Properties

Label 1200.4.a.bo.1.1
Level $1200$
Weight $4$
Character 1200.1
Self dual yes
Analytic conductor $70.802$
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] = [1200,4,Mod(1,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, 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("1200.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.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: \(70.8022920069\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{129}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-5.17891\) of defining polynomial
Character \(\chi\) \(=\) 1200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -33.0735 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -33.0735 q^{7} +9.00000 q^{9} -48.3578 q^{11} +60.3578 q^{13} +17.7891 q^{17} +130.863 q^{19} +99.2204 q^{21} +70.8625 q^{23} -27.0000 q^{27} +104.505 q^{29} +210.441 q^{31} +145.073 q^{33} -300.945 q^{37} -181.073 q^{39} +240.147 q^{41} -108.000 q^{43} -278.991 q^{47} +750.853 q^{49} -53.3673 q^{51} -328.358 q^{53} -392.588 q^{57} -889.533 q^{59} -241.450 q^{61} -297.661 q^{63} +103.834 q^{67} -212.588 q^{69} +277.597 q^{71} -274.403 q^{73} +1599.36 q^{77} -366.991 q^{79} +81.0000 q^{81} +57.7251 q^{83} -313.514 q^{87} -203.175 q^{89} -1996.24 q^{91} -631.322 q^{93} -1283.45 q^{97} -435.220 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} + 2 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} + 2 q^{7} + 18 q^{9} - 74 q^{11} + 98 q^{13} - 78 q^{17} + 80 q^{19} - 6 q^{21} - 40 q^{23} - 54 q^{27} + 50 q^{29} + 12 q^{31} + 222 q^{33} - 34 q^{37} - 294 q^{39} + 344 q^{41} - 216 q^{43} - 876 q^{47} + 1638 q^{49} + 234 q^{51} - 634 q^{53} - 240 q^{57} - 666 q^{59} + 244 q^{61} + 18 q^{63} + 980 q^{67} + 120 q^{69} - 308 q^{71} - 1412 q^{73} + 700 q^{77} - 1052 q^{79} + 162 q^{81} - 248 q^{83} - 150 q^{87} + 684 q^{89} - 676 q^{91} - 36 q^{93} - 1840 q^{97} - 666 q^{99}+O(q^{100}) \) 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
\(5\) 0 0
\(6\) 0 0
\(7\) −33.0735 −1.78580 −0.892899 0.450257i \(-0.851333\pi\)
−0.892899 + 0.450257i \(0.851333\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −48.3578 −1.32549 −0.662747 0.748844i \(-0.730609\pi\)
−0.662747 + 0.748844i \(0.730609\pi\)
\(12\) 0 0
\(13\) 60.3578 1.28771 0.643856 0.765147i \(-0.277334\pi\)
0.643856 + 0.765147i \(0.277334\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 17.7891 0.253793 0.126897 0.991916i \(-0.459498\pi\)
0.126897 + 0.991916i \(0.459498\pi\)
\(18\) 0 0
\(19\) 130.863 1.58010 0.790051 0.613042i \(-0.210054\pi\)
0.790051 + 0.613042i \(0.210054\pi\)
\(20\) 0 0
\(21\) 99.2204 1.03103
\(22\) 0 0
\(23\) 70.8625 0.642429 0.321214 0.947007i \(-0.395909\pi\)
0.321214 + 0.947007i \(0.395909\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 104.505 0.669174 0.334587 0.942365i \(-0.391403\pi\)
0.334587 + 0.942365i \(0.391403\pi\)
\(30\) 0 0
\(31\) 210.441 1.21923 0.609617 0.792696i \(-0.291323\pi\)
0.609617 + 0.792696i \(0.291323\pi\)
\(32\) 0 0
\(33\) 145.073 0.765274
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −300.945 −1.33717 −0.668583 0.743638i \(-0.733099\pi\)
−0.668583 + 0.743638i \(0.733099\pi\)
\(38\) 0 0
\(39\) −181.073 −0.743460
\(40\) 0 0
\(41\) 240.147 0.914747 0.457374 0.889275i \(-0.348790\pi\)
0.457374 + 0.889275i \(0.348790\pi\)
\(42\) 0 0
\(43\) −108.000 −0.383020 −0.191510 0.981491i \(-0.561338\pi\)
−0.191510 + 0.981491i \(0.561338\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −278.991 −0.865850 −0.432925 0.901430i \(-0.642519\pi\)
−0.432925 + 0.901430i \(0.642519\pi\)
\(48\) 0 0
\(49\) 750.853 2.18908
\(50\) 0 0
\(51\) −53.3673 −0.146528
\(52\) 0 0
\(53\) −328.358 −0.851008 −0.425504 0.904957i \(-0.639903\pi\)
−0.425504 + 0.904957i \(0.639903\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −392.588 −0.912272
\(58\) 0 0
\(59\) −889.533 −1.96284 −0.981418 0.191882i \(-0.938541\pi\)
−0.981418 + 0.191882i \(0.938541\pi\)
\(60\) 0 0
\(61\) −241.450 −0.506795 −0.253398 0.967362i \(-0.581548\pi\)
−0.253398 + 0.967362i \(0.581548\pi\)
\(62\) 0 0
\(63\) −297.661 −0.595266
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 103.834 0.189334 0.0946669 0.995509i \(-0.469821\pi\)
0.0946669 + 0.995509i \(0.469821\pi\)
\(68\) 0 0
\(69\) −212.588 −0.370906
\(70\) 0 0
\(71\) 277.597 0.464010 0.232005 0.972715i \(-0.425471\pi\)
0.232005 + 0.972715i \(0.425471\pi\)
\(72\) 0 0
\(73\) −274.403 −0.439951 −0.219976 0.975505i \(-0.570598\pi\)
−0.219976 + 0.975505i \(0.570598\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1599.36 2.36706
\(78\) 0 0
\(79\) −366.991 −0.522654 −0.261327 0.965250i \(-0.584160\pi\)
−0.261327 + 0.965250i \(0.584160\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 57.7251 0.0763391 0.0381696 0.999271i \(-0.487847\pi\)
0.0381696 + 0.999271i \(0.487847\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −313.514 −0.386348
\(88\) 0 0
\(89\) −203.175 −0.241983 −0.120992 0.992654i \(-0.538607\pi\)
−0.120992 + 0.992654i \(0.538607\pi\)
\(90\) 0 0
\(91\) −1996.24 −2.29959
\(92\) 0 0
\(93\) −631.322 −0.703925
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1283.45 −1.34345 −0.671725 0.740801i \(-0.734446\pi\)
−0.671725 + 0.740801i \(0.734446\pi\)
\(98\) 0 0
\(99\) −435.220 −0.441831
\(100\) 0 0
\(101\) 886.908 0.873768 0.436884 0.899518i \(-0.356082\pi\)
0.436884 + 0.899518i \(0.356082\pi\)
\(102\) 0 0
\(103\) −783.055 −0.749094 −0.374547 0.927208i \(-0.622202\pi\)
−0.374547 + 0.927208i \(0.622202\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1301.51 1.17590 0.587950 0.808897i \(-0.299935\pi\)
0.587950 + 0.808897i \(0.299935\pi\)
\(108\) 0 0
\(109\) 1161.91 1.02102 0.510508 0.859873i \(-0.329457\pi\)
0.510508 + 0.859873i \(0.329457\pi\)
\(110\) 0 0
\(111\) 902.836 0.772013
\(112\) 0 0
\(113\) −507.808 −0.422748 −0.211374 0.977405i \(-0.567794\pi\)
−0.211374 + 0.977405i \(0.567794\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 543.220 0.429237
\(118\) 0 0
\(119\) −588.346 −0.453224
\(120\) 0 0
\(121\) 1007.48 0.756933
\(122\) 0 0
\(123\) −720.441 −0.528130
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 796.064 0.556215 0.278107 0.960550i \(-0.410293\pi\)
0.278107 + 0.960550i \(0.410293\pi\)
\(128\) 0 0
\(129\) 324.000 0.221137
\(130\) 0 0
\(131\) 91.4764 0.0610102 0.0305051 0.999535i \(-0.490288\pi\)
0.0305051 + 0.999535i \(0.490288\pi\)
\(132\) 0 0
\(133\) −4328.08 −2.82174
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2273.28 −1.41766 −0.708829 0.705380i \(-0.750776\pi\)
−0.708829 + 0.705380i \(0.750776\pi\)
\(138\) 0 0
\(139\) 738.735 0.450782 0.225391 0.974268i \(-0.427634\pi\)
0.225391 + 0.974268i \(0.427634\pi\)
\(140\) 0 0
\(141\) 836.972 0.499899
\(142\) 0 0
\(143\) −2918.77 −1.70685
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −2252.56 −1.26386
\(148\) 0 0
\(149\) 1507.15 0.828661 0.414330 0.910127i \(-0.364016\pi\)
0.414330 + 0.910127i \(0.364016\pi\)
\(150\) 0 0
\(151\) −154.365 −0.0831925 −0.0415962 0.999135i \(-0.513244\pi\)
−0.0415962 + 0.999135i \(0.513244\pi\)
\(152\) 0 0
\(153\) 160.102 0.0845978
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 315.514 0.160387 0.0801935 0.996779i \(-0.474446\pi\)
0.0801935 + 0.996779i \(0.474446\pi\)
\(158\) 0 0
\(159\) 985.073 0.491330
\(160\) 0 0
\(161\) −2343.67 −1.14725
\(162\) 0 0
\(163\) −1457.85 −0.700539 −0.350270 0.936649i \(-0.613910\pi\)
−0.350270 + 0.936649i \(0.613910\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2198.61 −1.01876 −0.509381 0.860541i \(-0.670126\pi\)
−0.509381 + 0.860541i \(0.670126\pi\)
\(168\) 0 0
\(169\) 1446.07 0.658200
\(170\) 0 0
\(171\) 1177.76 0.526700
\(172\) 0 0
\(173\) 2030.49 0.892343 0.446171 0.894948i \(-0.352787\pi\)
0.446171 + 0.894948i \(0.352787\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2668.60 1.13324
\(178\) 0 0
\(179\) −427.220 −0.178391 −0.0891954 0.996014i \(-0.528430\pi\)
−0.0891954 + 0.996014i \(0.528430\pi\)
\(180\) 0 0
\(181\) −3779.97 −1.55228 −0.776140 0.630561i \(-0.782825\pi\)
−0.776140 + 0.630561i \(0.782825\pi\)
\(182\) 0 0
\(183\) 724.350 0.292598
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −860.241 −0.336401
\(188\) 0 0
\(189\) 892.983 0.343677
\(190\) 0 0
\(191\) −1565.60 −0.593103 −0.296551 0.955017i \(-0.595837\pi\)
−0.296551 + 0.955017i \(0.595837\pi\)
\(192\) 0 0
\(193\) 2642.63 0.985599 0.492800 0.870143i \(-0.335974\pi\)
0.492800 + 0.870143i \(0.335974\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −98.4522 −0.0356062 −0.0178031 0.999842i \(-0.505667\pi\)
−0.0178031 + 0.999842i \(0.505667\pi\)
\(198\) 0 0
\(199\) −1131.62 −0.403106 −0.201553 0.979478i \(-0.564599\pi\)
−0.201553 + 0.979478i \(0.564599\pi\)
\(200\) 0 0
\(201\) −311.503 −0.109312
\(202\) 0 0
\(203\) −3456.33 −1.19501
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 637.763 0.214143
\(208\) 0 0
\(209\) −6328.23 −2.09441
\(210\) 0 0
\(211\) 1902.85 0.620841 0.310420 0.950599i \(-0.399530\pi\)
0.310420 + 0.950599i \(0.399530\pi\)
\(212\) 0 0
\(213\) −832.791 −0.267896
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −6960.00 −2.17731
\(218\) 0 0
\(219\) 823.209 0.254006
\(220\) 0 0
\(221\) 1073.71 0.326813
\(222\) 0 0
\(223\) 4855.59 1.45809 0.729046 0.684465i \(-0.239964\pi\)
0.729046 + 0.684465i \(0.239964\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −6536.19 −1.91111 −0.955555 0.294812i \(-0.904743\pi\)
−0.955555 + 0.294812i \(0.904743\pi\)
\(228\) 0 0
\(229\) −5510.35 −1.59011 −0.795053 0.606539i \(-0.792557\pi\)
−0.795053 + 0.606539i \(0.792557\pi\)
\(230\) 0 0
\(231\) −4798.08 −1.36663
\(232\) 0 0
\(233\) −5915.65 −1.66329 −0.831646 0.555306i \(-0.812601\pi\)
−0.831646 + 0.555306i \(0.812601\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1100.97 0.301754
\(238\) 0 0
\(239\) −2263.25 −0.612543 −0.306272 0.951944i \(-0.599082\pi\)
−0.306272 + 0.951944i \(0.599082\pi\)
\(240\) 0 0
\(241\) 772.493 0.206476 0.103238 0.994657i \(-0.467080\pi\)
0.103238 + 0.994657i \(0.467080\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 7898.58 2.03471
\(248\) 0 0
\(249\) −173.175 −0.0440744
\(250\) 0 0
\(251\) −596.192 −0.149926 −0.0749628 0.997186i \(-0.523884\pi\)
−0.0749628 + 0.997186i \(0.523884\pi\)
\(252\) 0 0
\(253\) −3426.76 −0.851535
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4139.19 −1.00465 −0.502326 0.864678i \(-0.667522\pi\)
−0.502326 + 0.864678i \(0.667522\pi\)
\(258\) 0 0
\(259\) 9953.30 2.38791
\(260\) 0 0
\(261\) 940.542 0.223058
\(262\) 0 0
\(263\) −1611.34 −0.377792 −0.188896 0.981997i \(-0.560491\pi\)
−0.188896 + 0.981997i \(0.560491\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 609.526 0.139709
\(268\) 0 0
\(269\) −7031.33 −1.59371 −0.796855 0.604171i \(-0.793504\pi\)
−0.796855 + 0.604171i \(0.793504\pi\)
\(270\) 0 0
\(271\) 5441.27 1.21968 0.609840 0.792524i \(-0.291234\pi\)
0.609840 + 0.792524i \(0.291234\pi\)
\(272\) 0 0
\(273\) 5988.72 1.32767
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1080.43 −0.234355 −0.117178 0.993111i \(-0.537385\pi\)
−0.117178 + 0.993111i \(0.537385\pi\)
\(278\) 0 0
\(279\) 1893.97 0.406411
\(280\) 0 0
\(281\) −1602.99 −0.340308 −0.170154 0.985417i \(-0.554427\pi\)
−0.170154 + 0.985417i \(0.554427\pi\)
\(282\) 0 0
\(283\) 334.810 0.0703265 0.0351632 0.999382i \(-0.488805\pi\)
0.0351632 + 0.999382i \(0.488805\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −7942.49 −1.63355
\(288\) 0 0
\(289\) −4596.55 −0.935589
\(290\) 0 0
\(291\) 3850.35 0.775641
\(292\) 0 0
\(293\) −539.250 −0.107520 −0.0537599 0.998554i \(-0.517121\pi\)
−0.0537599 + 0.998554i \(0.517121\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1305.66 0.255091
\(298\) 0 0
\(299\) 4277.11 0.827263
\(300\) 0 0
\(301\) 3571.93 0.683996
\(302\) 0 0
\(303\) −2660.72 −0.504470
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 8477.58 1.57603 0.788015 0.615656i \(-0.211109\pi\)
0.788015 + 0.615656i \(0.211109\pi\)
\(308\) 0 0
\(309\) 2349.16 0.432490
\(310\) 0 0
\(311\) −3646.92 −0.664945 −0.332473 0.943113i \(-0.607883\pi\)
−0.332473 + 0.943113i \(0.607883\pi\)
\(312\) 0 0
\(313\) −7537.05 −1.36108 −0.680542 0.732709i \(-0.738255\pi\)
−0.680542 + 0.732709i \(0.738255\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10721.2 1.89956 0.949781 0.312916i \(-0.101306\pi\)
0.949781 + 0.312916i \(0.101306\pi\)
\(318\) 0 0
\(319\) −5053.62 −0.886986
\(320\) 0 0
\(321\) −3904.52 −0.678907
\(322\) 0 0
\(323\) 2327.92 0.401019
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −3485.73 −0.589484
\(328\) 0 0
\(329\) 9227.18 1.54623
\(330\) 0 0
\(331\) 4405.14 0.731505 0.365753 0.930712i \(-0.380812\pi\)
0.365753 + 0.930712i \(0.380812\pi\)
\(332\) 0 0
\(333\) −2708.51 −0.445722
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 186.117 0.0300844 0.0150422 0.999887i \(-0.495212\pi\)
0.0150422 + 0.999887i \(0.495212\pi\)
\(338\) 0 0
\(339\) 1523.42 0.244074
\(340\) 0 0
\(341\) −10176.5 −1.61609
\(342\) 0 0
\(343\) −13489.1 −2.12345
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5547.38 0.858211 0.429105 0.903254i \(-0.358829\pi\)
0.429105 + 0.903254i \(0.358829\pi\)
\(348\) 0 0
\(349\) 9078.82 1.39249 0.696244 0.717805i \(-0.254853\pi\)
0.696244 + 0.717805i \(0.254853\pi\)
\(350\) 0 0
\(351\) −1629.66 −0.247820
\(352\) 0 0
\(353\) −10678.1 −1.61002 −0.805009 0.593262i \(-0.797840\pi\)
−0.805009 + 0.593262i \(0.797840\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1765.04 0.261669
\(358\) 0 0
\(359\) 265.733 0.0390665 0.0195332 0.999809i \(-0.493782\pi\)
0.0195332 + 0.999809i \(0.493782\pi\)
\(360\) 0 0
\(361\) 10266.0 1.49672
\(362\) 0 0
\(363\) −3022.44 −0.437016
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −5854.21 −0.832663 −0.416332 0.909213i \(-0.636684\pi\)
−0.416332 + 0.909213i \(0.636684\pi\)
\(368\) 0 0
\(369\) 2161.32 0.304916
\(370\) 0 0
\(371\) 10859.9 1.51973
\(372\) 0 0
\(373\) 10134.5 1.40682 0.703408 0.710787i \(-0.251661\pi\)
0.703408 + 0.710787i \(0.251661\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6307.68 0.861703
\(378\) 0 0
\(379\) 4235.70 0.574072 0.287036 0.957920i \(-0.407330\pi\)
0.287036 + 0.957920i \(0.407330\pi\)
\(380\) 0 0
\(381\) −2388.19 −0.321131
\(382\) 0 0
\(383\) 8100.84 1.08077 0.540383 0.841419i \(-0.318279\pi\)
0.540383 + 0.841419i \(0.318279\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −972.000 −0.127673
\(388\) 0 0
\(389\) −13820.2 −1.80131 −0.900656 0.434532i \(-0.856914\pi\)
−0.900656 + 0.434532i \(0.856914\pi\)
\(390\) 0 0
\(391\) 1260.58 0.163044
\(392\) 0 0
\(393\) −274.429 −0.0352242
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −11523.9 −1.45685 −0.728425 0.685125i \(-0.759748\pi\)
−0.728425 + 0.685125i \(0.759748\pi\)
\(398\) 0 0
\(399\) 12984.2 1.62913
\(400\) 0 0
\(401\) 700.915 0.0872869 0.0436434 0.999047i \(-0.486103\pi\)
0.0436434 + 0.999047i \(0.486103\pi\)
\(402\) 0 0
\(403\) 12701.7 1.57002
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 14553.1 1.77240
\(408\) 0 0
\(409\) 6650.69 0.804047 0.402024 0.915629i \(-0.368307\pi\)
0.402024 + 0.915629i \(0.368307\pi\)
\(410\) 0 0
\(411\) 6819.83 0.818485
\(412\) 0 0
\(413\) 29419.9 3.50523
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −2216.20 −0.260259
\(418\) 0 0
\(419\) −13844.4 −1.61418 −0.807091 0.590426i \(-0.798960\pi\)
−0.807091 + 0.590426i \(0.798960\pi\)
\(420\) 0 0
\(421\) 12576.3 1.45590 0.727949 0.685631i \(-0.240474\pi\)
0.727949 + 0.685631i \(0.240474\pi\)
\(422\) 0 0
\(423\) −2510.92 −0.288617
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 7985.59 0.905035
\(428\) 0 0
\(429\) 8756.32 0.985452
\(430\) 0 0
\(431\) −13440.6 −1.50212 −0.751059 0.660236i \(-0.770456\pi\)
−0.751059 + 0.660236i \(0.770456\pi\)
\(432\) 0 0
\(433\) −3012.87 −0.334387 −0.167193 0.985924i \(-0.553470\pi\)
−0.167193 + 0.985924i \(0.553470\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 9273.25 1.01510
\(438\) 0 0
\(439\) −10675.0 −1.16057 −0.580283 0.814415i \(-0.697058\pi\)
−0.580283 + 0.814415i \(0.697058\pi\)
\(440\) 0 0
\(441\) 6757.68 0.729692
\(442\) 0 0
\(443\) 125.868 0.0134992 0.00674962 0.999977i \(-0.497852\pi\)
0.00674962 + 0.999977i \(0.497852\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −4521.45 −0.478428
\(448\) 0 0
\(449\) −9707.21 −1.02029 −0.510146 0.860088i \(-0.670409\pi\)
−0.510146 + 0.860088i \(0.670409\pi\)
\(450\) 0 0
\(451\) −11613.0 −1.21249
\(452\) 0 0
\(453\) 463.096 0.0480312
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1279.97 0.131016 0.0655082 0.997852i \(-0.479133\pi\)
0.0655082 + 0.997852i \(0.479133\pi\)
\(458\) 0 0
\(459\) −480.305 −0.0488425
\(460\) 0 0
\(461\) 3080.48 0.311220 0.155610 0.987819i \(-0.450266\pi\)
0.155610 + 0.987819i \(0.450266\pi\)
\(462\) 0 0
\(463\) 18017.3 1.80850 0.904248 0.427008i \(-0.140432\pi\)
0.904248 + 0.427008i \(0.140432\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7236.77 −0.717083 −0.358541 0.933514i \(-0.616726\pi\)
−0.358541 + 0.933514i \(0.616726\pi\)
\(468\) 0 0
\(469\) −3434.16 −0.338112
\(470\) 0 0
\(471\) −946.542 −0.0925995
\(472\) 0 0
\(473\) 5222.64 0.507690
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −2955.22 −0.283669
\(478\) 0 0
\(479\) −4932.78 −0.470532 −0.235266 0.971931i \(-0.575596\pi\)
−0.235266 + 0.971931i \(0.575596\pi\)
\(480\) 0 0
\(481\) −18164.4 −1.72188
\(482\) 0 0
\(483\) 7031.01 0.662364
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −5937.91 −0.552510 −0.276255 0.961084i \(-0.589093\pi\)
−0.276255 + 0.961084i \(0.589093\pi\)
\(488\) 0 0
\(489\) 4373.56 0.404456
\(490\) 0 0
\(491\) −15703.5 −1.44336 −0.721678 0.692229i \(-0.756629\pi\)
−0.721678 + 0.692229i \(0.756629\pi\)
\(492\) 0 0
\(493\) 1859.04 0.169832
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9181.09 −0.828628
\(498\) 0 0
\(499\) 5656.77 0.507478 0.253739 0.967273i \(-0.418340\pi\)
0.253739 + 0.967273i \(0.418340\pi\)
\(500\) 0 0
\(501\) 6595.82 0.588183
\(502\) 0 0
\(503\) 290.441 0.0257457 0.0128729 0.999917i \(-0.495902\pi\)
0.0128729 + 0.999917i \(0.495902\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −4338.20 −0.380012
\(508\) 0 0
\(509\) 17330.6 1.50916 0.754582 0.656206i \(-0.227840\pi\)
0.754582 + 0.656206i \(0.227840\pi\)
\(510\) 0 0
\(511\) 9075.45 0.785664
\(512\) 0 0
\(513\) −3533.29 −0.304091
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 13491.4 1.14768
\(518\) 0 0
\(519\) −6091.47 −0.515194
\(520\) 0 0
\(521\) 6174.36 0.519201 0.259601 0.965716i \(-0.416409\pi\)
0.259601 + 0.965716i \(0.416409\pi\)
\(522\) 0 0
\(523\) −13389.4 −1.11946 −0.559730 0.828675i \(-0.689095\pi\)
−0.559730 + 0.828675i \(0.689095\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3743.55 0.309434
\(528\) 0 0
\(529\) −7145.50 −0.587285
\(530\) 0 0
\(531\) −8005.80 −0.654279
\(532\) 0 0
\(533\) 14494.7 1.17793
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1281.66 0.102994
\(538\) 0 0
\(539\) −36309.6 −2.90161
\(540\) 0 0
\(541\) −14355.5 −1.14084 −0.570418 0.821354i \(-0.693219\pi\)
−0.570418 + 0.821354i \(0.693219\pi\)
\(542\) 0 0
\(543\) 11339.9 0.896209
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −21133.9 −1.65195 −0.825977 0.563704i \(-0.809376\pi\)
−0.825977 + 0.563704i \(0.809376\pi\)
\(548\) 0 0
\(549\) −2173.05 −0.168932
\(550\) 0 0
\(551\) 13675.8 1.05736
\(552\) 0 0
\(553\) 12137.6 0.933355
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −3098.60 −0.235713 −0.117856 0.993031i \(-0.537602\pi\)
−0.117856 + 0.993031i \(0.537602\pi\)
\(558\) 0 0
\(559\) −6518.64 −0.493219
\(560\) 0 0
\(561\) 2580.72 0.194221
\(562\) 0 0
\(563\) 7908.77 0.592033 0.296017 0.955183i \(-0.404342\pi\)
0.296017 + 0.955183i \(0.404342\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2678.95 −0.198422
\(568\) 0 0
\(569\) −10740.7 −0.791345 −0.395673 0.918392i \(-0.629488\pi\)
−0.395673 + 0.918392i \(0.629488\pi\)
\(570\) 0 0
\(571\) −14701.2 −1.07745 −0.538725 0.842482i \(-0.681094\pi\)
−0.538725 + 0.842482i \(0.681094\pi\)
\(572\) 0 0
\(573\) 4696.79 0.342428
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −15788.1 −1.13911 −0.569554 0.821954i \(-0.692884\pi\)
−0.569554 + 0.821954i \(0.692884\pi\)
\(578\) 0 0
\(579\) −7927.89 −0.569036
\(580\) 0 0
\(581\) −1909.17 −0.136326
\(582\) 0 0
\(583\) 15878.7 1.12801
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14778.5 1.03913 0.519567 0.854430i \(-0.326093\pi\)
0.519567 + 0.854430i \(0.326093\pi\)
\(588\) 0 0
\(589\) 27538.8 1.92651
\(590\) 0 0
\(591\) 295.356 0.0205573
\(592\) 0 0
\(593\) −11143.9 −0.771713 −0.385857 0.922559i \(-0.626094\pi\)
−0.385857 + 0.922559i \(0.626094\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 3394.85 0.232733
\(598\) 0 0
\(599\) 7707.87 0.525768 0.262884 0.964827i \(-0.415326\pi\)
0.262884 + 0.964827i \(0.415326\pi\)
\(600\) 0 0
\(601\) −13681.4 −0.928580 −0.464290 0.885683i \(-0.653691\pi\)
−0.464290 + 0.885683i \(0.653691\pi\)
\(602\) 0 0
\(603\) 934.508 0.0631113
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 11552.4 0.772481 0.386241 0.922398i \(-0.373773\pi\)
0.386241 + 0.922398i \(0.373773\pi\)
\(608\) 0 0
\(609\) 10369.0 0.689939
\(610\) 0 0
\(611\) −16839.3 −1.11496
\(612\) 0 0
\(613\) 9904.66 0.652603 0.326301 0.945266i \(-0.394198\pi\)
0.326301 + 0.945266i \(0.394198\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −20323.5 −1.32608 −0.663042 0.748582i \(-0.730735\pi\)
−0.663042 + 0.748582i \(0.730735\pi\)
\(618\) 0 0
\(619\) 9223.99 0.598940 0.299470 0.954106i \(-0.403190\pi\)
0.299470 + 0.954106i \(0.403190\pi\)
\(620\) 0 0
\(621\) −1913.29 −0.123635
\(622\) 0 0
\(623\) 6719.70 0.432134
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 18984.7 1.20921
\(628\) 0 0
\(629\) −5353.54 −0.339364
\(630\) 0 0
\(631\) 16916.0 1.06722 0.533610 0.845730i \(-0.320835\pi\)
0.533610 + 0.845730i \(0.320835\pi\)
\(632\) 0 0
\(633\) −5708.54 −0.358443
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 45319.9 2.81890
\(638\) 0 0
\(639\) 2498.37 0.154670
\(640\) 0 0
\(641\) −5811.35 −0.358088 −0.179044 0.983841i \(-0.557300\pi\)
−0.179044 + 0.983841i \(0.557300\pi\)
\(642\) 0 0
\(643\) 27931.7 1.71309 0.856547 0.516069i \(-0.172605\pi\)
0.856547 + 0.516069i \(0.172605\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 25437.8 1.54569 0.772845 0.634595i \(-0.218833\pi\)
0.772845 + 0.634595i \(0.218833\pi\)
\(648\) 0 0
\(649\) 43015.9 2.60173
\(650\) 0 0
\(651\) 20880.0 1.25707
\(652\) 0 0
\(653\) −22647.7 −1.35723 −0.678617 0.734493i \(-0.737420\pi\)
−0.678617 + 0.734493i \(0.737420\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −2469.63 −0.146650
\(658\) 0 0
\(659\) −25158.5 −1.48716 −0.743579 0.668649i \(-0.766873\pi\)
−0.743579 + 0.668649i \(0.766873\pi\)
\(660\) 0 0
\(661\) 23441.0 1.37935 0.689675 0.724119i \(-0.257753\pi\)
0.689675 + 0.724119i \(0.257753\pi\)
\(662\) 0 0
\(663\) −3221.13 −0.188685
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7405.47 0.429896
\(668\) 0 0
\(669\) −14566.8 −0.841829
\(670\) 0 0
\(671\) 11676.0 0.671754
\(672\) 0 0
\(673\) 3145.33 0.180154 0.0900770 0.995935i \(-0.471289\pi\)
0.0900770 + 0.995935i \(0.471289\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10606.2 0.602109 0.301054 0.953607i \(-0.402661\pi\)
0.301054 + 0.953607i \(0.402661\pi\)
\(678\) 0 0
\(679\) 42448.1 2.39913
\(680\) 0 0
\(681\) 19608.6 1.10338
\(682\) 0 0
\(683\) −7825.06 −0.438386 −0.219193 0.975682i \(-0.570342\pi\)
−0.219193 + 0.975682i \(0.570342\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 16531.1 0.918049
\(688\) 0 0
\(689\) −19819.0 −1.09585
\(690\) 0 0
\(691\) −22750.9 −1.25251 −0.626256 0.779618i \(-0.715413\pi\)
−0.626256 + 0.779618i \(0.715413\pi\)
\(692\) 0 0
\(693\) 14394.2 0.789022
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 4271.99 0.232157
\(698\) 0 0
\(699\) 17747.0 0.960302
\(700\) 0 0
\(701\) −133.598 −0.00719817 −0.00359908 0.999994i \(-0.501146\pi\)
−0.00359908 + 0.999994i \(0.501146\pi\)
\(702\) 0 0
\(703\) −39382.5 −2.11286
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −29333.1 −1.56037
\(708\) 0 0
\(709\) 6886.26 0.364766 0.182383 0.983228i \(-0.441619\pi\)
0.182383 + 0.983228i \(0.441619\pi\)
\(710\) 0 0
\(711\) −3302.92 −0.174218
\(712\) 0 0
\(713\) 14912.4 0.783271
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6789.76 0.353652
\(718\) 0 0
\(719\) 1570.18 0.0814436 0.0407218 0.999171i \(-0.487034\pi\)
0.0407218 + 0.999171i \(0.487034\pi\)
\(720\) 0 0
\(721\) 25898.3 1.33773
\(722\) 0 0
\(723\) −2317.48 −0.119209
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −3399.18 −0.173409 −0.0867047 0.996234i \(-0.527634\pi\)
−0.0867047 + 0.996234i \(0.527634\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −1921.22 −0.0972078
\(732\) 0 0
\(733\) 14152.1 0.713125 0.356562 0.934272i \(-0.383949\pi\)
0.356562 + 0.934272i \(0.383949\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5021.20 −0.250961
\(738\) 0 0
\(739\) 14919.5 0.742655 0.371328 0.928502i \(-0.378903\pi\)
0.371328 + 0.928502i \(0.378903\pi\)
\(740\) 0 0
\(741\) −23695.7 −1.17474
\(742\) 0 0
\(743\) 7287.29 0.359818 0.179909 0.983683i \(-0.442420\pi\)
0.179909 + 0.983683i \(0.442420\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 519.526 0.0254464
\(748\) 0 0
\(749\) −43045.3 −2.09992
\(750\) 0 0
\(751\) −18783.4 −0.912670 −0.456335 0.889808i \(-0.650838\pi\)
−0.456335 + 0.889808i \(0.650838\pi\)
\(752\) 0 0
\(753\) 1788.58 0.0865595
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 30614.3 1.46988 0.734939 0.678134i \(-0.237211\pi\)
0.734939 + 0.678134i \(0.237211\pi\)
\(758\) 0 0
\(759\) 10280.3 0.491634
\(760\) 0 0
\(761\) −15277.2 −0.727723 −0.363861 0.931453i \(-0.618542\pi\)
−0.363861 + 0.931453i \(0.618542\pi\)
\(762\) 0 0
\(763\) −38428.4 −1.82333
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −53690.3 −2.52757
\(768\) 0 0
\(769\) −17700.1 −0.830016 −0.415008 0.909818i \(-0.636221\pi\)
−0.415008 + 0.909818i \(0.636221\pi\)
\(770\) 0 0
\(771\) 12417.6 0.580036
\(772\) 0 0
\(773\) −29362.5 −1.36623 −0.683116 0.730310i \(-0.739375\pi\)
−0.683116 + 0.730310i \(0.739375\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −29859.9 −1.37866
\(778\) 0 0
\(779\) 31426.2 1.44539
\(780\) 0 0
\(781\) −13424.0 −0.615042
\(782\) 0 0
\(783\) −2821.63 −0.128783
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −2816.92 −0.127589 −0.0637943 0.997963i \(-0.520320\pi\)
−0.0637943 + 0.997963i \(0.520320\pi\)
\(788\) 0 0
\(789\) 4834.01 0.218118
\(790\) 0 0
\(791\) 16795.0 0.754943
\(792\) 0 0
\(793\) −14573.4 −0.652606
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18433.9 0.819276 0.409638 0.912248i \(-0.365655\pi\)
0.409638 + 0.912248i \(0.365655\pi\)
\(798\) 0 0
\(799\) −4962.99 −0.219747
\(800\) 0 0
\(801\) −1828.58 −0.0806612
\(802\) 0 0
\(803\) 13269.5 0.583153
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 21094.0 0.920128
\(808\) 0 0
\(809\) −13276.6 −0.576983 −0.288492 0.957482i \(-0.593154\pi\)
−0.288492 + 0.957482i \(0.593154\pi\)
\(810\) 0 0
\(811\) 18842.5 0.815845 0.407923 0.913016i \(-0.366253\pi\)
0.407923 + 0.913016i \(0.366253\pi\)
\(812\) 0 0
\(813\) −16323.8 −0.704183
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −14133.2 −0.605210
\(818\) 0 0
\(819\) −17966.2 −0.766531
\(820\) 0 0
\(821\) 5863.62 0.249259 0.124629 0.992203i \(-0.460226\pi\)
0.124629 + 0.992203i \(0.460226\pi\)
\(822\) 0 0
\(823\) 5018.31 0.212548 0.106274 0.994337i \(-0.466108\pi\)
0.106274 + 0.994337i \(0.466108\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 20303.9 0.853730 0.426865 0.904315i \(-0.359618\pi\)
0.426865 + 0.904315i \(0.359618\pi\)
\(828\) 0 0
\(829\) −38156.2 −1.59857 −0.799287 0.600949i \(-0.794789\pi\)
−0.799287 + 0.600949i \(0.794789\pi\)
\(830\) 0 0
\(831\) 3241.28 0.135305
\(832\) 0 0
\(833\) 13357.0 0.555573
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −5681.90 −0.234642
\(838\) 0 0
\(839\) −41004.6 −1.68729 −0.843645 0.536901i \(-0.819595\pi\)
−0.843645 + 0.536901i \(0.819595\pi\)
\(840\) 0 0
\(841\) −13467.8 −0.552206
\(842\) 0 0
\(843\) 4808.98 0.196477
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −33320.8 −1.35173
\(848\) 0 0
\(849\) −1004.43 −0.0406030
\(850\) 0 0
\(851\) −21325.8 −0.859033
\(852\) 0 0
\(853\) 22451.8 0.901214 0.450607 0.892722i \(-0.351208\pi\)
0.450607 + 0.892722i \(0.351208\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −39858.0 −1.58871 −0.794354 0.607455i \(-0.792191\pi\)
−0.794354 + 0.607455i \(0.792191\pi\)
\(858\) 0 0
\(859\) −32585.0 −1.29428 −0.647139 0.762372i \(-0.724035\pi\)
−0.647139 + 0.762372i \(0.724035\pi\)
\(860\) 0 0
\(861\) 23827.5 0.943133
\(862\) 0 0
\(863\) −15593.6 −0.615078 −0.307539 0.951535i \(-0.599505\pi\)
−0.307539 + 0.951535i \(0.599505\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13789.6 0.540163
\(868\) 0 0
\(869\) 17746.9 0.692775
\(870\) 0 0
\(871\) 6267.21 0.243807
\(872\) 0 0
\(873\) −11551.1 −0.447817
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1804.75 0.0694892 0.0347446 0.999396i \(-0.488938\pi\)
0.0347446 + 0.999396i \(0.488938\pi\)
\(878\) 0 0
\(879\) 1617.75 0.0620766
\(880\) 0 0
\(881\) −14788.0 −0.565515 −0.282758 0.959191i \(-0.591249\pi\)
−0.282758 + 0.959191i \(0.591249\pi\)
\(882\) 0 0
\(883\) 1999.24 0.0761947 0.0380973 0.999274i \(-0.487870\pi\)
0.0380973 + 0.999274i \(0.487870\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 32474.4 1.22929 0.614647 0.788802i \(-0.289299\pi\)
0.614647 + 0.788802i \(0.289299\pi\)
\(888\) 0 0
\(889\) −26328.6 −0.993287
\(890\) 0 0
\(891\) −3916.98 −0.147277
\(892\) 0 0
\(893\) −36509.4 −1.36813
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −12831.3 −0.477620
\(898\) 0 0
\(899\) 21992.0 0.815880
\(900\) 0 0
\(901\) −5841.18 −0.215980
\(902\) 0 0
\(903\) −10715.8 −0.394905
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −27597.9 −1.01033 −0.505167 0.863022i \(-0.668569\pi\)
−0.505167 + 0.863022i \(0.668569\pi\)
\(908\) 0 0
\(909\) 7982.17 0.291256
\(910\) 0 0
\(911\) 19861.4 0.722325 0.361163 0.932503i \(-0.382380\pi\)
0.361163 + 0.932503i \(0.382380\pi\)
\(912\) 0 0
\(913\) −2791.46 −0.101187
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3025.44 −0.108952
\(918\) 0 0
\(919\) 20886.0 0.749691 0.374845 0.927087i \(-0.377696\pi\)
0.374845 + 0.927087i \(0.377696\pi\)
\(920\) 0 0
\(921\) −25432.7 −0.909921
\(922\) 0 0
\(923\) 16755.2 0.597511
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −7047.49 −0.249698
\(928\) 0 0
\(929\) 37461.2 1.32299 0.661497 0.749948i \(-0.269922\pi\)
0.661497 + 0.749948i \(0.269922\pi\)
\(930\) 0 0
\(931\) 98258.5 3.45896
\(932\) 0 0
\(933\) 10940.8 0.383906
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −27149.0 −0.946552 −0.473276 0.880914i \(-0.656929\pi\)
−0.473276 + 0.880914i \(0.656929\pi\)
\(938\) 0 0
\(939\) 22611.1 0.785822
\(940\) 0 0
\(941\) −43068.3 −1.49202 −0.746008 0.665937i \(-0.768032\pi\)
−0.746008 + 0.665937i \(0.768032\pi\)
\(942\) 0 0
\(943\) 17017.4 0.587660
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 30021.1 1.03015 0.515076 0.857145i \(-0.327764\pi\)
0.515076 + 0.857145i \(0.327764\pi\)
\(948\) 0 0
\(949\) −16562.4 −0.566530
\(950\) 0 0
\(951\) −32163.5 −1.09671
\(952\) 0 0
\(953\) 53820.4 1.82939 0.914697 0.404140i \(-0.132429\pi\)
0.914697 + 0.404140i \(0.132429\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 15160.9 0.512101
\(958\) 0 0
\(959\) 75185.1 2.53165
\(960\) 0 0
\(961\) 14494.3 0.486532
\(962\) 0 0
\(963\) 11713.6 0.391967
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −32210.2 −1.07116 −0.535580 0.844485i \(-0.679907\pi\)
−0.535580 + 0.844485i \(0.679907\pi\)
\(968\) 0 0
\(969\) −6983.77 −0.231529
\(970\) 0 0
\(971\) 21616.1 0.714411 0.357206 0.934026i \(-0.383730\pi\)
0.357206 + 0.934026i \(0.383730\pi\)
\(972\) 0 0
\(973\) −24432.5 −0.805005
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −46002.8 −1.50641 −0.753204 0.657787i \(-0.771493\pi\)
−0.753204 + 0.657787i \(0.771493\pi\)
\(978\) 0 0
\(979\) 9825.11 0.320748
\(980\) 0 0
\(981\) 10457.2 0.340339
\(982\) 0 0
\(983\) 3926.43 0.127400 0.0636998 0.997969i \(-0.479710\pi\)
0.0636998 + 0.997969i \(0.479710\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −27681.5 −0.892719
\(988\) 0 0
\(989\) −7653.15 −0.246063
\(990\) 0 0
\(991\) −55802.3 −1.78872 −0.894359 0.447351i \(-0.852368\pi\)
−0.894359 + 0.447351i \(0.852368\pi\)
\(992\) 0 0
\(993\) −13215.4 −0.422335
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 43880.8 1.39390 0.696951 0.717119i \(-0.254540\pi\)
0.696951 + 0.717119i \(0.254540\pi\)
\(998\) 0 0
\(999\) 8125.53 0.257338
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 1200.4.a.bo.1.1 2
4.3 odd 2 600.4.a.v.1.2 2
5.2 odd 4 240.4.f.g.49.4 4
5.3 odd 4 240.4.f.g.49.2 4
5.4 even 2 1200.4.a.bq.1.2 2
12.11 even 2 1800.4.a.bl.1.2 2
15.2 even 4 720.4.f.i.289.2 4
15.8 even 4 720.4.f.i.289.1 4
20.3 even 4 120.4.f.d.49.4 yes 4
20.7 even 4 120.4.f.d.49.2 4
20.19 odd 2 600.4.a.t.1.1 2
40.3 even 4 960.4.f.n.769.1 4
40.13 odd 4 960.4.f.o.769.3 4
40.27 even 4 960.4.f.n.769.3 4
40.37 odd 4 960.4.f.o.769.1 4
60.23 odd 4 360.4.f.d.289.1 4
60.47 odd 4 360.4.f.d.289.2 4
60.59 even 2 1800.4.a.bn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.f.d.49.2 4 20.7 even 4
120.4.f.d.49.4 yes 4 20.3 even 4
240.4.f.g.49.2 4 5.3 odd 4
240.4.f.g.49.4 4 5.2 odd 4
360.4.f.d.289.1 4 60.23 odd 4
360.4.f.d.289.2 4 60.47 odd 4
600.4.a.t.1.1 2 20.19 odd 2
600.4.a.v.1.2 2 4.3 odd 2
720.4.f.i.289.1 4 15.8 even 4
720.4.f.i.289.2 4 15.2 even 4
960.4.f.n.769.1 4 40.3 even 4
960.4.f.n.769.3 4 40.27 even 4
960.4.f.o.769.1 4 40.37 odd 4
960.4.f.o.769.3 4 40.13 odd 4
1200.4.a.bo.1.1 2 1.1 even 1 trivial
1200.4.a.bq.1.2 2 5.4 even 2
1800.4.a.bl.1.2 2 12.11 even 2
1800.4.a.bn.1.1 2 60.59 even 2