Properties

Label 768.4.a.f.1.1
Level $768$
Weight $4$
Character 768.1
Self dual yes
Analytic conductor $45.313$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,4,Mod(1,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, 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("768.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.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: \(45.3134668844\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 192)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 768.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -3.46410 q^{5} -24.2487 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -3.46410 q^{5} -24.2487 q^{7} +9.00000 q^{9} -48.0000 q^{11} -41.5692 q^{13} +10.3923 q^{15} +54.0000 q^{17} +4.00000 q^{19} +72.7461 q^{21} -173.205 q^{23} -113.000 q^{25} -27.0000 q^{27} -162.813 q^{29} +58.8897 q^{31} +144.000 q^{33} +84.0000 q^{35} +325.626 q^{37} +124.708 q^{39} +294.000 q^{41} -188.000 q^{43} -31.1769 q^{45} +505.759 q^{47} +245.000 q^{49} -162.000 q^{51} -744.782 q^{53} +166.277 q^{55} -12.0000 q^{57} -252.000 q^{59} +90.0666 q^{61} -218.238 q^{63} +144.000 q^{65} -628.000 q^{67} +519.615 q^{69} +6.92820 q^{71} +1006.00 q^{73} +339.000 q^{75} +1163.94 q^{77} -1340.61 q^{79} +81.0000 q^{81} +720.000 q^{83} -187.061 q^{85} +488.438 q^{87} +1482.00 q^{89} +1008.00 q^{91} -176.669 q^{93} -13.8564 q^{95} +1822.00 q^{97} -432.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 18 q^{9} - 96 q^{11} + 108 q^{17} + 8 q^{19} - 226 q^{25} - 54 q^{27} + 288 q^{33} + 168 q^{35} + 588 q^{41} - 376 q^{43} + 490 q^{49} - 324 q^{51} - 24 q^{57} - 504 q^{59} + 288 q^{65} - 1256 q^{67} + 2012 q^{73} + 678 q^{75} + 162 q^{81} + 1440 q^{83} + 2964 q^{89} + 2016 q^{91} + 3644 q^{97} - 864 q^{99}+O(q^{100}) \) 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
\(5\) −3.46410 −0.309839 −0.154919 0.987927i \(-0.549512\pi\)
−0.154919 + 0.987927i \(0.549512\pi\)
\(6\) 0 0
\(7\) −24.2487 −1.30931 −0.654654 0.755929i \(-0.727186\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −48.0000 −1.31569 −0.657843 0.753155i \(-0.728531\pi\)
−0.657843 + 0.753155i \(0.728531\pi\)
\(12\) 0 0
\(13\) −41.5692 −0.886864 −0.443432 0.896308i \(-0.646239\pi\)
−0.443432 + 0.896308i \(0.646239\pi\)
\(14\) 0 0
\(15\) 10.3923 0.178885
\(16\) 0 0
\(17\) 54.0000 0.770407 0.385204 0.922832i \(-0.374131\pi\)
0.385204 + 0.922832i \(0.374131\pi\)
\(18\) 0 0
\(19\) 4.00000 0.0482980 0.0241490 0.999708i \(-0.492312\pi\)
0.0241490 + 0.999708i \(0.492312\pi\)
\(20\) 0 0
\(21\) 72.7461 0.755929
\(22\) 0 0
\(23\) −173.205 −1.57025 −0.785125 0.619337i \(-0.787401\pi\)
−0.785125 + 0.619337i \(0.787401\pi\)
\(24\) 0 0
\(25\) −113.000 −0.904000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −162.813 −1.04254 −0.521269 0.853393i \(-0.674541\pi\)
−0.521269 + 0.853393i \(0.674541\pi\)
\(30\) 0 0
\(31\) 58.8897 0.341191 0.170595 0.985341i \(-0.445431\pi\)
0.170595 + 0.985341i \(0.445431\pi\)
\(32\) 0 0
\(33\) 144.000 0.759612
\(34\) 0 0
\(35\) 84.0000 0.405674
\(36\) 0 0
\(37\) 325.626 1.44682 0.723412 0.690416i \(-0.242573\pi\)
0.723412 + 0.690416i \(0.242573\pi\)
\(38\) 0 0
\(39\) 124.708 0.512031
\(40\) 0 0
\(41\) 294.000 1.11988 0.559940 0.828533i \(-0.310824\pi\)
0.559940 + 0.828533i \(0.310824\pi\)
\(42\) 0 0
\(43\) −188.000 −0.666738 −0.333369 0.942796i \(-0.608185\pi\)
−0.333369 + 0.942796i \(0.608185\pi\)
\(44\) 0 0
\(45\) −31.1769 −0.103280
\(46\) 0 0
\(47\) 505.759 1.56963 0.784814 0.619731i \(-0.212758\pi\)
0.784814 + 0.619731i \(0.212758\pi\)
\(48\) 0 0
\(49\) 245.000 0.714286
\(50\) 0 0
\(51\) −162.000 −0.444795
\(52\) 0 0
\(53\) −744.782 −1.93026 −0.965129 0.261775i \(-0.915692\pi\)
−0.965129 + 0.261775i \(0.915692\pi\)
\(54\) 0 0
\(55\) 166.277 0.407650
\(56\) 0 0
\(57\) −12.0000 −0.0278849
\(58\) 0 0
\(59\) −252.000 −0.556061 −0.278031 0.960572i \(-0.589682\pi\)
−0.278031 + 0.960572i \(0.589682\pi\)
\(60\) 0 0
\(61\) 90.0666 0.189047 0.0945234 0.995523i \(-0.469867\pi\)
0.0945234 + 0.995523i \(0.469867\pi\)
\(62\) 0 0
\(63\) −218.238 −0.436436
\(64\) 0 0
\(65\) 144.000 0.274785
\(66\) 0 0
\(67\) −628.000 −1.14511 −0.572555 0.819866i \(-0.694048\pi\)
−0.572555 + 0.819866i \(0.694048\pi\)
\(68\) 0 0
\(69\) 519.615 0.906584
\(70\) 0 0
\(71\) 6.92820 0.0115807 0.00579033 0.999983i \(-0.498157\pi\)
0.00579033 + 0.999983i \(0.498157\pi\)
\(72\) 0 0
\(73\) 1006.00 1.61292 0.806462 0.591286i \(-0.201380\pi\)
0.806462 + 0.591286i \(0.201380\pi\)
\(74\) 0 0
\(75\) 339.000 0.521925
\(76\) 0 0
\(77\) 1163.94 1.72264
\(78\) 0 0
\(79\) −1340.61 −1.90924 −0.954621 0.297824i \(-0.903739\pi\)
−0.954621 + 0.297824i \(0.903739\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 720.000 0.952172 0.476086 0.879399i \(-0.342055\pi\)
0.476086 + 0.879399i \(0.342055\pi\)
\(84\) 0 0
\(85\) −187.061 −0.238702
\(86\) 0 0
\(87\) 488.438 0.601909
\(88\) 0 0
\(89\) 1482.00 1.76508 0.882538 0.470242i \(-0.155833\pi\)
0.882538 + 0.470242i \(0.155833\pi\)
\(90\) 0 0
\(91\) 1008.00 1.16118
\(92\) 0 0
\(93\) −176.669 −0.196986
\(94\) 0 0
\(95\) −13.8564 −0.0149646
\(96\) 0 0
\(97\) 1822.00 1.90718 0.953588 0.301114i \(-0.0973586\pi\)
0.953588 + 0.301114i \(0.0973586\pi\)
\(98\) 0 0
\(99\) −432.000 −0.438562
\(100\) 0 0
\(101\) −911.059 −0.897562 −0.448781 0.893642i \(-0.648142\pi\)
−0.448781 + 0.893642i \(0.648142\pi\)
\(102\) 0 0
\(103\) 453.797 0.434116 0.217058 0.976159i \(-0.430354\pi\)
0.217058 + 0.976159i \(0.430354\pi\)
\(104\) 0 0
\(105\) −252.000 −0.234216
\(106\) 0 0
\(107\) 1188.00 1.07335 0.536674 0.843790i \(-0.319680\pi\)
0.536674 + 0.843790i \(0.319680\pi\)
\(108\) 0 0
\(109\) −471.118 −0.413990 −0.206995 0.978342i \(-0.566368\pi\)
−0.206995 + 0.978342i \(0.566368\pi\)
\(110\) 0 0
\(111\) −976.877 −0.835325
\(112\) 0 0
\(113\) −390.000 −0.324674 −0.162337 0.986735i \(-0.551903\pi\)
−0.162337 + 0.986735i \(0.551903\pi\)
\(114\) 0 0
\(115\) 600.000 0.486524
\(116\) 0 0
\(117\) −374.123 −0.295621
\(118\) 0 0
\(119\) −1309.43 −1.00870
\(120\) 0 0
\(121\) 973.000 0.731029
\(122\) 0 0
\(123\) −882.000 −0.646563
\(124\) 0 0
\(125\) 824.456 0.589933
\(126\) 0 0
\(127\) 606.218 0.423568 0.211784 0.977317i \(-0.432073\pi\)
0.211784 + 0.977317i \(0.432073\pi\)
\(128\) 0 0
\(129\) 564.000 0.384941
\(130\) 0 0
\(131\) 1380.00 0.920391 0.460195 0.887818i \(-0.347779\pi\)
0.460195 + 0.887818i \(0.347779\pi\)
\(132\) 0 0
\(133\) −96.9948 −0.0632370
\(134\) 0 0
\(135\) 93.5307 0.0596285
\(136\) 0 0
\(137\) 1158.00 0.722150 0.361075 0.932537i \(-0.382410\pi\)
0.361075 + 0.932537i \(0.382410\pi\)
\(138\) 0 0
\(139\) −1180.00 −0.720045 −0.360023 0.932944i \(-0.617231\pi\)
−0.360023 + 0.932944i \(0.617231\pi\)
\(140\) 0 0
\(141\) −1517.28 −0.906225
\(142\) 0 0
\(143\) 1995.32 1.16683
\(144\) 0 0
\(145\) 564.000 0.323018
\(146\) 0 0
\(147\) −735.000 −0.412393
\(148\) 0 0
\(149\) 2171.99 1.19420 0.597102 0.802165i \(-0.296319\pi\)
0.597102 + 0.802165i \(0.296319\pi\)
\(150\) 0 0
\(151\) 142.028 0.0765436 0.0382718 0.999267i \(-0.487815\pi\)
0.0382718 + 0.999267i \(0.487815\pi\)
\(152\) 0 0
\(153\) 486.000 0.256802
\(154\) 0 0
\(155\) −204.000 −0.105714
\(156\) 0 0
\(157\) −1337.14 −0.679717 −0.339859 0.940476i \(-0.610379\pi\)
−0.339859 + 0.940476i \(0.610379\pi\)
\(158\) 0 0
\(159\) 2234.35 1.11443
\(160\) 0 0
\(161\) 4200.00 2.05594
\(162\) 0 0
\(163\) −1748.00 −0.839963 −0.419981 0.907533i \(-0.637963\pi\)
−0.419981 + 0.907533i \(0.637963\pi\)
\(164\) 0 0
\(165\) −498.831 −0.235357
\(166\) 0 0
\(167\) −13.8564 −0.00642060 −0.00321030 0.999995i \(-0.501022\pi\)
−0.00321030 + 0.999995i \(0.501022\pi\)
\(168\) 0 0
\(169\) −469.000 −0.213473
\(170\) 0 0
\(171\) 36.0000 0.0160993
\(172\) 0 0
\(173\) −599.290 −0.263371 −0.131685 0.991292i \(-0.542039\pi\)
−0.131685 + 0.991292i \(0.542039\pi\)
\(174\) 0 0
\(175\) 2740.10 1.18361
\(176\) 0 0
\(177\) 756.000 0.321042
\(178\) 0 0
\(179\) 3228.00 1.34789 0.673944 0.738782i \(-0.264599\pi\)
0.673944 + 0.738782i \(0.264599\pi\)
\(180\) 0 0
\(181\) 2023.04 0.830779 0.415390 0.909644i \(-0.363645\pi\)
0.415390 + 0.909644i \(0.363645\pi\)
\(182\) 0 0
\(183\) −270.200 −0.109146
\(184\) 0 0
\(185\) −1128.00 −0.448282
\(186\) 0 0
\(187\) −2592.00 −1.01361
\(188\) 0 0
\(189\) 654.715 0.251976
\(190\) 0 0
\(191\) −3477.96 −1.31757 −0.658786 0.752330i \(-0.728930\pi\)
−0.658786 + 0.752330i \(0.728930\pi\)
\(192\) 0 0
\(193\) −766.000 −0.285689 −0.142844 0.989745i \(-0.545625\pi\)
−0.142844 + 0.989745i \(0.545625\pi\)
\(194\) 0 0
\(195\) −432.000 −0.158647
\(196\) 0 0
\(197\) 2899.45 1.04862 0.524308 0.851529i \(-0.324324\pi\)
0.524308 + 0.851529i \(0.324324\pi\)
\(198\) 0 0
\(199\) −1735.51 −0.618228 −0.309114 0.951025i \(-0.600032\pi\)
−0.309114 + 0.951025i \(0.600032\pi\)
\(200\) 0 0
\(201\) 1884.00 0.661130
\(202\) 0 0
\(203\) 3948.00 1.36500
\(204\) 0 0
\(205\) −1018.45 −0.346982
\(206\) 0 0
\(207\) −1558.85 −0.523417
\(208\) 0 0
\(209\) −192.000 −0.0635451
\(210\) 0 0
\(211\) 1100.00 0.358896 0.179448 0.983767i \(-0.442569\pi\)
0.179448 + 0.983767i \(0.442569\pi\)
\(212\) 0 0
\(213\) −20.7846 −0.00668609
\(214\) 0 0
\(215\) 651.251 0.206581
\(216\) 0 0
\(217\) −1428.00 −0.446723
\(218\) 0 0
\(219\) −3018.00 −0.931222
\(220\) 0 0
\(221\) −2244.74 −0.683246
\(222\) 0 0
\(223\) −391.443 −0.117547 −0.0587735 0.998271i \(-0.518719\pi\)
−0.0587735 + 0.998271i \(0.518719\pi\)
\(224\) 0 0
\(225\) −1017.00 −0.301333
\(226\) 0 0
\(227\) −3336.00 −0.975410 −0.487705 0.873008i \(-0.662166\pi\)
−0.487705 + 0.873008i \(0.662166\pi\)
\(228\) 0 0
\(229\) −5999.82 −1.73135 −0.865676 0.500605i \(-0.833111\pi\)
−0.865676 + 0.500605i \(0.833111\pi\)
\(230\) 0 0
\(231\) −3491.81 −0.994565
\(232\) 0 0
\(233\) −318.000 −0.0894115 −0.0447057 0.999000i \(-0.514235\pi\)
−0.0447057 + 0.999000i \(0.514235\pi\)
\(234\) 0 0
\(235\) −1752.00 −0.486331
\(236\) 0 0
\(237\) 4021.82 1.10230
\(238\) 0 0
\(239\) −859.097 −0.232512 −0.116256 0.993219i \(-0.537089\pi\)
−0.116256 + 0.993219i \(0.537089\pi\)
\(240\) 0 0
\(241\) −2710.00 −0.724342 −0.362171 0.932112i \(-0.617964\pi\)
−0.362171 + 0.932112i \(0.617964\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) −848.705 −0.221313
\(246\) 0 0
\(247\) −166.277 −0.0428338
\(248\) 0 0
\(249\) −2160.00 −0.549737
\(250\) 0 0
\(251\) 5136.00 1.29156 0.645780 0.763524i \(-0.276532\pi\)
0.645780 + 0.763524i \(0.276532\pi\)
\(252\) 0 0
\(253\) 8313.84 2.06596
\(254\) 0 0
\(255\) 561.184 0.137815
\(256\) 0 0
\(257\) −4398.00 −1.06747 −0.533735 0.845652i \(-0.679212\pi\)
−0.533735 + 0.845652i \(0.679212\pi\)
\(258\) 0 0
\(259\) −7896.00 −1.89434
\(260\) 0 0
\(261\) −1465.31 −0.347512
\(262\) 0 0
\(263\) 6817.35 1.59839 0.799194 0.601073i \(-0.205260\pi\)
0.799194 + 0.601073i \(0.205260\pi\)
\(264\) 0 0
\(265\) 2580.00 0.598068
\(266\) 0 0
\(267\) −4446.00 −1.01907
\(268\) 0 0
\(269\) −4624.58 −1.04820 −0.524099 0.851657i \(-0.675598\pi\)
−0.524099 + 0.851657i \(0.675598\pi\)
\(270\) 0 0
\(271\) −3883.26 −0.870447 −0.435223 0.900322i \(-0.643331\pi\)
−0.435223 + 0.900322i \(0.643331\pi\)
\(272\) 0 0
\(273\) −3024.00 −0.670406
\(274\) 0 0
\(275\) 5424.00 1.18938
\(276\) 0 0
\(277\) 1524.20 0.330616 0.165308 0.986242i \(-0.447138\pi\)
0.165308 + 0.986242i \(0.447138\pi\)
\(278\) 0 0
\(279\) 530.008 0.113730
\(280\) 0 0
\(281\) −4398.00 −0.933675 −0.466838 0.884343i \(-0.654607\pi\)
−0.466838 + 0.884343i \(0.654607\pi\)
\(282\) 0 0
\(283\) −4372.00 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(284\) 0 0
\(285\) 41.5692 0.00863982
\(286\) 0 0
\(287\) −7129.12 −1.46627
\(288\) 0 0
\(289\) −1997.00 −0.406473
\(290\) 0 0
\(291\) −5466.00 −1.10111
\(292\) 0 0
\(293\) −3571.49 −0.712111 −0.356056 0.934465i \(-0.615879\pi\)
−0.356056 + 0.934465i \(0.615879\pi\)
\(294\) 0 0
\(295\) 872.954 0.172289
\(296\) 0 0
\(297\) 1296.00 0.253204
\(298\) 0 0
\(299\) 7200.00 1.39260
\(300\) 0 0
\(301\) 4558.76 0.872965
\(302\) 0 0
\(303\) 2733.18 0.518207
\(304\) 0 0
\(305\) −312.000 −0.0585740
\(306\) 0 0
\(307\) 4172.00 0.775598 0.387799 0.921744i \(-0.373235\pi\)
0.387799 + 0.921744i \(0.373235\pi\)
\(308\) 0 0
\(309\) −1361.39 −0.250637
\(310\) 0 0
\(311\) 6470.94 1.17985 0.589925 0.807458i \(-0.299157\pi\)
0.589925 + 0.807458i \(0.299157\pi\)
\(312\) 0 0
\(313\) 74.0000 0.0133633 0.00668167 0.999978i \(-0.497873\pi\)
0.00668167 + 0.999978i \(0.497873\pi\)
\(314\) 0 0
\(315\) 756.000 0.135225
\(316\) 0 0
\(317\) −1964.15 −0.348004 −0.174002 0.984745i \(-0.555670\pi\)
−0.174002 + 0.984745i \(0.555670\pi\)
\(318\) 0 0
\(319\) 7815.01 1.37165
\(320\) 0 0
\(321\) −3564.00 −0.619698
\(322\) 0 0
\(323\) 216.000 0.0372092
\(324\) 0 0
\(325\) 4697.32 0.801725
\(326\) 0 0
\(327\) 1413.35 0.239017
\(328\) 0 0
\(329\) −12264.0 −2.05513
\(330\) 0 0
\(331\) 7556.00 1.25473 0.627365 0.778726i \(-0.284134\pi\)
0.627365 + 0.778726i \(0.284134\pi\)
\(332\) 0 0
\(333\) 2930.63 0.482275
\(334\) 0 0
\(335\) 2175.46 0.354800
\(336\) 0 0
\(337\) −4106.00 −0.663703 −0.331852 0.943332i \(-0.607673\pi\)
−0.331852 + 0.943332i \(0.607673\pi\)
\(338\) 0 0
\(339\) 1170.00 0.187450
\(340\) 0 0
\(341\) −2826.71 −0.448900
\(342\) 0 0
\(343\) 2376.37 0.374088
\(344\) 0 0
\(345\) −1800.00 −0.280895
\(346\) 0 0
\(347\) 5256.00 0.813132 0.406566 0.913621i \(-0.366726\pi\)
0.406566 + 0.913621i \(0.366726\pi\)
\(348\) 0 0
\(349\) −10385.4 −1.59288 −0.796442 0.604715i \(-0.793287\pi\)
−0.796442 + 0.604715i \(0.793287\pi\)
\(350\) 0 0
\(351\) 1122.37 0.170677
\(352\) 0 0
\(353\) −3942.00 −0.594367 −0.297183 0.954820i \(-0.596047\pi\)
−0.297183 + 0.954820i \(0.596047\pi\)
\(354\) 0 0
\(355\) −24.0000 −0.00358813
\(356\) 0 0
\(357\) 3928.29 0.582373
\(358\) 0 0
\(359\) −6644.15 −0.976782 −0.488391 0.872625i \(-0.662416\pi\)
−0.488391 + 0.872625i \(0.662416\pi\)
\(360\) 0 0
\(361\) −6843.00 −0.997667
\(362\) 0 0
\(363\) −2919.00 −0.422060
\(364\) 0 0
\(365\) −3484.89 −0.499746
\(366\) 0 0
\(367\) −2906.38 −0.413384 −0.206692 0.978406i \(-0.566270\pi\)
−0.206692 + 0.978406i \(0.566270\pi\)
\(368\) 0 0
\(369\) 2646.00 0.373293
\(370\) 0 0
\(371\) 18060.0 2.52730
\(372\) 0 0
\(373\) −10246.8 −1.42241 −0.711206 0.702983i \(-0.751851\pi\)
−0.711206 + 0.702983i \(0.751851\pi\)
\(374\) 0 0
\(375\) −2473.37 −0.340598
\(376\) 0 0
\(377\) 6768.00 0.924588
\(378\) 0 0
\(379\) −13844.0 −1.87630 −0.938151 0.346226i \(-0.887463\pi\)
−0.938151 + 0.346226i \(0.887463\pi\)
\(380\) 0 0
\(381\) −1818.65 −0.244547
\(382\) 0 0
\(383\) 7163.76 0.955747 0.477874 0.878429i \(-0.341408\pi\)
0.477874 + 0.878429i \(0.341408\pi\)
\(384\) 0 0
\(385\) −4032.00 −0.533740
\(386\) 0 0
\(387\) −1692.00 −0.222246
\(388\) 0 0
\(389\) 12993.8 1.69361 0.846805 0.531904i \(-0.178523\pi\)
0.846805 + 0.531904i \(0.178523\pi\)
\(390\) 0 0
\(391\) −9353.07 −1.20973
\(392\) 0 0
\(393\) −4140.00 −0.531388
\(394\) 0 0
\(395\) 4644.00 0.591557
\(396\) 0 0
\(397\) 117.779 0.0148896 0.00744481 0.999972i \(-0.497630\pi\)
0.00744481 + 0.999972i \(0.497630\pi\)
\(398\) 0 0
\(399\) 290.985 0.0365099
\(400\) 0 0
\(401\) −5418.00 −0.674718 −0.337359 0.941376i \(-0.609534\pi\)
−0.337359 + 0.941376i \(0.609534\pi\)
\(402\) 0 0
\(403\) −2448.00 −0.302589
\(404\) 0 0
\(405\) −280.592 −0.0344265
\(406\) 0 0
\(407\) −15630.0 −1.90357
\(408\) 0 0
\(409\) −11450.0 −1.38427 −0.692135 0.721768i \(-0.743330\pi\)
−0.692135 + 0.721768i \(0.743330\pi\)
\(410\) 0 0
\(411\) −3474.00 −0.416934
\(412\) 0 0
\(413\) 6110.68 0.728055
\(414\) 0 0
\(415\) −2494.15 −0.295020
\(416\) 0 0
\(417\) 3540.00 0.415718
\(418\) 0 0
\(419\) −1176.00 −0.137115 −0.0685577 0.997647i \(-0.521840\pi\)
−0.0685577 + 0.997647i \(0.521840\pi\)
\(420\) 0 0
\(421\) 10032.0 1.16136 0.580679 0.814133i \(-0.302787\pi\)
0.580679 + 0.814133i \(0.302787\pi\)
\(422\) 0 0
\(423\) 4551.83 0.523209
\(424\) 0 0
\(425\) −6102.00 −0.696448
\(426\) 0 0
\(427\) −2184.00 −0.247520
\(428\) 0 0
\(429\) −5985.97 −0.673672
\(430\) 0 0
\(431\) −838.313 −0.0936893 −0.0468447 0.998902i \(-0.514917\pi\)
−0.0468447 + 0.998902i \(0.514917\pi\)
\(432\) 0 0
\(433\) 4318.00 0.479237 0.239619 0.970867i \(-0.422978\pi\)
0.239619 + 0.970867i \(0.422978\pi\)
\(434\) 0 0
\(435\) −1692.00 −0.186495
\(436\) 0 0
\(437\) −692.820 −0.0758400
\(438\) 0 0
\(439\) −1610.81 −0.175124 −0.0875622 0.996159i \(-0.527908\pi\)
−0.0875622 + 0.996159i \(0.527908\pi\)
\(440\) 0 0
\(441\) 2205.00 0.238095
\(442\) 0 0
\(443\) 1032.00 0.110681 0.0553406 0.998468i \(-0.482376\pi\)
0.0553406 + 0.998468i \(0.482376\pi\)
\(444\) 0 0
\(445\) −5133.80 −0.546889
\(446\) 0 0
\(447\) −6515.98 −0.689474
\(448\) 0 0
\(449\) 726.000 0.0763075 0.0381537 0.999272i \(-0.487852\pi\)
0.0381537 + 0.999272i \(0.487852\pi\)
\(450\) 0 0
\(451\) −14112.0 −1.47341
\(452\) 0 0
\(453\) −426.084 −0.0441925
\(454\) 0 0
\(455\) −3491.81 −0.359778
\(456\) 0 0
\(457\) −8666.00 −0.887042 −0.443521 0.896264i \(-0.646271\pi\)
−0.443521 + 0.896264i \(0.646271\pi\)
\(458\) 0 0
\(459\) −1458.00 −0.148265
\(460\) 0 0
\(461\) 14684.3 1.48355 0.741776 0.670648i \(-0.233984\pi\)
0.741776 + 0.670648i \(0.233984\pi\)
\(462\) 0 0
\(463\) 4998.70 0.501748 0.250874 0.968020i \(-0.419282\pi\)
0.250874 + 0.968020i \(0.419282\pi\)
\(464\) 0 0
\(465\) 612.000 0.0610340
\(466\) 0 0
\(467\) 16824.0 1.66707 0.833535 0.552466i \(-0.186313\pi\)
0.833535 + 0.552466i \(0.186313\pi\)
\(468\) 0 0
\(469\) 15228.2 1.49930
\(470\) 0 0
\(471\) 4011.43 0.392435
\(472\) 0 0
\(473\) 9024.00 0.877218
\(474\) 0 0
\(475\) −452.000 −0.0436614
\(476\) 0 0
\(477\) −6703.04 −0.643419
\(478\) 0 0
\(479\) 10953.5 1.04484 0.522419 0.852689i \(-0.325030\pi\)
0.522419 + 0.852689i \(0.325030\pi\)
\(480\) 0 0
\(481\) −13536.0 −1.28314
\(482\) 0 0
\(483\) −12600.0 −1.18700
\(484\) 0 0
\(485\) −6311.59 −0.590917
\(486\) 0 0
\(487\) 10714.5 0.996959 0.498479 0.866902i \(-0.333892\pi\)
0.498479 + 0.866902i \(0.333892\pi\)
\(488\) 0 0
\(489\) 5244.00 0.484953
\(490\) 0 0
\(491\) −852.000 −0.0783100 −0.0391550 0.999233i \(-0.512467\pi\)
−0.0391550 + 0.999233i \(0.512467\pi\)
\(492\) 0 0
\(493\) −8791.89 −0.803178
\(494\) 0 0
\(495\) 1496.49 0.135883
\(496\) 0 0
\(497\) −168.000 −0.0151626
\(498\) 0 0
\(499\) 11156.0 1.00082 0.500412 0.865787i \(-0.333182\pi\)
0.500412 + 0.865787i \(0.333182\pi\)
\(500\) 0 0
\(501\) 41.5692 0.00370694
\(502\) 0 0
\(503\) 14999.6 1.32962 0.664808 0.747014i \(-0.268513\pi\)
0.664808 + 0.747014i \(0.268513\pi\)
\(504\) 0 0
\(505\) 3156.00 0.278099
\(506\) 0 0
\(507\) 1407.00 0.123249
\(508\) 0 0
\(509\) 9287.26 0.808743 0.404372 0.914595i \(-0.367490\pi\)
0.404372 + 0.914595i \(0.367490\pi\)
\(510\) 0 0
\(511\) −24394.2 −2.11181
\(512\) 0 0
\(513\) −108.000 −0.00929496
\(514\) 0 0
\(515\) −1572.00 −0.134506
\(516\) 0 0
\(517\) −24276.4 −2.06514
\(518\) 0 0
\(519\) 1797.87 0.152057
\(520\) 0 0
\(521\) 2766.00 0.232592 0.116296 0.993215i \(-0.462898\pi\)
0.116296 + 0.993215i \(0.462898\pi\)
\(522\) 0 0
\(523\) 18988.0 1.58755 0.793774 0.608213i \(-0.208113\pi\)
0.793774 + 0.608213i \(0.208113\pi\)
\(524\) 0 0
\(525\) −8220.31 −0.683360
\(526\) 0 0
\(527\) 3180.05 0.262856
\(528\) 0 0
\(529\) 17833.0 1.46569
\(530\) 0 0
\(531\) −2268.00 −0.185354
\(532\) 0 0
\(533\) −12221.4 −0.993181
\(534\) 0 0
\(535\) −4115.35 −0.332565
\(536\) 0 0
\(537\) −9684.00 −0.778204
\(538\) 0 0
\(539\) −11760.0 −0.939776
\(540\) 0 0
\(541\) 12997.3 1.03290 0.516449 0.856318i \(-0.327254\pi\)
0.516449 + 0.856318i \(0.327254\pi\)
\(542\) 0 0
\(543\) −6069.11 −0.479651
\(544\) 0 0
\(545\) 1632.00 0.128270
\(546\) 0 0
\(547\) −21188.0 −1.65619 −0.828093 0.560591i \(-0.810574\pi\)
−0.828093 + 0.560591i \(0.810574\pi\)
\(548\) 0 0
\(549\) 810.600 0.0630156
\(550\) 0 0
\(551\) −651.251 −0.0503525
\(552\) 0 0
\(553\) 32508.0 2.49978
\(554\) 0 0
\(555\) 3384.00 0.258816
\(556\) 0 0
\(557\) 12231.7 0.930477 0.465238 0.885185i \(-0.345969\pi\)
0.465238 + 0.885185i \(0.345969\pi\)
\(558\) 0 0
\(559\) 7815.01 0.591306
\(560\) 0 0
\(561\) 7776.00 0.585210
\(562\) 0 0
\(563\) −504.000 −0.0377284 −0.0188642 0.999822i \(-0.506005\pi\)
−0.0188642 + 0.999822i \(0.506005\pi\)
\(564\) 0 0
\(565\) 1351.00 0.100596
\(566\) 0 0
\(567\) −1964.15 −0.145479
\(568\) 0 0
\(569\) 20358.0 1.49992 0.749958 0.661486i \(-0.230074\pi\)
0.749958 + 0.661486i \(0.230074\pi\)
\(570\) 0 0
\(571\) −13300.0 −0.974760 −0.487380 0.873190i \(-0.662047\pi\)
−0.487380 + 0.873190i \(0.662047\pi\)
\(572\) 0 0
\(573\) 10433.9 0.760700
\(574\) 0 0
\(575\) 19572.2 1.41951
\(576\) 0 0
\(577\) 4606.00 0.332323 0.166161 0.986099i \(-0.446863\pi\)
0.166161 + 0.986099i \(0.446863\pi\)
\(578\) 0 0
\(579\) 2298.00 0.164942
\(580\) 0 0
\(581\) −17459.1 −1.24669
\(582\) 0 0
\(583\) 35749.5 2.53961
\(584\) 0 0
\(585\) 1296.00 0.0915949
\(586\) 0 0
\(587\) 13980.0 0.982992 0.491496 0.870880i \(-0.336450\pi\)
0.491496 + 0.870880i \(0.336450\pi\)
\(588\) 0 0
\(589\) 235.559 0.0164788
\(590\) 0 0
\(591\) −8698.36 −0.605419
\(592\) 0 0
\(593\) −12486.0 −0.864652 −0.432326 0.901717i \(-0.642307\pi\)
−0.432326 + 0.901717i \(0.642307\pi\)
\(594\) 0 0
\(595\) 4536.00 0.312534
\(596\) 0 0
\(597\) 5206.54 0.356934
\(598\) 0 0
\(599\) −8778.03 −0.598766 −0.299383 0.954133i \(-0.596781\pi\)
−0.299383 + 0.954133i \(0.596781\pi\)
\(600\) 0 0
\(601\) −6986.00 −0.474151 −0.237076 0.971491i \(-0.576189\pi\)
−0.237076 + 0.971491i \(0.576189\pi\)
\(602\) 0 0
\(603\) −5652.00 −0.381704
\(604\) 0 0
\(605\) −3370.57 −0.226501
\(606\) 0 0
\(607\) 4596.86 0.307382 0.153691 0.988119i \(-0.450884\pi\)
0.153691 + 0.988119i \(0.450884\pi\)
\(608\) 0 0
\(609\) −11844.0 −0.788084
\(610\) 0 0
\(611\) −21024.0 −1.39205
\(612\) 0 0
\(613\) 11092.1 0.730838 0.365419 0.930843i \(-0.380926\pi\)
0.365419 + 0.930843i \(0.380926\pi\)
\(614\) 0 0
\(615\) 3055.34 0.200330
\(616\) 0 0
\(617\) 2850.00 0.185959 0.0929795 0.995668i \(-0.470361\pi\)
0.0929795 + 0.995668i \(0.470361\pi\)
\(618\) 0 0
\(619\) −20116.0 −1.30619 −0.653094 0.757277i \(-0.726529\pi\)
−0.653094 + 0.757277i \(0.726529\pi\)
\(620\) 0 0
\(621\) 4676.54 0.302195
\(622\) 0 0
\(623\) −35936.6 −2.31103
\(624\) 0 0
\(625\) 11269.0 0.721216
\(626\) 0 0
\(627\) 576.000 0.0366878
\(628\) 0 0
\(629\) 17583.8 1.11464
\(630\) 0 0
\(631\) −7271.15 −0.458732 −0.229366 0.973340i \(-0.573665\pi\)
−0.229366 + 0.973340i \(0.573665\pi\)
\(632\) 0 0
\(633\) −3300.00 −0.207209
\(634\) 0 0
\(635\) −2100.00 −0.131238
\(636\) 0 0
\(637\) −10184.5 −0.633474
\(638\) 0 0
\(639\) 62.3538 0.00386022
\(640\) 0 0
\(641\) 7230.00 0.445504 0.222752 0.974875i \(-0.428496\pi\)
0.222752 + 0.974875i \(0.428496\pi\)
\(642\) 0 0
\(643\) −2948.00 −0.180805 −0.0904026 0.995905i \(-0.528815\pi\)
−0.0904026 + 0.995905i \(0.528815\pi\)
\(644\) 0 0
\(645\) −1953.75 −0.119270
\(646\) 0 0
\(647\) −17161.2 −1.04277 −0.521387 0.853320i \(-0.674585\pi\)
−0.521387 + 0.853320i \(0.674585\pi\)
\(648\) 0 0
\(649\) 12096.0 0.731602
\(650\) 0 0
\(651\) 4284.00 0.257916
\(652\) 0 0
\(653\) −10381.9 −0.622168 −0.311084 0.950382i \(-0.600692\pi\)
−0.311084 + 0.950382i \(0.600692\pi\)
\(654\) 0 0
\(655\) −4780.46 −0.285173
\(656\) 0 0
\(657\) 9054.00 0.537641
\(658\) 0 0
\(659\) 10308.0 0.609321 0.304661 0.952461i \(-0.401457\pi\)
0.304661 + 0.952461i \(0.401457\pi\)
\(660\) 0 0
\(661\) −15803.2 −0.929916 −0.464958 0.885333i \(-0.653931\pi\)
−0.464958 + 0.885333i \(0.653931\pi\)
\(662\) 0 0
\(663\) 6734.21 0.394472
\(664\) 0 0
\(665\) 336.000 0.0195933
\(666\) 0 0
\(667\) 28200.0 1.63704
\(668\) 0 0
\(669\) 1174.33 0.0678658
\(670\) 0 0
\(671\) −4323.20 −0.248726
\(672\) 0 0
\(673\) 30910.0 1.77042 0.885210 0.465191i \(-0.154014\pi\)
0.885210 + 0.465191i \(0.154014\pi\)
\(674\) 0 0
\(675\) 3051.00 0.173975
\(676\) 0 0
\(677\) −14802.1 −0.840312 −0.420156 0.907452i \(-0.638025\pi\)
−0.420156 + 0.907452i \(0.638025\pi\)
\(678\) 0 0
\(679\) −44181.2 −2.49708
\(680\) 0 0
\(681\) 10008.0 0.563153
\(682\) 0 0
\(683\) −528.000 −0.0295803 −0.0147902 0.999891i \(-0.504708\pi\)
−0.0147902 + 0.999891i \(0.504708\pi\)
\(684\) 0 0
\(685\) −4011.43 −0.223750
\(686\) 0 0
\(687\) 17999.5 0.999596
\(688\) 0 0
\(689\) 30960.0 1.71188
\(690\) 0 0
\(691\) 9052.00 0.498342 0.249171 0.968460i \(-0.419842\pi\)
0.249171 + 0.968460i \(0.419842\pi\)
\(692\) 0 0
\(693\) 10475.4 0.574212
\(694\) 0 0
\(695\) 4087.64 0.223098
\(696\) 0 0
\(697\) 15876.0 0.862764
\(698\) 0 0
\(699\) 954.000 0.0516217
\(700\) 0 0
\(701\) −32600.7 −1.75650 −0.878252 0.478197i \(-0.841290\pi\)
−0.878252 + 0.478197i \(0.841290\pi\)
\(702\) 0 0
\(703\) 1302.50 0.0698788
\(704\) 0 0
\(705\) 5256.00 0.280784
\(706\) 0 0
\(707\) 22092.0 1.17518
\(708\) 0 0
\(709\) 27227.8 1.44226 0.721130 0.692799i \(-0.243623\pi\)
0.721130 + 0.692799i \(0.243623\pi\)
\(710\) 0 0
\(711\) −12065.5 −0.636414
\(712\) 0 0
\(713\) −10200.0 −0.535755
\(714\) 0 0
\(715\) −6912.00 −0.361530
\(716\) 0 0
\(717\) 2577.29 0.134241
\(718\) 0 0
\(719\) 685.892 0.0355764 0.0177882 0.999842i \(-0.494338\pi\)
0.0177882 + 0.999842i \(0.494338\pi\)
\(720\) 0 0
\(721\) −11004.0 −0.568392
\(722\) 0 0
\(723\) 8130.00 0.418199
\(724\) 0 0
\(725\) 18397.8 0.942453
\(726\) 0 0
\(727\) 20192.2 1.03011 0.515054 0.857158i \(-0.327772\pi\)
0.515054 + 0.857158i \(0.327772\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −10152.0 −0.513660
\(732\) 0 0
\(733\) 35236.8 1.77558 0.887792 0.460246i \(-0.152239\pi\)
0.887792 + 0.460246i \(0.152239\pi\)
\(734\) 0 0
\(735\) 2546.11 0.127775
\(736\) 0 0
\(737\) 30144.0 1.50661
\(738\) 0 0
\(739\) 13940.0 0.693899 0.346949 0.937884i \(-0.387218\pi\)
0.346949 + 0.937884i \(0.387218\pi\)
\(740\) 0 0
\(741\) 498.831 0.0247301
\(742\) 0 0
\(743\) 11002.0 0.543235 0.271618 0.962405i \(-0.412441\pi\)
0.271618 + 0.962405i \(0.412441\pi\)
\(744\) 0 0
\(745\) −7524.00 −0.370011
\(746\) 0 0
\(747\) 6480.00 0.317391
\(748\) 0 0
\(749\) −28807.5 −1.40534
\(750\) 0 0
\(751\) 33342.0 1.62006 0.810031 0.586388i \(-0.199450\pi\)
0.810031 + 0.586388i \(0.199450\pi\)
\(752\) 0 0
\(753\) −15408.0 −0.745682
\(754\) 0 0
\(755\) −492.000 −0.0237162
\(756\) 0 0
\(757\) 17445.2 0.837592 0.418796 0.908080i \(-0.362452\pi\)
0.418796 + 0.908080i \(0.362452\pi\)
\(758\) 0 0
\(759\) −24941.5 −1.19278
\(760\) 0 0
\(761\) 30222.0 1.43961 0.719807 0.694174i \(-0.244230\pi\)
0.719807 + 0.694174i \(0.244230\pi\)
\(762\) 0 0
\(763\) 11424.0 0.542040
\(764\) 0 0
\(765\) −1683.55 −0.0795673
\(766\) 0 0
\(767\) 10475.4 0.493150
\(768\) 0 0
\(769\) −11758.0 −0.551371 −0.275686 0.961248i \(-0.588905\pi\)
−0.275686 + 0.961248i \(0.588905\pi\)
\(770\) 0 0
\(771\) 13194.0 0.616304
\(772\) 0 0
\(773\) 1874.08 0.0872004 0.0436002 0.999049i \(-0.486117\pi\)
0.0436002 + 0.999049i \(0.486117\pi\)
\(774\) 0 0
\(775\) −6654.54 −0.308436
\(776\) 0 0
\(777\) 23688.0 1.09370
\(778\) 0 0
\(779\) 1176.00 0.0540880
\(780\) 0 0
\(781\) −332.554 −0.0152365
\(782\) 0 0
\(783\) 4395.94 0.200636
\(784\) 0 0
\(785\) 4632.00 0.210603
\(786\) 0 0
\(787\) 31012.0 1.40465 0.702324 0.711857i \(-0.252146\pi\)
0.702324 + 0.711857i \(0.252146\pi\)
\(788\) 0 0
\(789\) −20452.1 −0.922829
\(790\) 0 0
\(791\) 9457.00 0.425097
\(792\) 0 0
\(793\) −3744.00 −0.167659
\(794\) 0 0
\(795\) −7740.00 −0.345295
\(796\) 0 0
\(797\) 7091.02 0.315153 0.157576 0.987507i \(-0.449632\pi\)
0.157576 + 0.987507i \(0.449632\pi\)
\(798\) 0 0
\(799\) 27311.0 1.20925
\(800\) 0 0
\(801\) 13338.0 0.588358
\(802\) 0 0
\(803\) −48288.0 −2.12210
\(804\) 0 0
\(805\) −14549.2 −0.637010
\(806\) 0 0
\(807\) 13873.7 0.605178
\(808\) 0 0
\(809\) −40650.0 −1.76660 −0.883299 0.468810i \(-0.844683\pi\)
−0.883299 + 0.468810i \(0.844683\pi\)
\(810\) 0 0
\(811\) −8372.00 −0.362492 −0.181246 0.983438i \(-0.558013\pi\)
−0.181246 + 0.983438i \(0.558013\pi\)
\(812\) 0 0
\(813\) 11649.8 0.502553
\(814\) 0 0
\(815\) 6055.25 0.260253
\(816\) 0 0
\(817\) −752.000 −0.0322021
\(818\) 0 0
\(819\) 9072.00 0.387059
\(820\) 0 0
\(821\) 9370.39 0.398330 0.199165 0.979966i \(-0.436177\pi\)
0.199165 + 0.979966i \(0.436177\pi\)
\(822\) 0 0
\(823\) −21668.0 −0.917737 −0.458868 0.888504i \(-0.651745\pi\)
−0.458868 + 0.888504i \(0.651745\pi\)
\(824\) 0 0
\(825\) −16272.0 −0.686689
\(826\) 0 0
\(827\) −6684.00 −0.281046 −0.140523 0.990077i \(-0.544878\pi\)
−0.140523 + 0.990077i \(0.544878\pi\)
\(828\) 0 0
\(829\) −24359.6 −1.02056 −0.510279 0.860009i \(-0.670458\pi\)
−0.510279 + 0.860009i \(0.670458\pi\)
\(830\) 0 0
\(831\) −4572.61 −0.190881
\(832\) 0 0
\(833\) 13230.0 0.550291
\(834\) 0 0
\(835\) 48.0000 0.00198935
\(836\) 0 0
\(837\) −1590.02 −0.0656622
\(838\) 0 0
\(839\) 19613.7 0.807082 0.403541 0.914962i \(-0.367779\pi\)
0.403541 + 0.914962i \(0.367779\pi\)
\(840\) 0 0
\(841\) 2119.00 0.0868834
\(842\) 0 0
\(843\) 13194.0 0.539058
\(844\) 0 0
\(845\) 1624.66 0.0661422
\(846\) 0 0
\(847\) −23594.0 −0.957142
\(848\) 0 0
\(849\) 13116.0 0.530200
\(850\) 0 0
\(851\) −56400.0 −2.27188
\(852\) 0 0
\(853\) −20569.8 −0.825671 −0.412836 0.910806i \(-0.635462\pi\)
−0.412836 + 0.910806i \(0.635462\pi\)
\(854\) 0 0
\(855\) −124.708 −0.00498820
\(856\) 0 0
\(857\) 27222.0 1.08505 0.542524 0.840040i \(-0.317469\pi\)
0.542524 + 0.840040i \(0.317469\pi\)
\(858\) 0 0
\(859\) −3548.00 −0.140927 −0.0704634 0.997514i \(-0.522448\pi\)
−0.0704634 + 0.997514i \(0.522448\pi\)
\(860\) 0 0
\(861\) 21387.4 0.846550
\(862\) 0 0
\(863\) −45047.2 −1.77685 −0.888426 0.459019i \(-0.848201\pi\)
−0.888426 + 0.459019i \(0.848201\pi\)
\(864\) 0 0
\(865\) 2076.00 0.0816024
\(866\) 0 0
\(867\) 5991.00 0.234677
\(868\) 0 0
\(869\) 64349.2 2.51196
\(870\) 0 0
\(871\) 26105.5 1.01556
\(872\) 0 0
\(873\) 16398.0 0.635725
\(874\) 0 0
\(875\) −19992.0 −0.772403
\(876\) 0 0
\(877\) 29022.2 1.11746 0.558729 0.829350i \(-0.311289\pi\)
0.558729 + 0.829350i \(0.311289\pi\)
\(878\) 0 0
\(879\) 10714.5 0.411138
\(880\) 0 0
\(881\) −48318.0 −1.84776 −0.923879 0.382685i \(-0.875000\pi\)
−0.923879 + 0.382685i \(0.875000\pi\)
\(882\) 0 0
\(883\) 14380.0 0.548047 0.274024 0.961723i \(-0.411645\pi\)
0.274024 + 0.961723i \(0.411645\pi\)
\(884\) 0 0
\(885\) −2618.86 −0.0994712
\(886\) 0 0
\(887\) 34086.8 1.29033 0.645164 0.764044i \(-0.276789\pi\)
0.645164 + 0.764044i \(0.276789\pi\)
\(888\) 0 0
\(889\) −14700.0 −0.554581
\(890\) 0 0
\(891\) −3888.00 −0.146187
\(892\) 0 0
\(893\) 2023.04 0.0758100
\(894\) 0 0
\(895\) −11182.1 −0.417628
\(896\) 0 0
\(897\) −21600.0 −0.804017
\(898\) 0 0
\(899\) −9588.00 −0.355704
\(900\) 0 0
\(901\) −40218.2 −1.48708
\(902\) 0 0
\(903\) −13676.3 −0.504007
\(904\) 0 0
\(905\) −7008.00 −0.257408
\(906\) 0 0
\(907\) 31252.0 1.14411 0.572054 0.820216i \(-0.306147\pi\)
0.572054 + 0.820216i \(0.306147\pi\)
\(908\) 0 0
\(909\) −8199.53 −0.299187
\(910\) 0 0
\(911\) 13080.4 0.475713 0.237857 0.971300i \(-0.423555\pi\)
0.237857 + 0.971300i \(0.423555\pi\)
\(912\) 0 0
\(913\) −34560.0 −1.25276
\(914\) 0 0
\(915\) 936.000 0.0338177
\(916\) 0 0
\(917\) −33463.2 −1.20507
\(918\) 0 0
\(919\) 8843.85 0.317445 0.158722 0.987323i \(-0.449263\pi\)
0.158722 + 0.987323i \(0.449263\pi\)
\(920\) 0 0
\(921\) −12516.0 −0.447792
\(922\) 0 0
\(923\) −288.000 −0.0102705
\(924\) 0 0
\(925\) −36795.7 −1.30793
\(926\) 0 0
\(927\) 4084.18 0.144705
\(928\) 0 0
\(929\) 29622.0 1.04614 0.523071 0.852289i \(-0.324786\pi\)
0.523071 + 0.852289i \(0.324786\pi\)
\(930\) 0 0
\(931\) 980.000 0.0344986
\(932\) 0 0
\(933\) −19412.8 −0.681187
\(934\) 0 0
\(935\) 8978.95 0.314057
\(936\) 0 0
\(937\) 23210.0 0.809218 0.404609 0.914490i \(-0.367408\pi\)
0.404609 + 0.914490i \(0.367408\pi\)
\(938\) 0 0
\(939\) −222.000 −0.00771533
\(940\) 0 0
\(941\) 19728.1 0.683439 0.341720 0.939802i \(-0.388991\pi\)
0.341720 + 0.939802i \(0.388991\pi\)
\(942\) 0 0
\(943\) −50922.3 −1.75849
\(944\) 0 0
\(945\) −2268.00 −0.0780720
\(946\) 0 0
\(947\) −1236.00 −0.0424125 −0.0212062 0.999775i \(-0.506751\pi\)
−0.0212062 + 0.999775i \(0.506751\pi\)
\(948\) 0 0
\(949\) −41818.6 −1.43044
\(950\) 0 0
\(951\) 5892.44 0.200920
\(952\) 0 0
\(953\) −18402.0 −0.625498 −0.312749 0.949836i \(-0.601250\pi\)
−0.312749 + 0.949836i \(0.601250\pi\)
\(954\) 0 0
\(955\) 12048.0 0.408235
\(956\) 0 0
\(957\) −23445.0 −0.791923
\(958\) 0 0
\(959\) −28080.0 −0.945517
\(960\) 0 0
\(961\) −26323.0 −0.883589
\(962\) 0 0
\(963\) 10692.0 0.357783
\(964\) 0 0
\(965\) 2653.50 0.0885174
\(966\) 0 0
\(967\) −41836.0 −1.39127 −0.695633 0.718398i \(-0.744876\pi\)
−0.695633 + 0.718398i \(0.744876\pi\)
\(968\) 0 0
\(969\) −648.000 −0.0214827
\(970\) 0 0
\(971\) 8832.00 0.291897 0.145949 0.989292i \(-0.453377\pi\)
0.145949 + 0.989292i \(0.453377\pi\)
\(972\) 0 0
\(973\) 28613.5 0.942761
\(974\) 0 0
\(975\) −14092.0 −0.462876
\(976\) 0 0
\(977\) −23034.0 −0.754271 −0.377136 0.926158i \(-0.623091\pi\)
−0.377136 + 0.926158i \(0.623091\pi\)
\(978\) 0 0
\(979\) −71136.0 −2.32228
\(980\) 0 0
\(981\) −4240.06 −0.137997
\(982\) 0 0
\(983\) −17791.6 −0.577278 −0.288639 0.957438i \(-0.593203\pi\)
−0.288639 + 0.957438i \(0.593203\pi\)
\(984\) 0 0
\(985\) −10044.0 −0.324902
\(986\) 0 0
\(987\) 36792.0 1.18653
\(988\) 0 0
\(989\) 32562.6 1.04695
\(990\) 0 0
\(991\) −3072.66 −0.0984926 −0.0492463 0.998787i \(-0.515682\pi\)
−0.0492463 + 0.998787i \(0.515682\pi\)
\(992\) 0 0
\(993\) −22668.0 −0.724418
\(994\) 0 0
\(995\) 6012.00 0.191551
\(996\) 0 0
\(997\) 52273.3 1.66049 0.830247 0.557396i \(-0.188200\pi\)
0.830247 + 0.557396i \(0.188200\pi\)
\(998\) 0 0
\(999\) −8791.89 −0.278442
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 768.4.a.f.1.1 2
3.2 odd 2 2304.4.a.bk.1.2 2
4.3 odd 2 768.4.a.o.1.1 2
8.3 odd 2 inner 768.4.a.f.1.2 2
8.5 even 2 768.4.a.o.1.2 2
12.11 even 2 2304.4.a.x.1.2 2
16.3 odd 4 192.4.d.c.97.4 yes 4
16.5 even 4 192.4.d.c.97.3 yes 4
16.11 odd 4 192.4.d.c.97.1 4
16.13 even 4 192.4.d.c.97.2 yes 4
24.5 odd 2 2304.4.a.x.1.1 2
24.11 even 2 2304.4.a.bk.1.1 2
48.5 odd 4 576.4.d.c.289.4 4
48.11 even 4 576.4.d.c.289.3 4
48.29 odd 4 576.4.d.c.289.2 4
48.35 even 4 576.4.d.c.289.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
192.4.d.c.97.1 4 16.11 odd 4
192.4.d.c.97.2 yes 4 16.13 even 4
192.4.d.c.97.3 yes 4 16.5 even 4
192.4.d.c.97.4 yes 4 16.3 odd 4
576.4.d.c.289.1 4 48.35 even 4
576.4.d.c.289.2 4 48.29 odd 4
576.4.d.c.289.3 4 48.11 even 4
576.4.d.c.289.4 4 48.5 odd 4
768.4.a.f.1.1 2 1.1 even 1 trivial
768.4.a.f.1.2 2 8.3 odd 2 inner
768.4.a.o.1.1 2 4.3 odd 2
768.4.a.o.1.2 2 8.5 even 2
2304.4.a.x.1.1 2 24.5 odd 2
2304.4.a.x.1.2 2 12.11 even 2
2304.4.a.bk.1.1 2 24.11 even 2
2304.4.a.bk.1.2 2 3.2 odd 2