Properties

Label 1088.4.a.y.1.1
Level $1088$
Weight $4$
Character 1088.1
Self dual yes
Analytic conductor $64.194$
Analytic rank $0$
Dimension $3$
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] = [1088,4,Mod(1,1088)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1088, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1088.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1088 = 2^{6} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1088.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: \(64.1940780862\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.8396.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 20x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 136)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.20657\) of defining polynomial
Character \(\chi\) \(=\) 1088.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.95089 q^{3} +20.3149 q^{5} +22.2885 q^{7} -18.2922 q^{9} +O(q^{10})\) \(q-2.95089 q^{3} +20.3149 q^{5} +22.2885 q^{7} -18.2922 q^{9} +41.8754 q^{11} +35.3376 q^{13} -59.9472 q^{15} -17.0000 q^{17} +67.1209 q^{19} -65.7710 q^{21} -64.6360 q^{23} +287.696 q^{25} +133.653 q^{27} +83.0810 q^{29} +239.181 q^{31} -123.570 q^{33} +452.790 q^{35} +49.0207 q^{37} -104.278 q^{39} -225.820 q^{41} +83.4282 q^{43} -371.605 q^{45} -54.9476 q^{47} +153.778 q^{49} +50.1652 q^{51} -641.241 q^{53} +850.695 q^{55} -198.067 q^{57} -727.129 q^{59} -868.730 q^{61} -407.707 q^{63} +717.881 q^{65} +1005.09 q^{67} +190.734 q^{69} +382.967 q^{71} -640.421 q^{73} -848.961 q^{75} +933.340 q^{77} +19.6094 q^{79} +99.4956 q^{81} +619.902 q^{83} -345.354 q^{85} -245.163 q^{87} -1302.88 q^{89} +787.623 q^{91} -705.798 q^{93} +1363.56 q^{95} +1685.31 q^{97} -765.994 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 4 q^{3} + 8 q^{5} - 2 q^{7} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 4 q^{3} + 8 q^{5} - 2 q^{7} - 5 q^{9} + 84 q^{11} + 50 q^{13} - 148 q^{15} - 51 q^{17} + 224 q^{19} + 16 q^{21} - 234 q^{23} + 161 q^{25} + 292 q^{27} + 72 q^{29} - 2 q^{31} + 148 q^{33} + 428 q^{35} - 100 q^{37} + 156 q^{39} + 218 q^{41} - 44 q^{43} - 768 q^{45} - 16 q^{47} + 403 q^{49} - 68 q^{51} - 462 q^{53} + 460 q^{55} + 688 q^{57} + 68 q^{59} - 460 q^{61} + 586 q^{63} + 408 q^{65} + 1008 q^{67} - 400 q^{69} + 518 q^{71} + 838 q^{73} - 732 q^{75} + 904 q^{77} + 1238 q^{79} + 455 q^{81} + 1148 q^{83} - 136 q^{85} + 1108 q^{87} - 2506 q^{89} + 1416 q^{91} - 648 q^{93} + 40 q^{95} + 2098 q^{97} + 384 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.95089 −0.567900 −0.283950 0.958839i \(-0.591645\pi\)
−0.283950 + 0.958839i \(0.591645\pi\)
\(4\) 0 0
\(5\) 20.3149 1.81702 0.908511 0.417861i \(-0.137220\pi\)
0.908511 + 0.417861i \(0.137220\pi\)
\(6\) 0 0
\(7\) 22.2885 1.20347 0.601733 0.798697i \(-0.294477\pi\)
0.601733 + 0.798697i \(0.294477\pi\)
\(8\) 0 0
\(9\) −18.2922 −0.677490
\(10\) 0 0
\(11\) 41.8754 1.14781 0.573905 0.818922i \(-0.305428\pi\)
0.573905 + 0.818922i \(0.305428\pi\)
\(12\) 0 0
\(13\) 35.3376 0.753915 0.376957 0.926231i \(-0.376970\pi\)
0.376957 + 0.926231i \(0.376970\pi\)
\(14\) 0 0
\(15\) −59.9472 −1.03189
\(16\) 0 0
\(17\) −17.0000 −0.242536
\(18\) 0 0
\(19\) 67.1209 0.810452 0.405226 0.914216i \(-0.367193\pi\)
0.405226 + 0.914216i \(0.367193\pi\)
\(20\) 0 0
\(21\) −65.7710 −0.683449
\(22\) 0 0
\(23\) −64.6360 −0.585980 −0.292990 0.956116i \(-0.594650\pi\)
−0.292990 + 0.956116i \(0.594650\pi\)
\(24\) 0 0
\(25\) 287.696 2.30157
\(26\) 0 0
\(27\) 133.653 0.952646
\(28\) 0 0
\(29\) 83.0810 0.531991 0.265996 0.963974i \(-0.414299\pi\)
0.265996 + 0.963974i \(0.414299\pi\)
\(30\) 0 0
\(31\) 239.181 1.38575 0.692874 0.721059i \(-0.256344\pi\)
0.692874 + 0.721059i \(0.256344\pi\)
\(32\) 0 0
\(33\) −123.570 −0.651841
\(34\) 0 0
\(35\) 452.790 2.18673
\(36\) 0 0
\(37\) 49.0207 0.217810 0.108905 0.994052i \(-0.465266\pi\)
0.108905 + 0.994052i \(0.465266\pi\)
\(38\) 0 0
\(39\) −104.278 −0.428148
\(40\) 0 0
\(41\) −225.820 −0.860173 −0.430087 0.902788i \(-0.641517\pi\)
−0.430087 + 0.902788i \(0.641517\pi\)
\(42\) 0 0
\(43\) 83.4282 0.295876 0.147938 0.988997i \(-0.452736\pi\)
0.147938 + 0.988997i \(0.452736\pi\)
\(44\) 0 0
\(45\) −371.605 −1.23101
\(46\) 0 0
\(47\) −54.9476 −0.170531 −0.0852653 0.996358i \(-0.527174\pi\)
−0.0852653 + 0.996358i \(0.527174\pi\)
\(48\) 0 0
\(49\) 153.778 0.448332
\(50\) 0 0
\(51\) 50.1652 0.137736
\(52\) 0 0
\(53\) −641.241 −1.66191 −0.830955 0.556339i \(-0.812206\pi\)
−0.830955 + 0.556339i \(0.812206\pi\)
\(54\) 0 0
\(55\) 850.695 2.08559
\(56\) 0 0
\(57\) −198.067 −0.460256
\(58\) 0 0
\(59\) −727.129 −1.60448 −0.802238 0.597004i \(-0.796358\pi\)
−0.802238 + 0.597004i \(0.796358\pi\)
\(60\) 0 0
\(61\) −868.730 −1.82343 −0.911717 0.410818i \(-0.865243\pi\)
−0.911717 + 0.410818i \(0.865243\pi\)
\(62\) 0 0
\(63\) −407.707 −0.815337
\(64\) 0 0
\(65\) 717.881 1.36988
\(66\) 0 0
\(67\) 1005.09 1.83270 0.916350 0.400379i \(-0.131122\pi\)
0.916350 + 0.400379i \(0.131122\pi\)
\(68\) 0 0
\(69\) 190.734 0.332778
\(70\) 0 0
\(71\) 382.967 0.640139 0.320069 0.947394i \(-0.396294\pi\)
0.320069 + 0.947394i \(0.396294\pi\)
\(72\) 0 0
\(73\) −640.421 −1.02679 −0.513395 0.858152i \(-0.671612\pi\)
−0.513395 + 0.858152i \(0.671612\pi\)
\(74\) 0 0
\(75\) −848.961 −1.30706
\(76\) 0 0
\(77\) 933.340 1.38135
\(78\) 0 0
\(79\) 19.6094 0.0279270 0.0139635 0.999903i \(-0.495555\pi\)
0.0139635 + 0.999903i \(0.495555\pi\)
\(80\) 0 0
\(81\) 99.4956 0.136482
\(82\) 0 0
\(83\) 619.902 0.819796 0.409898 0.912131i \(-0.365564\pi\)
0.409898 + 0.912131i \(0.365564\pi\)
\(84\) 0 0
\(85\) −345.354 −0.440693
\(86\) 0 0
\(87\) −245.163 −0.302118
\(88\) 0 0
\(89\) −1302.88 −1.55174 −0.775871 0.630891i \(-0.782689\pi\)
−0.775871 + 0.630891i \(0.782689\pi\)
\(90\) 0 0
\(91\) 787.623 0.907312
\(92\) 0 0
\(93\) −705.798 −0.786966
\(94\) 0 0
\(95\) 1363.56 1.47261
\(96\) 0 0
\(97\) 1685.31 1.76410 0.882050 0.471155i \(-0.156163\pi\)
0.882050 + 0.471155i \(0.156163\pi\)
\(98\) 0 0
\(99\) −765.994 −0.777629
\(100\) 0 0
\(101\) 880.978 0.867927 0.433963 0.900931i \(-0.357115\pi\)
0.433963 + 0.900931i \(0.357115\pi\)
\(102\) 0 0
\(103\) −1305.59 −1.24897 −0.624484 0.781037i \(-0.714691\pi\)
−0.624484 + 0.781037i \(0.714691\pi\)
\(104\) 0 0
\(105\) −1336.13 −1.24184
\(106\) 0 0
\(107\) −1123.58 −1.01514 −0.507572 0.861609i \(-0.669457\pi\)
−0.507572 + 0.861609i \(0.669457\pi\)
\(108\) 0 0
\(109\) 1876.51 1.64896 0.824481 0.565890i \(-0.191467\pi\)
0.824481 + 0.565890i \(0.191467\pi\)
\(110\) 0 0
\(111\) −144.655 −0.123694
\(112\) 0 0
\(113\) −652.720 −0.543387 −0.271693 0.962384i \(-0.587584\pi\)
−0.271693 + 0.962384i \(0.587584\pi\)
\(114\) 0 0
\(115\) −1313.07 −1.06474
\(116\) 0 0
\(117\) −646.404 −0.510770
\(118\) 0 0
\(119\) −378.905 −0.291884
\(120\) 0 0
\(121\) 422.547 0.317466
\(122\) 0 0
\(123\) 666.370 0.488492
\(124\) 0 0
\(125\) 3305.16 2.36498
\(126\) 0 0
\(127\) −1500.95 −1.04872 −0.524362 0.851495i \(-0.675696\pi\)
−0.524362 + 0.851495i \(0.675696\pi\)
\(128\) 0 0
\(129\) −246.188 −0.168028
\(130\) 0 0
\(131\) 1608.92 1.07307 0.536533 0.843879i \(-0.319734\pi\)
0.536533 + 0.843879i \(0.319734\pi\)
\(132\) 0 0
\(133\) 1496.03 0.975352
\(134\) 0 0
\(135\) 2715.14 1.73098
\(136\) 0 0
\(137\) −452.630 −0.282269 −0.141134 0.989990i \(-0.545075\pi\)
−0.141134 + 0.989990i \(0.545075\pi\)
\(138\) 0 0
\(139\) −416.188 −0.253961 −0.126981 0.991905i \(-0.540529\pi\)
−0.126981 + 0.991905i \(0.540529\pi\)
\(140\) 0 0
\(141\) 162.145 0.0968443
\(142\) 0 0
\(143\) 1479.78 0.865351
\(144\) 0 0
\(145\) 1687.78 0.966640
\(146\) 0 0
\(147\) −453.783 −0.254608
\(148\) 0 0
\(149\) −2758.63 −1.51675 −0.758376 0.651817i \(-0.774007\pi\)
−0.758376 + 0.651817i \(0.774007\pi\)
\(150\) 0 0
\(151\) −14.1640 −0.00763344 −0.00381672 0.999993i \(-0.501215\pi\)
−0.00381672 + 0.999993i \(0.501215\pi\)
\(152\) 0 0
\(153\) 310.968 0.164315
\(154\) 0 0
\(155\) 4858.95 2.51794
\(156\) 0 0
\(157\) −1150.42 −0.584802 −0.292401 0.956296i \(-0.594454\pi\)
−0.292401 + 0.956296i \(0.594454\pi\)
\(158\) 0 0
\(159\) 1892.24 0.943799
\(160\) 0 0
\(161\) −1440.64 −0.705207
\(162\) 0 0
\(163\) 3172.29 1.52437 0.762187 0.647357i \(-0.224126\pi\)
0.762187 + 0.647357i \(0.224126\pi\)
\(164\) 0 0
\(165\) −2510.31 −1.18441
\(166\) 0 0
\(167\) 1037.92 0.480936 0.240468 0.970657i \(-0.422699\pi\)
0.240468 + 0.970657i \(0.422699\pi\)
\(168\) 0 0
\(169\) −948.252 −0.431612
\(170\) 0 0
\(171\) −1227.79 −0.549073
\(172\) 0 0
\(173\) 1253.09 0.550695 0.275348 0.961345i \(-0.411207\pi\)
0.275348 + 0.961345i \(0.411207\pi\)
\(174\) 0 0
\(175\) 6412.32 2.76986
\(176\) 0 0
\(177\) 2145.68 0.911182
\(178\) 0 0
\(179\) 4287.22 1.79018 0.895090 0.445886i \(-0.147111\pi\)
0.895090 + 0.445886i \(0.147111\pi\)
\(180\) 0 0
\(181\) 321.513 0.132032 0.0660162 0.997819i \(-0.478971\pi\)
0.0660162 + 0.997819i \(0.478971\pi\)
\(182\) 0 0
\(183\) 2563.53 1.03553
\(184\) 0 0
\(185\) 995.853 0.395765
\(186\) 0 0
\(187\) −711.881 −0.278385
\(188\) 0 0
\(189\) 2978.92 1.14648
\(190\) 0 0
\(191\) −1202.62 −0.455594 −0.227797 0.973709i \(-0.573152\pi\)
−0.227797 + 0.973709i \(0.573152\pi\)
\(192\) 0 0
\(193\) 2423.56 0.903895 0.451948 0.892045i \(-0.350729\pi\)
0.451948 + 0.892045i \(0.350729\pi\)
\(194\) 0 0
\(195\) −2118.39 −0.777955
\(196\) 0 0
\(197\) 3049.26 1.10280 0.551398 0.834243i \(-0.314095\pi\)
0.551398 + 0.834243i \(0.314095\pi\)
\(198\) 0 0
\(199\) 1722.08 0.613444 0.306722 0.951799i \(-0.400768\pi\)
0.306722 + 0.951799i \(0.400768\pi\)
\(200\) 0 0
\(201\) −2965.90 −1.04079
\(202\) 0 0
\(203\) 1851.75 0.640234
\(204\) 0 0
\(205\) −4587.51 −1.56295
\(206\) 0 0
\(207\) 1182.34 0.396995
\(208\) 0 0
\(209\) 2810.71 0.930245
\(210\) 0 0
\(211\) 1259.46 0.410924 0.205462 0.978665i \(-0.434130\pi\)
0.205462 + 0.978665i \(0.434130\pi\)
\(212\) 0 0
\(213\) −1130.10 −0.363535
\(214\) 0 0
\(215\) 1694.84 0.537613
\(216\) 0 0
\(217\) 5330.99 1.66770
\(218\) 0 0
\(219\) 1889.82 0.583114
\(220\) 0 0
\(221\) −600.740 −0.182851
\(222\) 0 0
\(223\) −3632.09 −1.09069 −0.545343 0.838213i \(-0.683600\pi\)
−0.545343 + 0.838213i \(0.683600\pi\)
\(224\) 0 0
\(225\) −5262.60 −1.55929
\(226\) 0 0
\(227\) −1304.97 −0.381558 −0.190779 0.981633i \(-0.561101\pi\)
−0.190779 + 0.981633i \(0.561101\pi\)
\(228\) 0 0
\(229\) −2633.32 −0.759889 −0.379944 0.925009i \(-0.624057\pi\)
−0.379944 + 0.925009i \(0.624057\pi\)
\(230\) 0 0
\(231\) −2754.19 −0.784469
\(232\) 0 0
\(233\) 554.016 0.155772 0.0778858 0.996962i \(-0.475183\pi\)
0.0778858 + 0.996962i \(0.475183\pi\)
\(234\) 0 0
\(235\) −1116.26 −0.309858
\(236\) 0 0
\(237\) −57.8653 −0.0158597
\(238\) 0 0
\(239\) 1655.66 0.448099 0.224049 0.974578i \(-0.428072\pi\)
0.224049 + 0.974578i \(0.428072\pi\)
\(240\) 0 0
\(241\) 5367.63 1.43469 0.717343 0.696720i \(-0.245358\pi\)
0.717343 + 0.696720i \(0.245358\pi\)
\(242\) 0 0
\(243\) −3902.22 −1.03015
\(244\) 0 0
\(245\) 3123.99 0.814630
\(246\) 0 0
\(247\) 2371.89 0.611012
\(248\) 0 0
\(249\) −1829.27 −0.465562
\(250\) 0 0
\(251\) 5880.71 1.47883 0.739417 0.673248i \(-0.235101\pi\)
0.739417 + 0.673248i \(0.235101\pi\)
\(252\) 0 0
\(253\) −2706.66 −0.672593
\(254\) 0 0
\(255\) 1019.10 0.250269
\(256\) 0 0
\(257\) 4933.74 1.19750 0.598751 0.800935i \(-0.295664\pi\)
0.598751 + 0.800935i \(0.295664\pi\)
\(258\) 0 0
\(259\) 1092.60 0.262127
\(260\) 0 0
\(261\) −1519.74 −0.360419
\(262\) 0 0
\(263\) −3159.19 −0.740700 −0.370350 0.928892i \(-0.620762\pi\)
−0.370350 + 0.928892i \(0.620762\pi\)
\(264\) 0 0
\(265\) −13026.8 −3.01973
\(266\) 0 0
\(267\) 3844.66 0.881234
\(268\) 0 0
\(269\) −5575.92 −1.26383 −0.631914 0.775038i \(-0.717731\pi\)
−0.631914 + 0.775038i \(0.717731\pi\)
\(270\) 0 0
\(271\) −5176.42 −1.16031 −0.580157 0.814504i \(-0.697009\pi\)
−0.580157 + 0.814504i \(0.697009\pi\)
\(272\) 0 0
\(273\) −2324.19 −0.515262
\(274\) 0 0
\(275\) 12047.4 2.64176
\(276\) 0 0
\(277\) −495.895 −0.107565 −0.0537823 0.998553i \(-0.517128\pi\)
−0.0537823 + 0.998553i \(0.517128\pi\)
\(278\) 0 0
\(279\) −4375.15 −0.938830
\(280\) 0 0
\(281\) 153.597 0.0326080 0.0163040 0.999867i \(-0.494810\pi\)
0.0163040 + 0.999867i \(0.494810\pi\)
\(282\) 0 0
\(283\) −838.842 −0.176198 −0.0880990 0.996112i \(-0.528079\pi\)
−0.0880990 + 0.996112i \(0.528079\pi\)
\(284\) 0 0
\(285\) −4023.71 −0.836295
\(286\) 0 0
\(287\) −5033.19 −1.03519
\(288\) 0 0
\(289\) 289.000 0.0588235
\(290\) 0 0
\(291\) −4973.18 −1.00183
\(292\) 0 0
\(293\) 719.532 0.143466 0.0717330 0.997424i \(-0.477147\pi\)
0.0717330 + 0.997424i \(0.477147\pi\)
\(294\) 0 0
\(295\) −14771.6 −2.91537
\(296\) 0 0
\(297\) 5596.75 1.09346
\(298\) 0 0
\(299\) −2284.08 −0.441779
\(300\) 0 0
\(301\) 1859.49 0.356077
\(302\) 0 0
\(303\) −2599.67 −0.492895
\(304\) 0 0
\(305\) −17648.2 −3.31322
\(306\) 0 0
\(307\) −5805.49 −1.07927 −0.539637 0.841898i \(-0.681438\pi\)
−0.539637 + 0.841898i \(0.681438\pi\)
\(308\) 0 0
\(309\) 3852.66 0.709289
\(310\) 0 0
\(311\) 8382.25 1.52834 0.764170 0.645015i \(-0.223149\pi\)
0.764170 + 0.645015i \(0.223149\pi\)
\(312\) 0 0
\(313\) 9017.59 1.62845 0.814224 0.580551i \(-0.197163\pi\)
0.814224 + 0.580551i \(0.197163\pi\)
\(314\) 0 0
\(315\) −8282.53 −1.48148
\(316\) 0 0
\(317\) −3387.27 −0.600152 −0.300076 0.953915i \(-0.597012\pi\)
−0.300076 + 0.953915i \(0.597012\pi\)
\(318\) 0 0
\(319\) 3479.05 0.610625
\(320\) 0 0
\(321\) 3315.56 0.576500
\(322\) 0 0
\(323\) −1141.06 −0.196564
\(324\) 0 0
\(325\) 10166.5 1.73519
\(326\) 0 0
\(327\) −5537.37 −0.936445
\(328\) 0 0
\(329\) −1224.70 −0.205228
\(330\) 0 0
\(331\) −7034.50 −1.16813 −0.584065 0.811707i \(-0.698539\pi\)
−0.584065 + 0.811707i \(0.698539\pi\)
\(332\) 0 0
\(333\) −896.698 −0.147564
\(334\) 0 0
\(335\) 20418.3 3.33005
\(336\) 0 0
\(337\) 726.807 0.117483 0.0587414 0.998273i \(-0.481291\pi\)
0.0587414 + 0.998273i \(0.481291\pi\)
\(338\) 0 0
\(339\) 1926.11 0.308589
\(340\) 0 0
\(341\) 10015.8 1.59057
\(342\) 0 0
\(343\) −4217.48 −0.663914
\(344\) 0 0
\(345\) 3874.74 0.604664
\(346\) 0 0
\(347\) −12717.6 −1.96748 −0.983740 0.179599i \(-0.942520\pi\)
−0.983740 + 0.179599i \(0.942520\pi\)
\(348\) 0 0
\(349\) 5731.11 0.879024 0.439512 0.898237i \(-0.355151\pi\)
0.439512 + 0.898237i \(0.355151\pi\)
\(350\) 0 0
\(351\) 4722.96 0.718214
\(352\) 0 0
\(353\) −10421.7 −1.57136 −0.785679 0.618634i \(-0.787686\pi\)
−0.785679 + 0.618634i \(0.787686\pi\)
\(354\) 0 0
\(355\) 7779.95 1.16315
\(356\) 0 0
\(357\) 1118.11 0.165761
\(358\) 0 0
\(359\) 2695.18 0.396229 0.198115 0.980179i \(-0.436518\pi\)
0.198115 + 0.980179i \(0.436518\pi\)
\(360\) 0 0
\(361\) −2353.78 −0.343167
\(362\) 0 0
\(363\) −1246.89 −0.180289
\(364\) 0 0
\(365\) −13010.1 −1.86570
\(366\) 0 0
\(367\) 4415.83 0.628078 0.314039 0.949410i \(-0.398318\pi\)
0.314039 + 0.949410i \(0.398318\pi\)
\(368\) 0 0
\(369\) 4130.74 0.582759
\(370\) 0 0
\(371\) −14292.3 −2.00005
\(372\) 0 0
\(373\) 1279.95 0.177676 0.0888381 0.996046i \(-0.471685\pi\)
0.0888381 + 0.996046i \(0.471685\pi\)
\(374\) 0 0
\(375\) −9753.18 −1.34307
\(376\) 0 0
\(377\) 2935.88 0.401076
\(378\) 0 0
\(379\) 11875.3 1.60948 0.804742 0.593625i \(-0.202304\pi\)
0.804742 + 0.593625i \(0.202304\pi\)
\(380\) 0 0
\(381\) 4429.15 0.595571
\(382\) 0 0
\(383\) −7769.82 −1.03660 −0.518302 0.855198i \(-0.673436\pi\)
−0.518302 + 0.855198i \(0.673436\pi\)
\(384\) 0 0
\(385\) 18960.7 2.50994
\(386\) 0 0
\(387\) −1526.09 −0.200453
\(388\) 0 0
\(389\) 8585.82 1.11907 0.559535 0.828807i \(-0.310980\pi\)
0.559535 + 0.828807i \(0.310980\pi\)
\(390\) 0 0
\(391\) 1098.81 0.142121
\(392\) 0 0
\(393\) −4747.74 −0.609394
\(394\) 0 0
\(395\) 398.364 0.0507439
\(396\) 0 0
\(397\) 4189.66 0.529655 0.264827 0.964296i \(-0.414685\pi\)
0.264827 + 0.964296i \(0.414685\pi\)
\(398\) 0 0
\(399\) −4414.61 −0.553902
\(400\) 0 0
\(401\) 2385.57 0.297082 0.148541 0.988906i \(-0.452542\pi\)
0.148541 + 0.988906i \(0.452542\pi\)
\(402\) 0 0
\(403\) 8452.09 1.04474
\(404\) 0 0
\(405\) 2021.25 0.247991
\(406\) 0 0
\(407\) 2052.76 0.250004
\(408\) 0 0
\(409\) −8595.38 −1.03916 −0.519578 0.854423i \(-0.673911\pi\)
−0.519578 + 0.854423i \(0.673911\pi\)
\(410\) 0 0
\(411\) 1335.66 0.160300
\(412\) 0 0
\(413\) −16206.6 −1.93093
\(414\) 0 0
\(415\) 12593.3 1.48959
\(416\) 0 0
\(417\) 1228.13 0.144224
\(418\) 0 0
\(419\) −7594.80 −0.885513 −0.442757 0.896642i \(-0.645999\pi\)
−0.442757 + 0.896642i \(0.645999\pi\)
\(420\) 0 0
\(421\) −13228.6 −1.53141 −0.765703 0.643194i \(-0.777609\pi\)
−0.765703 + 0.643194i \(0.777609\pi\)
\(422\) 0 0
\(423\) 1005.11 0.115533
\(424\) 0 0
\(425\) −4890.83 −0.558213
\(426\) 0 0
\(427\) −19362.7 −2.19444
\(428\) 0 0
\(429\) −4366.66 −0.491432
\(430\) 0 0
\(431\) −11945.2 −1.33498 −0.667492 0.744617i \(-0.732632\pi\)
−0.667492 + 0.744617i \(0.732632\pi\)
\(432\) 0 0
\(433\) −1088.93 −0.120856 −0.0604282 0.998173i \(-0.519247\pi\)
−0.0604282 + 0.998173i \(0.519247\pi\)
\(434\) 0 0
\(435\) −4980.47 −0.548955
\(436\) 0 0
\(437\) −4338.43 −0.474909
\(438\) 0 0
\(439\) −4884.36 −0.531020 −0.265510 0.964108i \(-0.585540\pi\)
−0.265510 + 0.964108i \(0.585540\pi\)
\(440\) 0 0
\(441\) −2812.94 −0.303741
\(442\) 0 0
\(443\) 3063.27 0.328534 0.164267 0.986416i \(-0.447474\pi\)
0.164267 + 0.986416i \(0.447474\pi\)
\(444\) 0 0
\(445\) −26467.9 −2.81955
\(446\) 0 0
\(447\) 8140.44 0.861364
\(448\) 0 0
\(449\) −7269.92 −0.764117 −0.382058 0.924138i \(-0.624785\pi\)
−0.382058 + 0.924138i \(0.624785\pi\)
\(450\) 0 0
\(451\) −9456.28 −0.987315
\(452\) 0 0
\(453\) 41.7964 0.00433503
\(454\) 0 0
\(455\) 16000.5 1.64861
\(456\) 0 0
\(457\) −3660.30 −0.374664 −0.187332 0.982297i \(-0.559984\pi\)
−0.187332 + 0.982297i \(0.559984\pi\)
\(458\) 0 0
\(459\) −2272.09 −0.231051
\(460\) 0 0
\(461\) 2683.39 0.271102 0.135551 0.990770i \(-0.456720\pi\)
0.135551 + 0.990770i \(0.456720\pi\)
\(462\) 0 0
\(463\) −16455.5 −1.65173 −0.825865 0.563868i \(-0.809313\pi\)
−0.825865 + 0.563868i \(0.809313\pi\)
\(464\) 0 0
\(465\) −14338.2 −1.42993
\(466\) 0 0
\(467\) −4902.37 −0.485770 −0.242885 0.970055i \(-0.578094\pi\)
−0.242885 + 0.970055i \(0.578094\pi\)
\(468\) 0 0
\(469\) 22401.9 2.20559
\(470\) 0 0
\(471\) 3394.78 0.332109
\(472\) 0 0
\(473\) 3493.59 0.339609
\(474\) 0 0
\(475\) 19310.4 1.86531
\(476\) 0 0
\(477\) 11729.7 1.12593
\(478\) 0 0
\(479\) −8325.55 −0.794163 −0.397081 0.917783i \(-0.629977\pi\)
−0.397081 + 0.917783i \(0.629977\pi\)
\(480\) 0 0
\(481\) 1732.28 0.164210
\(482\) 0 0
\(483\) 4251.18 0.400487
\(484\) 0 0
\(485\) 34237.0 3.20541
\(486\) 0 0
\(487\) 5436.03 0.505811 0.252905 0.967491i \(-0.418614\pi\)
0.252905 + 0.967491i \(0.418614\pi\)
\(488\) 0 0
\(489\) −9361.09 −0.865692
\(490\) 0 0
\(491\) 7108.10 0.653328 0.326664 0.945141i \(-0.394075\pi\)
0.326664 + 0.945141i \(0.394075\pi\)
\(492\) 0 0
\(493\) −1412.38 −0.129027
\(494\) 0 0
\(495\) −15561.1 −1.41297
\(496\) 0 0
\(497\) 8535.77 0.770386
\(498\) 0 0
\(499\) 7795.04 0.699306 0.349653 0.936879i \(-0.386299\pi\)
0.349653 + 0.936879i \(0.386299\pi\)
\(500\) 0 0
\(501\) −3062.78 −0.273124
\(502\) 0 0
\(503\) 20257.3 1.79568 0.897840 0.440323i \(-0.145136\pi\)
0.897840 + 0.440323i \(0.145136\pi\)
\(504\) 0 0
\(505\) 17897.0 1.57704
\(506\) 0 0
\(507\) 2798.19 0.245113
\(508\) 0 0
\(509\) −1940.28 −0.168961 −0.0844806 0.996425i \(-0.526923\pi\)
−0.0844806 + 0.996425i \(0.526923\pi\)
\(510\) 0 0
\(511\) −14274.0 −1.23571
\(512\) 0 0
\(513\) 8970.88 0.772074
\(514\) 0 0
\(515\) −26523.0 −2.26940
\(516\) 0 0
\(517\) −2300.95 −0.195737
\(518\) 0 0
\(519\) −3697.72 −0.312740
\(520\) 0 0
\(521\) −10297.8 −0.865943 −0.432971 0.901408i \(-0.642535\pi\)
−0.432971 + 0.901408i \(0.642535\pi\)
\(522\) 0 0
\(523\) −1895.90 −0.158512 −0.0792560 0.996854i \(-0.525254\pi\)
−0.0792560 + 0.996854i \(0.525254\pi\)
\(524\) 0 0
\(525\) −18922.1 −1.57300
\(526\) 0 0
\(527\) −4066.08 −0.336093
\(528\) 0 0
\(529\) −7989.19 −0.656628
\(530\) 0 0
\(531\) 13300.8 1.08702
\(532\) 0 0
\(533\) −7979.93 −0.648497
\(534\) 0 0
\(535\) −22825.4 −1.84454
\(536\) 0 0
\(537\) −12651.1 −1.01664
\(538\) 0 0
\(539\) 6439.51 0.514600
\(540\) 0 0
\(541\) −2926.34 −0.232557 −0.116278 0.993217i \(-0.537096\pi\)
−0.116278 + 0.993217i \(0.537096\pi\)
\(542\) 0 0
\(543\) −948.750 −0.0749811
\(544\) 0 0
\(545\) 38121.1 2.99620
\(546\) 0 0
\(547\) 16514.5 1.29088 0.645440 0.763811i \(-0.276674\pi\)
0.645440 + 0.763811i \(0.276674\pi\)
\(548\) 0 0
\(549\) 15891.0 1.23536
\(550\) 0 0
\(551\) 5576.47 0.431154
\(552\) 0 0
\(553\) 437.065 0.0336092
\(554\) 0 0
\(555\) −2938.66 −0.224755
\(556\) 0 0
\(557\) 2601.51 0.197899 0.0989494 0.995092i \(-0.468452\pi\)
0.0989494 + 0.995092i \(0.468452\pi\)
\(558\) 0 0
\(559\) 2948.15 0.223065
\(560\) 0 0
\(561\) 2100.69 0.158095
\(562\) 0 0
\(563\) 15095.3 1.13000 0.565001 0.825090i \(-0.308876\pi\)
0.565001 + 0.825090i \(0.308876\pi\)
\(564\) 0 0
\(565\) −13260.0 −0.987346
\(566\) 0 0
\(567\) 2217.61 0.164252
\(568\) 0 0
\(569\) 10564.9 0.778391 0.389195 0.921155i \(-0.372753\pi\)
0.389195 + 0.921155i \(0.372753\pi\)
\(570\) 0 0
\(571\) −2783.27 −0.203987 −0.101993 0.994785i \(-0.532522\pi\)
−0.101993 + 0.994785i \(0.532522\pi\)
\(572\) 0 0
\(573\) 3548.80 0.258732
\(574\) 0 0
\(575\) −18595.5 −1.34867
\(576\) 0 0
\(577\) 2965.98 0.213995 0.106998 0.994259i \(-0.465876\pi\)
0.106998 + 0.994259i \(0.465876\pi\)
\(578\) 0 0
\(579\) −7151.67 −0.513322
\(580\) 0 0
\(581\) 13816.7 0.986598
\(582\) 0 0
\(583\) −26852.2 −1.90756
\(584\) 0 0
\(585\) −13131.6 −0.928080
\(586\) 0 0
\(587\) −17710.3 −1.24528 −0.622642 0.782507i \(-0.713941\pi\)
−0.622642 + 0.782507i \(0.713941\pi\)
\(588\) 0 0
\(589\) 16054.1 1.12308
\(590\) 0 0
\(591\) −8998.04 −0.626277
\(592\) 0 0
\(593\) 10029.3 0.694527 0.347264 0.937768i \(-0.387111\pi\)
0.347264 + 0.937768i \(0.387111\pi\)
\(594\) 0 0
\(595\) −7697.42 −0.530359
\(596\) 0 0
\(597\) −5081.69 −0.348375
\(598\) 0 0
\(599\) −7889.70 −0.538171 −0.269086 0.963116i \(-0.586721\pi\)
−0.269086 + 0.963116i \(0.586721\pi\)
\(600\) 0 0
\(601\) 4972.07 0.337463 0.168731 0.985662i \(-0.446033\pi\)
0.168731 + 0.985662i \(0.446033\pi\)
\(602\) 0 0
\(603\) −18385.3 −1.24164
\(604\) 0 0
\(605\) 8584.02 0.576843
\(606\) 0 0
\(607\) 6455.66 0.431676 0.215838 0.976429i \(-0.430752\pi\)
0.215838 + 0.976429i \(0.430752\pi\)
\(608\) 0 0
\(609\) −5464.32 −0.363589
\(610\) 0 0
\(611\) −1941.72 −0.128566
\(612\) 0 0
\(613\) −9457.73 −0.623155 −0.311577 0.950221i \(-0.600857\pi\)
−0.311577 + 0.950221i \(0.600857\pi\)
\(614\) 0 0
\(615\) 13537.3 0.887601
\(616\) 0 0
\(617\) −96.7142 −0.00631048 −0.00315524 0.999995i \(-0.501004\pi\)
−0.00315524 + 0.999995i \(0.501004\pi\)
\(618\) 0 0
\(619\) 14409.5 0.935649 0.467824 0.883821i \(-0.345038\pi\)
0.467824 + 0.883821i \(0.345038\pi\)
\(620\) 0 0
\(621\) −8638.76 −0.558231
\(622\) 0 0
\(623\) −29039.3 −1.86747
\(624\) 0 0
\(625\) 31182.1 1.99565
\(626\) 0 0
\(627\) −8294.12 −0.528286
\(628\) 0 0
\(629\) −833.353 −0.0528266
\(630\) 0 0
\(631\) −13888.8 −0.876236 −0.438118 0.898918i \(-0.644355\pi\)
−0.438118 + 0.898918i \(0.644355\pi\)
\(632\) 0 0
\(633\) −3716.54 −0.233364
\(634\) 0 0
\(635\) −30491.7 −1.90556
\(636\) 0 0
\(637\) 5434.15 0.338004
\(638\) 0 0
\(639\) −7005.32 −0.433688
\(640\) 0 0
\(641\) 14430.5 0.889190 0.444595 0.895732i \(-0.353348\pi\)
0.444595 + 0.895732i \(0.353348\pi\)
\(642\) 0 0
\(643\) −29147.1 −1.78763 −0.893816 0.448433i \(-0.851982\pi\)
−0.893816 + 0.448433i \(0.851982\pi\)
\(644\) 0 0
\(645\) −5001.28 −0.305311
\(646\) 0 0
\(647\) 19933.0 1.21120 0.605602 0.795768i \(-0.292932\pi\)
0.605602 + 0.795768i \(0.292932\pi\)
\(648\) 0 0
\(649\) −30448.8 −1.84163
\(650\) 0 0
\(651\) −15731.2 −0.947088
\(652\) 0 0
\(653\) −10675.7 −0.639774 −0.319887 0.947456i \(-0.603645\pi\)
−0.319887 + 0.947456i \(0.603645\pi\)
\(654\) 0 0
\(655\) 32685.0 1.94979
\(656\) 0 0
\(657\) 11714.7 0.695640
\(658\) 0 0
\(659\) −16145.1 −0.954363 −0.477181 0.878805i \(-0.658342\pi\)
−0.477181 + 0.878805i \(0.658342\pi\)
\(660\) 0 0
\(661\) −600.194 −0.0353175 −0.0176587 0.999844i \(-0.505621\pi\)
−0.0176587 + 0.999844i \(0.505621\pi\)
\(662\) 0 0
\(663\) 1772.72 0.103841
\(664\) 0 0
\(665\) 30391.6 1.77224
\(666\) 0 0
\(667\) −5370.02 −0.311736
\(668\) 0 0
\(669\) 10717.9 0.619400
\(670\) 0 0
\(671\) −36378.4 −2.09296
\(672\) 0 0
\(673\) −18470.6 −1.05793 −0.528966 0.848643i \(-0.677420\pi\)
−0.528966 + 0.848643i \(0.677420\pi\)
\(674\) 0 0
\(675\) 38451.3 2.19258
\(676\) 0 0
\(677\) −22195.9 −1.26006 −0.630028 0.776572i \(-0.716957\pi\)
−0.630028 + 0.776572i \(0.716957\pi\)
\(678\) 0 0
\(679\) 37563.2 2.12304
\(680\) 0 0
\(681\) 3850.81 0.216686
\(682\) 0 0
\(683\) 6407.71 0.358982 0.179491 0.983760i \(-0.442555\pi\)
0.179491 + 0.983760i \(0.442555\pi\)
\(684\) 0 0
\(685\) −9195.15 −0.512888
\(686\) 0 0
\(687\) 7770.64 0.431541
\(688\) 0 0
\(689\) −22659.9 −1.25294
\(690\) 0 0
\(691\) 8163.28 0.449415 0.224708 0.974426i \(-0.427857\pi\)
0.224708 + 0.974426i \(0.427857\pi\)
\(692\) 0 0
\(693\) −17072.9 −0.935851
\(694\) 0 0
\(695\) −8454.82 −0.461453
\(696\) 0 0
\(697\) 3838.93 0.208623
\(698\) 0 0
\(699\) −1634.84 −0.0884627
\(700\) 0 0
\(701\) −20637.0 −1.11191 −0.555956 0.831212i \(-0.687648\pi\)
−0.555956 + 0.831212i \(0.687648\pi\)
\(702\) 0 0
\(703\) 3290.32 0.176524
\(704\) 0 0
\(705\) 3293.96 0.175968
\(706\) 0 0
\(707\) 19635.7 1.04452
\(708\) 0 0
\(709\) 13995.2 0.741326 0.370663 0.928767i \(-0.379131\pi\)
0.370663 + 0.928767i \(0.379131\pi\)
\(710\) 0 0
\(711\) −358.700 −0.0189202
\(712\) 0 0
\(713\) −15459.7 −0.812020
\(714\) 0 0
\(715\) 30061.5 1.57236
\(716\) 0 0
\(717\) −4885.67 −0.254475
\(718\) 0 0
\(719\) 31872.3 1.65318 0.826590 0.562804i \(-0.190278\pi\)
0.826590 + 0.562804i \(0.190278\pi\)
\(720\) 0 0
\(721\) −29099.7 −1.50309
\(722\) 0 0
\(723\) −15839.3 −0.814758
\(724\) 0 0
\(725\) 23902.1 1.22441
\(726\) 0 0
\(727\) −18002.0 −0.918371 −0.459186 0.888340i \(-0.651859\pi\)
−0.459186 + 0.888340i \(0.651859\pi\)
\(728\) 0 0
\(729\) 8828.65 0.448542
\(730\) 0 0
\(731\) −1418.28 −0.0717605
\(732\) 0 0
\(733\) 11807.6 0.594983 0.297491 0.954725i \(-0.403850\pi\)
0.297491 + 0.954725i \(0.403850\pi\)
\(734\) 0 0
\(735\) −9218.56 −0.462628
\(736\) 0 0
\(737\) 42088.4 2.10359
\(738\) 0 0
\(739\) −26955.2 −1.34176 −0.670882 0.741564i \(-0.734084\pi\)
−0.670882 + 0.741564i \(0.734084\pi\)
\(740\) 0 0
\(741\) −6999.21 −0.346994
\(742\) 0 0
\(743\) −34852.2 −1.72087 −0.860433 0.509564i \(-0.829807\pi\)
−0.860433 + 0.509564i \(0.829807\pi\)
\(744\) 0 0
\(745\) −56041.5 −2.75597
\(746\) 0 0
\(747\) −11339.4 −0.555404
\(748\) 0 0
\(749\) −25042.9 −1.22169
\(750\) 0 0
\(751\) 15904.0 0.772764 0.386382 0.922339i \(-0.373725\pi\)
0.386382 + 0.922339i \(0.373725\pi\)
\(752\) 0 0
\(753\) −17353.4 −0.839830
\(754\) 0 0
\(755\) −287.740 −0.0138701
\(756\) 0 0
\(757\) −9077.01 −0.435812 −0.217906 0.975970i \(-0.569923\pi\)
−0.217906 + 0.975970i \(0.569923\pi\)
\(758\) 0 0
\(759\) 7987.05 0.381965
\(760\) 0 0
\(761\) −15418.6 −0.734458 −0.367229 0.930130i \(-0.619694\pi\)
−0.367229 + 0.930130i \(0.619694\pi\)
\(762\) 0 0
\(763\) 41824.6 1.98447
\(764\) 0 0
\(765\) 6317.29 0.298565
\(766\) 0 0
\(767\) −25695.0 −1.20964
\(768\) 0 0
\(769\) 7493.32 0.351386 0.175693 0.984445i \(-0.443783\pi\)
0.175693 + 0.984445i \(0.443783\pi\)
\(770\) 0 0
\(771\) −14558.9 −0.680061
\(772\) 0 0
\(773\) 3589.28 0.167008 0.0835041 0.996507i \(-0.473389\pi\)
0.0835041 + 0.996507i \(0.473389\pi\)
\(774\) 0 0
\(775\) 68811.5 3.18940
\(776\) 0 0
\(777\) −3224.15 −0.148862
\(778\) 0 0
\(779\) −15157.2 −0.697129
\(780\) 0 0
\(781\) 16036.9 0.734757
\(782\) 0 0
\(783\) 11104.0 0.506799
\(784\) 0 0
\(785\) −23370.8 −1.06260
\(786\) 0 0
\(787\) 23754.7 1.07594 0.537970 0.842964i \(-0.319191\pi\)
0.537970 + 0.842964i \(0.319191\pi\)
\(788\) 0 0
\(789\) 9322.44 0.420644
\(790\) 0 0
\(791\) −14548.2 −0.653948
\(792\) 0 0
\(793\) −30698.9 −1.37471
\(794\) 0 0
\(795\) 38440.6 1.71490
\(796\) 0 0
\(797\) −9472.15 −0.420980 −0.210490 0.977596i \(-0.567506\pi\)
−0.210490 + 0.977596i \(0.567506\pi\)
\(798\) 0 0
\(799\) 934.110 0.0413597
\(800\) 0 0
\(801\) 23832.6 1.05129
\(802\) 0 0
\(803\) −26817.9 −1.17856
\(804\) 0 0
\(805\) −29266.5 −1.28138
\(806\) 0 0
\(807\) 16454.0 0.717728
\(808\) 0 0
\(809\) 5852.20 0.254329 0.127165 0.991882i \(-0.459412\pi\)
0.127165 + 0.991882i \(0.459412\pi\)
\(810\) 0 0
\(811\) 27780.9 1.20286 0.601430 0.798925i \(-0.294598\pi\)
0.601430 + 0.798925i \(0.294598\pi\)
\(812\) 0 0
\(813\) 15275.1 0.658942
\(814\) 0 0
\(815\) 64444.9 2.76982
\(816\) 0 0
\(817\) 5599.77 0.239793
\(818\) 0 0
\(819\) −14407.4 −0.614694
\(820\) 0 0
\(821\) 31092.6 1.32173 0.660865 0.750505i \(-0.270190\pi\)
0.660865 + 0.750505i \(0.270190\pi\)
\(822\) 0 0
\(823\) 21769.4 0.922032 0.461016 0.887392i \(-0.347485\pi\)
0.461016 + 0.887392i \(0.347485\pi\)
\(824\) 0 0
\(825\) −35550.6 −1.50026
\(826\) 0 0
\(827\) −22163.7 −0.931931 −0.465966 0.884803i \(-0.654293\pi\)
−0.465966 + 0.884803i \(0.654293\pi\)
\(828\) 0 0
\(829\) 28760.0 1.20492 0.602459 0.798150i \(-0.294188\pi\)
0.602459 + 0.798150i \(0.294188\pi\)
\(830\) 0 0
\(831\) 1463.33 0.0610860
\(832\) 0 0
\(833\) −2614.23 −0.108737
\(834\) 0 0
\(835\) 21085.2 0.873872
\(836\) 0 0
\(837\) 31967.2 1.32013
\(838\) 0 0
\(839\) −44987.0 −1.85116 −0.925580 0.378551i \(-0.876422\pi\)
−0.925580 + 0.378551i \(0.876422\pi\)
\(840\) 0 0
\(841\) −17486.6 −0.716985
\(842\) 0 0
\(843\) −453.249 −0.0185181
\(844\) 0 0
\(845\) −19263.7 −0.784249
\(846\) 0 0
\(847\) 9417.96 0.382060
\(848\) 0 0
\(849\) 2475.34 0.100063
\(850\) 0 0
\(851\) −3168.50 −0.127632
\(852\) 0 0
\(853\) −39205.2 −1.57369 −0.786846 0.617150i \(-0.788287\pi\)
−0.786846 + 0.617150i \(0.788287\pi\)
\(854\) 0 0
\(855\) −24942.5 −0.997678
\(856\) 0 0
\(857\) 2604.07 0.103796 0.0518981 0.998652i \(-0.483473\pi\)
0.0518981 + 0.998652i \(0.483473\pi\)
\(858\) 0 0
\(859\) 6875.54 0.273097 0.136549 0.990633i \(-0.456399\pi\)
0.136549 + 0.990633i \(0.456399\pi\)
\(860\) 0 0
\(861\) 14852.4 0.587884
\(862\) 0 0
\(863\) 16188.0 0.638524 0.319262 0.947666i \(-0.396565\pi\)
0.319262 + 0.947666i \(0.396565\pi\)
\(864\) 0 0
\(865\) 25456.3 1.00063
\(866\) 0 0
\(867\) −852.808 −0.0334059
\(868\) 0 0
\(869\) 821.152 0.0320548
\(870\) 0 0
\(871\) 35517.4 1.38170
\(872\) 0 0
\(873\) −30828.2 −1.19516
\(874\) 0 0
\(875\) 73667.1 2.84618
\(876\) 0 0
\(877\) 4950.76 0.190622 0.0953110 0.995448i \(-0.469615\pi\)
0.0953110 + 0.995448i \(0.469615\pi\)
\(878\) 0 0
\(879\) −2123.26 −0.0814743
\(880\) 0 0
\(881\) 37393.6 1.42999 0.714995 0.699129i \(-0.246429\pi\)
0.714995 + 0.699129i \(0.246429\pi\)
\(882\) 0 0
\(883\) 15390.9 0.586574 0.293287 0.956024i \(-0.405251\pi\)
0.293287 + 0.956024i \(0.405251\pi\)
\(884\) 0 0
\(885\) 43589.3 1.65564
\(886\) 0 0
\(887\) 7165.69 0.271252 0.135626 0.990760i \(-0.456695\pi\)
0.135626 + 0.990760i \(0.456695\pi\)
\(888\) 0 0
\(889\) −33454.0 −1.26211
\(890\) 0 0
\(891\) 4166.42 0.156656
\(892\) 0 0
\(893\) −3688.14 −0.138207
\(894\) 0 0
\(895\) 87094.6 3.25280
\(896\) 0 0
\(897\) 6740.08 0.250886
\(898\) 0 0
\(899\) 19871.4 0.737206
\(900\) 0 0
\(901\) 10901.1 0.403073
\(902\) 0 0
\(903\) −5487.16 −0.202216
\(904\) 0 0
\(905\) 6531.51 0.239906
\(906\) 0 0
\(907\) 23524.1 0.861196 0.430598 0.902544i \(-0.358303\pi\)
0.430598 + 0.902544i \(0.358303\pi\)
\(908\) 0 0
\(909\) −16115.0 −0.588011
\(910\) 0 0
\(911\) −14148.4 −0.514553 −0.257276 0.966338i \(-0.582825\pi\)
−0.257276 + 0.966338i \(0.582825\pi\)
\(912\) 0 0
\(913\) 25958.6 0.940970
\(914\) 0 0
\(915\) 52077.9 1.88158
\(916\) 0 0
\(917\) 35860.4 1.29140
\(918\) 0 0
\(919\) −30972.7 −1.11175 −0.555874 0.831267i \(-0.687616\pi\)
−0.555874 + 0.831267i \(0.687616\pi\)
\(920\) 0 0
\(921\) 17131.4 0.612919
\(922\) 0 0
\(923\) 13533.2 0.482610
\(924\) 0 0
\(925\) 14103.1 0.501304
\(926\) 0 0
\(927\) 23882.2 0.846164
\(928\) 0 0
\(929\) −38998.3 −1.37728 −0.688640 0.725103i \(-0.741792\pi\)
−0.688640 + 0.725103i \(0.741792\pi\)
\(930\) 0 0
\(931\) 10321.7 0.363352
\(932\) 0 0
\(933\) −24735.1 −0.867944
\(934\) 0 0
\(935\) −14461.8 −0.505831
\(936\) 0 0
\(937\) 42772.3 1.49126 0.745629 0.666361i \(-0.232149\pi\)
0.745629 + 0.666361i \(0.232149\pi\)
\(938\) 0 0
\(939\) −26609.9 −0.924795
\(940\) 0 0
\(941\) −14143.3 −0.489967 −0.244983 0.969527i \(-0.578783\pi\)
−0.244983 + 0.969527i \(0.578783\pi\)
\(942\) 0 0
\(943\) 14596.1 0.504044
\(944\) 0 0
\(945\) 60516.5 2.08318
\(946\) 0 0
\(947\) −2085.11 −0.0715491 −0.0357746 0.999360i \(-0.511390\pi\)
−0.0357746 + 0.999360i \(0.511390\pi\)
\(948\) 0 0
\(949\) −22631.0 −0.774112
\(950\) 0 0
\(951\) 9995.48 0.340826
\(952\) 0 0
\(953\) −26684.3 −0.907018 −0.453509 0.891252i \(-0.649828\pi\)
−0.453509 + 0.891252i \(0.649828\pi\)
\(954\) 0 0
\(955\) −24431.1 −0.827825
\(956\) 0 0
\(957\) −10266.3 −0.346774
\(958\) 0 0
\(959\) −10088.5 −0.339701
\(960\) 0 0
\(961\) 27416.6 0.920298
\(962\) 0 0
\(963\) 20552.8 0.687750
\(964\) 0 0
\(965\) 49234.5 1.64240
\(966\) 0 0
\(967\) 11405.5 0.379292 0.189646 0.981852i \(-0.439266\pi\)
0.189646 + 0.981852i \(0.439266\pi\)
\(968\) 0 0
\(969\) 3367.13 0.111628
\(970\) 0 0
\(971\) −58306.3 −1.92702 −0.963510 0.267672i \(-0.913746\pi\)
−0.963510 + 0.267672i \(0.913746\pi\)
\(972\) 0 0
\(973\) −9276.21 −0.305634
\(974\) 0 0
\(975\) −30000.3 −0.985413
\(976\) 0 0
\(977\) −3492.44 −0.114363 −0.0571817 0.998364i \(-0.518211\pi\)
−0.0571817 + 0.998364i \(0.518211\pi\)
\(978\) 0 0
\(979\) −54558.6 −1.78110
\(980\) 0 0
\(981\) −34325.5 −1.11715
\(982\) 0 0
\(983\) 13554.6 0.439800 0.219900 0.975522i \(-0.429427\pi\)
0.219900 + 0.975522i \(0.429427\pi\)
\(984\) 0 0
\(985\) 61945.4 2.00380
\(986\) 0 0
\(987\) 3613.96 0.116549
\(988\) 0 0
\(989\) −5392.46 −0.173377
\(990\) 0 0
\(991\) −18929.4 −0.606772 −0.303386 0.952868i \(-0.598117\pi\)
−0.303386 + 0.952868i \(0.598117\pi\)
\(992\) 0 0
\(993\) 20758.1 0.663381
\(994\) 0 0
\(995\) 34984.0 1.11464
\(996\) 0 0
\(997\) −47042.8 −1.49434 −0.747171 0.664632i \(-0.768588\pi\)
−0.747171 + 0.664632i \(0.768588\pi\)
\(998\) 0 0
\(999\) 6551.75 0.207496
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 1088.4.a.y.1.1 3
4.3 odd 2 1088.4.a.u.1.3 3
8.3 odd 2 272.4.a.j.1.1 3
8.5 even 2 136.4.a.b.1.3 3
24.5 odd 2 1224.4.a.i.1.3 3
24.11 even 2 2448.4.a.bj.1.3 3
136.101 even 2 2312.4.a.d.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.4.a.b.1.3 3 8.5 even 2
272.4.a.j.1.1 3 8.3 odd 2
1088.4.a.u.1.3 3 4.3 odd 2
1088.4.a.y.1.1 3 1.1 even 1 trivial
1224.4.a.i.1.3 3 24.5 odd 2
2312.4.a.d.1.1 3 136.101 even 2
2448.4.a.bj.1.3 3 24.11 even 2