497
\(628\) 0 0
\(629\) 45886.5 2.90877
\(630\) 0 0
\(631\) −4126.33 −0.260327 −0.130164 0.991493i \(-0.541550\pi\)
−0.130164 + 0.991493i \(0.541550\pi\)
\(632\) 0 0
\(633\) 1840.33 0.115555
\(634\) 0 0
\(635\) 1942.41 0.121389
\(636\) 0 0
\(637\) 16.8799 0.00104993
\(638\) 0 0
\(639\) −3829.95 −0.237106
\(640\) 0 0
\(641\) −21550.1 −1.32789 −0.663946 0.747780i \(-0.731120\pi\)
−0.663946 + 0.747780i \(0.731120\pi\)
\(642\) 0 0
\(643\) 13528.4 0.829718 0.414859 0.909886i \(-0.363831\pi\)
0.414859 + 0.909886i \(0.363831\pi\)
\(644\) 0 0
\(645\) −1114.88 −0.0680592
\(646\) 0 0
\(647\) −3157.14 −0.191839 −0.0959197 0.995389i \(-0.530579\pi\)
−0.0959197 + 0.995389i \(0.530579\pi\)
\(648\) 0 0
\(649\) −21798.4 −1.31843
\(650\) 0 0
\(651\) −10943.5 −0.658846
\(652\) 0 0
\(653\) 9644.38 0.577969 0.288984 0.957334i \(-0.406682\pi\)
0.288984 + 0.957334i \(0.406682\pi\)
\(654\) 0 0
\(655\) −13885.3 −0.828311
\(656\) 0 0
\(657\) −853.537 −0.0506844
\(658\) 0 0
\(659\) 19276.7 1.13948 0.569738 0.821826i \(-0.307045\pi\)
0.569738 + 0.821826i \(0.307045\pi\)
\(660\) 0 0
\(661\) −7366.06 −0.433444 −0.216722 0.976233i \(-0.569537\pi\)
−0.216722 + 0.976233i \(0.569537\pi\)
\(662\) 0 0
\(663\) −641.624 −0.0375846
\(664\) 0 0
\(665\) −9145.12 −0.533282
\(666\) 0 0
\(667\) 8554.91 0.496623
\(668\) 0 0
\(669\) −3459.77 −0.199944
\(670\) 0 0
\(671\) 326.400 0.0187787
\(672\) 0 0
\(673\) 3283.74 0.188081 0.0940407 0.995568i \(-0.470022\pi\)
0.0940407 + 0.995568i \(0.470022\pi\)
\(674\) 0 0
\(675\) −675.000 −0.0384900
\(676\) 0 0
\(677\) −8895.32 −0.504985 −0.252493 0.967599i \(-0.581250\pi\)
−0.252493 + 0.967599i \(0.581250\pi\)
\(678\) 0 0
\(679\) 2859.49 0.161616
\(680\) 0 0
\(681\) −14620.9 −0.822725
\(682\) 0 0
\(683\) 14632.3 0.819751 0.409875 0.912141i \(-0.365572\pi\)
0.409875 + 0.912141i \(0.365572\pi\)
\(684\) 0 0
\(685\) −865.593 −0.0482812
\(686\) 0 0
\(687\) −1791.49 −0.0994898
\(688\) 0 0
\(689\) 1049.74 0.0580432
\(690\) 0 0
\(691\) 7677.28 0.422659 0.211330 0.977415i \(-0.432221\pi\)
0.211330 + 0.977415i \(0.432221\pi\)
\(692\) 0 0
\(693\) −10701.2 −0.586589
\(694\) 0 0
\(695\) −2254.03 −0.123022
\(696\) 0 0
\(697\) −66300.9 −3.60305
\(698\) 0 0
\(699\) 15907.8 0.860785
\(700\) 0 0
\(701\) 2348.58 0.126540 0.0632700 0.997996i \(-0.479847\pi\)
0.0632700 + 0.997996i \(0.479847\pi\)
\(702\) 0 0
\(703\) −32994.8 −1.77016
\(704\) 0 0
\(705\) −8167.22 −0.436305
\(706\) 0 0
\(707\) 9108.34 0.484518
\(708\) 0 0
\(709\) −30897.1 −1.63662 −0.818310 0.574777i \(-0.805089\pi\)
−0.818310 + 0.574777i \(0.805089\pi\)
\(710\) 0 0
\(711\) −6936.13 −0.365858
\(712\) 0 0
\(713\) 7990.30 0.419690
\(714\) 0 0
\(715\) −499.874 −0.0261458
\(716\) 0 0
\(717\) 10713.9 0.558047
\(718\) 0 0
\(719\) 7448.66 0.386353 0.193177 0.981164i \(-0.438121\pi\)
0.193177 + 0.981164i \(0.438121\pi\)
\(720\) 0 0
\(721\) 30419.2 1.57125
\(722\) 0 0
\(723\) −14390.9 −0.740255
\(724\) 0 0
\(725\) −5191.87 −0.265961
\(726\) 0 0
\(727\) 5301.58 0.270460 0.135230 0.990814i \(-0.456823\pi\)
0.135230 + 0.990814i \(0.456823\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −10052.9 −0.508647
\(732\) 0 0
\(733\) −18909.0 −0.952825 −0.476413 0.879222i \(-0.658063\pi\)
−0.476413 + 0.879222i \(0.658063\pi\)
\(734\) 0 0
\(735\) 160.125 0.00803577
\(736\) 0 0
\(737\) 42454.0 2.12186
\(738\) 0 0
\(739\) 30354.5 1.51097 0.755486 0.655165i \(-0.227401\pi\)
0.755486 + 0.655165i \(0.227401\pi\)
\(740\) 0 0
\(741\) 461.361 0.0228725
\(742\) 0 0
\(743\) 26539.9 1.31044 0.655219 0.755439i \(-0.272576\pi\)
0.655219 + 0.755439i \(0.272576\pi\)
\(744\) 0 0
\(745\) 5295.12 0.260400
\(746\) 0 0
\(747\) −5308.20 −0.259996
\(748\) 0 0
\(749\) 19001.1 0.926951
\(750\) 0 0
\(751\) 23097.1 1.12227 0.561136 0.827724i \(-0.310364\pi\)
0.561136 + 0.827724i \(0.310364\pi\)
\(752\) 0 0
\(753\) 7324.31 0.354466
\(754\) 0 0
\(755\) −6368.16 −0.306968
\(756\) 0 0
\(757\) 25979.5 1.24734 0.623672 0.781686i \(-0.285640\pi\)
0.623672 + 0.781686i \(0.285640\pi\)
\(758\) 0 0
\(759\) 7813.42 0.373662
\(760\) 0 0
\(761\) −21314.3 −1.01530 −0.507649 0.861564i \(-0.669485\pi\)
−0.507649 + 0.861564i \(0.669485\pi\)
\(762\) 0 0
\(763\) −32.4384 −0.00153912
\(764\) 0 0
\(765\) −6086.53 −0.287659
\(766\) 0 0
\(767\) −545.177 −0.0256652
\(768\) 0 0
\(769\) −2318.70 −0.108731 −0.0543657 0.998521i \(-0.517314\pi\)
−0.0543657 + 0.998521i \(0.517314\pi\)
\(770\) 0 0
\(771\) 12489.5 0.583396
\(772\) 0 0
\(773\) −13988.2 −0.650868 −0.325434 0.945565i \(-0.605510\pi\)
−0.325434 + 0.945565i \(0.605510\pi\)
\(774\) 0 0
\(775\) −4849.22 −0.224760
\(776\) 0 0
\(777\) 19140.4 0.883730
\(778\) 0 0
\(779\) 47673.8 2.19267
\(780\) 0 0
\(781\) −26905.4 −1.23272
\(782\) 0 0
\(783\) 5607.22 0.255921
\(784\) 0 0
\(785\) 7213.03 0.327954
\(786\) 0 0
\(787\) −9663.71 −0.437705 −0.218853 0.975758i \(-0.570231\pi\)
−0.218853 + 0.975758i \(0.570231\pi\)
\(788\) 0 0
\(789\) −1631.08 −0.0735970
\(790\) 0 0
\(791\) 13712.0 0.616362
\(792\) 0 0
\(793\) 8.16324 0.000365555 0
\(794\) 0 0
\(795\) 9957.94 0.444241
\(796\) 0 0
\(797\) 35878.9 1.59460 0.797299 0.603584i \(-0.206261\pi\)
0.797299 + 0.603584i \(0.206261\pi\)
\(798\) 0 0
\(799\) −73644.5 −3.26077
\(800\) 0 0
\(801\) −3688.31 −0.162697
\(802\) 0 0
\(803\) −5996.10 −0.263509
\(804\) 0 0
\(805\) −3873.50 −0.169594
\(806\) 0 0
\(807\) −2976.98 −0.129857
\(808\) 0 0
\(809\) −18021.0 −0.783172 −0.391586 0.920142i \(-0.628073\pi\)
−0.391586 + 0.920142i \(0.628073\pi\)
\(810\) 0 0
\(811\) 31290.5 1.35482 0.677409 0.735606i \(-0.263103\pi\)
0.677409 + 0.735606i \(0.263103\pi\)
\(812\) 0 0
\(813\) −15906.3 −0.686173
\(814\) 0 0
\(815\) 12201.7 0.524428
\(816\) 0 0
\(817\) 7228.57 0.309542
\(818\) 0 0
\(819\) −267.637 −0.0114188
\(820\) 0 0
\(821\) 17983.2 0.764458 0.382229 0.924068i \(-0.375157\pi\)
0.382229 + 0.924068i \(0.375157\pi\)
\(822\) 0 0
\(823\) 2663.27 0.112802 0.0564008 0.998408i \(-0.482038\pi\)
0.0564008 + 0.998408i \(0.482038\pi\)
\(824\) 0 0
\(825\) −4741.87 −0.200110
\(826\) 0 0
\(827\) −21334.0 −0.897043 −0.448522 0.893772i \(-0.648049\pi\)
−0.448522 + 0.893772i \(0.648049\pi\)
\(828\) 0 0
\(829\) −34240.2 −1.43451 −0.717256 0.696810i \(-0.754602\pi\)
−0.717256 + 0.696810i \(0.754602\pi\)
\(830\) 0 0
\(831\) 2787.32 0.116355
\(832\) 0 0
\(833\) 1443.86 0.0600561
\(834\) 0 0
\(835\) −18275.9 −0.757442
\(836\) 0 0
\(837\) 5237.16 0.216276
\(838\) 0 0
\(839\) −23674.2 −0.974165 −0.487082 0.873356i \(-0.661939\pi\)
−0.487082 + 0.873356i \(0.661939\pi\)
\(840\) 0 0
\(841\) 18739.9 0.768375
\(842\) 0 0
\(843\) 22116.7 0.903606
\(844\) 0 0
\(845\) 10972.5 0.446705
\(846\) 0 0
\(847\) −50145.0 −2.03424
\(848\) 0 0
\(849\) 13530.1 0.546942
\(850\) 0 0
\(851\) −13975.2 −0.562944
\(852\) 0 0
\(853\) −6935.77 −0.278401 −0.139201 0.990264i \(-0.544453\pi\)
−0.139201 + 0.990264i \(0.544453\pi\)
\(854\) 0 0
\(855\) 4376.53 0.175058
\(856\) 0 0
\(857\) 8482.56 0.338108 0.169054 0.985607i \(-0.445929\pi\)
0.169054 + 0.985607i \(0.445929\pi\)
\(858\) 0 0
\(859\) 37384.8 1.48493 0.742464 0.669886i \(-0.233657\pi\)
0.742464 + 0.669886i \(0.233657\pi\)
\(860\) 0 0
\(861\) −27655.8 −1.09466
\(862\) 0 0
\(863\) 16182.0 0.638286 0.319143 0.947707i \(-0.396605\pi\)
0.319143 + 0.947707i \(0.396605\pi\)
\(864\) 0 0
\(865\) 15073.7 0.592512
\(866\) 0 0
\(867\) −40143.7 −1.57250
\(868\) 0 0
\(869\) −48726.3 −1.90210
\(870\) 0 0
\(871\) 1061.77 0.0413052
\(872\) 0 0
\(873\) −1368.45 −0.0530527
\(874\) 0 0
\(875\) 2350.78 0.0908239
\(876\) 0 0
\(877\) −15908.8 −0.612545 −0.306273 0.951944i \(-0.599082\pi\)
−0.306273 + 0.951944i \(0.599082\pi\)
\(878\) 0 0
\(879\) −13076.8 −0.501786
\(880\) 0 0
\(881\) −32957.3 −1.26034 −0.630170 0.776457i \(-0.717015\pi\)
−0.630170 + 0.776457i \(0.717015\pi\)
\(882\) 0 0
\(883\) −2932.72 −0.111771 −0.0558856 0.998437i \(-0.517798\pi\)
−0.0558856 + 0.998437i \(0.517798\pi\)
\(884\) 0 0
\(885\) −5171.63 −0.196432
\(886\) 0 0
\(887\) −10991.8 −0.416086 −0.208043 0.978120i \(-0.566709\pi\)
−0.208043 + 0.978120i \(0.566709\pi\)
\(888\) 0 0
\(889\) 7305.87 0.275626
\(890\) 0 0
\(891\) 5121.22 0.192556
\(892\) 0 0
\(893\) 52954.2 1.98437
\(894\) 0 0
\(895\) −8415.19 −0.314289
\(896\) 0 0
\(897\) 195.413 0.00727387
\(898\) 0 0
\(899\) 40282.5 1.49443
\(900\) 0 0
\(901\) 89791.5 3.32008
\(902\) 0 0
\(903\) −4193.32 −0.154535
\(904\) 0 0
\(905\) 12535.6 0.460440
\(906\) 0 0
\(907\) −15117.6 −0.553440 −0.276720 0.960951i \(-0.589248\pi\)
−0.276720 + 0.960951i \(0.589248\pi\)
\(908\) 0 0
\(909\) −4358.93 −0.159050
\(910\) 0 0
\(911\) −25259.2 −0.918633 −0.459317 0.888273i \(-0.651906\pi\)
−0.459317 + 0.888273i \(0.651906\pi\)
\(912\) 0 0
\(913\) −37290.1 −1.35172
\(914\) 0 0
\(915\) 77.4376 0.00279782
\(916\) 0 0
\(917\) −52226.1 −1.88076
\(918\) 0 0
\(919\) 43485.1 1.56087 0.780436 0.625235i \(-0.214997\pi\)
0.780436 + 0.625235i \(0.214997\pi\)
\(920\) 0 0
\(921\) −2366.89 −0.0846814
\(922\) 0 0
\(923\) −672.903 −0.0239966
\(924\) 0 0
\(925\) 8481.41 0.301478
\(926\) 0 0
\(927\) −14557.6 −0.515785
\(928\) 0 0
\(929\) 22312.2 0.787988 0.393994 0.919113i \(-0.371093\pi\)
0.393994 + 0.919113i \(0.371093\pi\)
\(930\) 0 0
\(931\) −1038.21 −0.0365477
\(932\) 0 0
\(933\) 2868.19 0.100643
\(934\) 0 0
\(935\) −42757.9 −1.49554
\(936\) 0 0
\(937\) −50965.7 −1.77692 −0.888461 0.458951i \(-0.848225\pi\)
−0.888461 + 0.458951i \(0.848225\pi\)
\(938\) 0 0
\(939\) 2186.66 0.0759947
\(940\) 0 0
\(941\) −29611.4 −1.02583 −0.512914 0.858440i \(-0.671434\pi\)
−0.512914 + 0.858440i \(0.671434\pi\)
\(942\) 0 0
\(943\) 20192.7 0.697310
\(944\) 0 0
\(945\) −2538.84 −0.0873953
\(946\) 0 0
\(947\) 16026.1 0.549925 0.274962 0.961455i \(-0.411335\pi\)
0.274962 + 0.961455i \(0.411335\pi\)
\(948\) 0 0
\(949\) −149.962 −0.00512959
\(950\) 0 0
\(951\) −2450.36 −0.0835525
\(952\) 0 0
\(953\) 41491.2 1.41032 0.705159 0.709050i \(-0.250876\pi\)
0.705159 + 0.709050i \(0.250876\pi\)
\(954\) 0 0
\(955\) 16255.9 0.550814
\(956\) 0 0
\(957\) 39390.7 1.33054
\(958\) 0 0
\(959\) −3255.71 −0.109627
\(960\) 0 0
\(961\) 7832.88 0.262928
\(962\) 0 0
\(963\) −9093.26 −0.304285
\(964\) 0 0
\(965\) −12344.4 −0.411794
\(966\) 0 0
\(967\) −16698.6 −0.555317 −0.277658 0.960680i \(-0.589558\pi\)
−0.277658 + 0.960680i \(0.589558\pi\)
\(968\) 0 0
\(969\) 39463.5 1.30831
\(970\) 0 0
\(971\) −44370.0 −1.46643 −0.733213 0.679999i \(-0.761980\pi\)
−0.733213 + 0.679999i \(0.761980\pi\)
\(972\) 0 0
\(973\) −8477.97 −0.279333
\(974\) 0 0
\(975\) −118.594 −0.00389544
\(976\) 0 0
\(977\) −12443.8 −0.407484 −0.203742 0.979025i \(-0.565310\pi\)
−0.203742 + 0.979025i \(0.565310\pi\)
\(978\) 0 0
\(979\) −25910.4 −0.845863
\(980\) 0 0
\(981\) 15.5238 0.000505238 0
\(982\) 0 0
\(983\) −45382.4 −1.47251 −0.736253 0.676707i \(-0.763406\pi\)
−0.736253 + 0.676707i \(0.763406\pi\)
\(984\) 0 0
\(985\) −776.062 −0.0251040
\(986\) 0 0
\(987\) −30718.9 −0.990674
\(988\) 0 0
\(989\) 3061.73 0.0984401
\(990\) 0 0
\(991\) −2468.08 −0.0791132 −0.0395566 0.999217i \(-0.512595\pi\)
−0.0395566 + 0.999217i \(0.512595\pi\)
\(992\) 0 0
\(993\) −11361.8 −0.363096
\(994\) 0 0
\(995\) −10589.9 −0.337410
\(996\) 0 0
\(997\) −8805.68 −0.279718 −0.139859 0.990171i \(-0.544665\pi\)
−0.139859 + 0.990171i \(0.544665\pi\)
\(998\) 0 0
\(999\) −9159.92 −0.290097
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 480.4.a.m.1.1 2
3.2 odd 2 1440.4.a.z.1.1 2
4.3 odd 2 480.4.a.q.1.2 yes 2
5.4 even 2 2400.4.a.bc.1.2 2
8.3 odd 2 960.4.a.bm.1.2 2
8.5 even 2 960.4.a.bo.1.1 2
12.11 even 2 1440.4.a.bg.1.2 2
20.19 odd 2 2400.4.a.x.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
480.4.a.m.1.1 2 1.1 even 1 trivial
480.4.a.q.1.2 yes 2 4.3 odd 2
960.4.a.bm.1.2 2 8.3 odd 2
960.4.a.bo.1.1 2 8.5 even 2
1440.4.a.z.1.1 2 3.2 odd 2
1440.4.a.bg.1.2 2 12.11 even 2
2400.4.a.x.1.1 2 20.19 odd 2
2400.4.a.bc.1.2 2 5.4 even 2