Properties

Label 441.4.p.b.215.3
Level $441$
Weight $4$
Character 441.215
Analytic conductor $26.020$
Analytic rank $0$
Dimension $8$
CM discriminant -7
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,4,Mod(80,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.80");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.p (of order \(6\), degree \(2\), not minimal)

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: no
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12745506816.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 215.3
Root \(-1.00781 - 0.581861i\) of defining polynomial
Character \(\chi\) \(=\) 441.215
Dual form 441.4.p.b.80.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.44168 + 0.832353i) q^{2} +(-2.61438 - 4.52824i) q^{4} -22.0220i q^{8} +O(q^{10})\) \(q+(1.44168 + 0.832353i) q^{2} +(-2.61438 - 4.52824i) q^{4} -22.0220i q^{8} +(-25.4395 + 14.6875i) q^{11} +(-2.58497 + 4.47731i) q^{16} -48.9007 q^{22} +(-157.350 - 90.8461i) q^{23} +(62.5000 + 108.253i) q^{25} +304.463i q^{29} +(-160.026 + 92.3911i) q^{32} +(5.29150 - 9.16515i) q^{37} -534.442 q^{43} +(133.017 + 76.7973i) q^{44} +(-151.232 - 261.942i) q^{46} +208.088i q^{50} +(-665.895 + 384.454i) q^{53} +(-253.420 + 438.937i) q^{58} -266.248 q^{64} +(-370.000 - 640.859i) q^{67} +1178.70i q^{71} +(15.2573 - 8.80879i) q^{74} +(692.000 - 1198.58i) q^{79} +(-770.492 - 444.844i) q^{86} +(323.448 + 560.228i) q^{88} +950.024i q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 32 q^{4} - 444 q^{16} - 1640 q^{22} + 500 q^{25} + 748 q^{46} - 260 q^{58} - 7104 q^{64} - 2960 q^{67} + 5536 q^{79} - 8260 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.44168 + 0.832353i 0.509710 + 0.294281i 0.732714 0.680536i \(-0.238253\pi\)
−0.223005 + 0.974817i \(0.571586\pi\)
\(3\) 0 0
\(4\) −2.61438 4.52824i −0.326797 0.566030i
\(5\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 22.0220i 0.973243i
\(9\) 0 0
\(10\) 0 0
\(11\) −25.4395 + 14.6875i −0.697299 + 0.402586i −0.806341 0.591451i \(-0.798555\pi\)
0.109042 + 0.994037i \(0.465222\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.58497 + 4.47731i −0.0403902 + 0.0699579i
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −48.9007 −0.473894
\(23\) −157.350 90.8461i −1.42651 0.823597i −0.429668 0.902987i \(-0.641369\pi\)
−0.996844 + 0.0793904i \(0.974703\pi\)
\(24\) 0 0
\(25\) 62.5000 + 108.253i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 304.463i 1.94956i 0.223165 + 0.974781i \(0.428361\pi\)
−0.223165 + 0.974781i \(0.571639\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) −160.026 + 92.3911i −0.884028 + 0.510394i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.29150 9.16515i 0.0235113 0.0407227i −0.854030 0.520223i \(-0.825849\pi\)
0.877542 + 0.479500i \(0.159182\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −534.442 −1.89539 −0.947693 0.319183i \(-0.896592\pi\)
−0.947693 + 0.319183i \(0.896592\pi\)
\(44\) 133.017 + 76.7973i 0.455751 + 0.263128i
\(45\) 0 0
\(46\) −151.232 261.942i −0.484738 0.839591i
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 208.088i 0.588562i
\(51\) 0 0
\(52\) 0 0
\(53\) −665.895 + 384.454i −1.72580 + 0.996394i −0.820478 + 0.571678i \(0.806293\pi\)
−0.905327 + 0.424716i \(0.860374\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −253.420 + 438.937i −0.573719 + 0.993711i
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −266.248 −0.520016
\(65\) 0 0
\(66\) 0 0
\(67\) −370.000 640.859i −0.674667 1.16856i −0.976566 0.215218i \(-0.930954\pi\)
0.301899 0.953340i \(-0.402379\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1178.70i 1.97022i 0.171931 + 0.985109i \(0.445000\pi\)
−0.171931 + 0.985109i \(0.555000\pi\)
\(72\) 0 0
\(73\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(74\) 15.2573 8.80879i 0.0239679 0.0138379i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 692.000 1198.58i 0.985520 1.70697i 0.345918 0.938265i \(-0.387568\pi\)
0.639602 0.768706i \(-0.279099\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −770.492 444.844i −0.966097 0.557776i
\(87\) 0 0
\(88\) 323.448 + 560.228i 0.391814 + 0.678642i
\(89\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 950.024i 1.07660i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 326.797 566.030i 0.326797 0.566030i
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −1280.01 −1.17288
\(107\) 18.1199 + 10.4615i 0.0163712 + 0.00945193i 0.508163 0.861261i \(-0.330325\pi\)
−0.491792 + 0.870713i \(0.663658\pi\)
\(108\) 0 0
\(109\) −1137.67 1970.51i −0.999718 1.73156i −0.520407 0.853919i \(-0.674220\pi\)
−0.479312 0.877645i \(-0.659114\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1157.60i 0.963699i −0.876254 0.481849i \(-0.839965\pi\)
0.876254 0.481849i \(-0.160035\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1378.68 795.980i 1.10351 0.637111i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −234.055 + 405.396i −0.175849 + 0.304580i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2000.00 1.39741 0.698706 0.715409i \(-0.253760\pi\)
0.698706 + 0.715409i \(0.253760\pi\)
\(128\) 896.365 + 517.517i 0.618970 + 0.357363i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1231.88i 0.794167i
\(135\) 0 0
\(136\) 0 0
\(137\) −2384.05 + 1376.43i −1.48674 + 0.858370i −0.999886 0.0151133i \(-0.995189\pi\)
−0.486854 + 0.873483i \(0.661856\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −981.091 + 1699.30i −0.579798 + 1.00424i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −55.3360 −0.0307337
\(149\) 1672.58 + 965.663i 0.919616 + 0.530941i 0.883513 0.468407i \(-0.155172\pi\)
0.0361037 + 0.999348i \(0.488505\pi\)
\(150\) 0 0
\(151\) 1124.44 + 1947.59i 0.606000 + 1.04962i 0.991893 + 0.127079i \(0.0405601\pi\)
−0.385893 + 0.922544i \(0.626107\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(158\) 1995.28 1151.98i 1.00466 0.580040i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 890.000 1541.53i 0.427670 0.740746i −0.568996 0.822340i \(-0.692668\pi\)
0.996666 + 0.0815946i \(0.0260013\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 2197.00 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 1397.23 + 2420.08i 0.619407 + 1.07284i
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 151.867i 0.0650421i
\(177\) 0 0
\(178\) 0 0
\(179\) 1364.72 787.920i 0.569854 0.329005i −0.187237 0.982315i \(-0.559953\pi\)
0.757091 + 0.653310i \(0.226620\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2000.61 + 3465.16i −0.801560 + 1.38834i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4551.22 2627.65i −1.72416 0.995444i −0.909755 0.415146i \(-0.863730\pi\)
−0.814405 0.580298i \(-0.802936\pi\)
\(192\) 0 0
\(193\) 1386.37 + 2401.27i 0.517064 + 0.895581i 0.999804 + 0.0198172i \(0.00630844\pi\)
−0.482740 + 0.875764i \(0.660358\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5147.21i 1.86154i −0.365603 0.930771i \(-0.619137\pi\)
0.365603 0.930771i \(-0.380863\pi\)
\(198\) 0 0
\(199\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(200\) 2383.95 1376.37i 0.842853 0.486622i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −1772.65 −0.578363 −0.289181 0.957274i \(-0.593383\pi\)
−0.289181 + 0.957274i \(0.593383\pi\)
\(212\) 3481.80 + 2010.22i 1.12798 + 0.651238i
\(213\) 0 0
\(214\) 17.4154 + 30.1644i 0.00556305 + 0.00963548i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 3787.78i 1.17679i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 963.532 1668.89i 0.283598 0.491207i
\(227\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(228\) 0 0
\(229\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6704.87 1.89740
\(233\) −356.810 206.004i −0.100324 0.0579219i 0.448999 0.893532i \(-0.351781\pi\)
−0.549322 + 0.835611i \(0.685114\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5533.66i 1.49767i −0.662757 0.748834i \(-0.730614\pi\)
0.662757 0.748834i \(-0.269386\pi\)
\(240\) 0 0
\(241\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(242\) −674.865 + 389.633i −0.179264 + 0.103498i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 5337.20 1.32627
\(254\) 2883.35 + 1664.71i 0.712274 + 0.411232i
\(255\) 0 0
\(256\) 1926.51 + 3336.81i 0.470338 + 0.814650i
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2139.12 + 1235.02i −0.501535 + 0.289561i −0.729347 0.684144i \(-0.760176\pi\)
0.227812 + 0.973705i \(0.426843\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1934.64 + 3350.89i −0.440959 + 0.763763i
\(269\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(270\) 0 0
\(271\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −4582.71 −1.01041
\(275\) −3179.93 1835.94i −0.697299 0.402586i
\(276\) 0 0
\(277\) −3655.00 6330.65i −0.792807 1.37318i −0.924222 0.381855i \(-0.875285\pi\)
0.131415 0.991327i \(-0.458048\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8982.08i 1.90685i 0.301625 + 0.953427i \(0.402471\pi\)
−0.301625 + 0.953427i \(0.597529\pi\)
\(282\) 0 0
\(283\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(284\) 5337.41 3081.56i 1.11520 0.643862i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2456.50 4254.78i 0.500000 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −201.835 116.529i −0.0396331 0.0228822i
\(297\) 0 0
\(298\) 1607.54 + 2784.35i 0.312492 + 0.541251i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 3743.74i 0.713337i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −7236.60 −1.28826
\(317\) 1169.11 + 674.983i 0.207140 + 0.119593i 0.599982 0.800014i \(-0.295175\pi\)
−0.392841 + 0.919606i \(0.628508\pi\)
\(318\) 0 0
\(319\) −4471.79 7745.37i −0.784866 1.35943i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 2566.19 1481.59i 0.435975 0.251710i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2553.15 4422.19i 0.423969 0.734336i −0.572354 0.820006i \(-0.693970\pi\)
0.996324 + 0.0856702i \(0.0273031\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −11916.5 −1.92621 −0.963103 0.269135i \(-0.913262\pi\)
−0.963103 + 0.269135i \(0.913262\pi\)
\(338\) 3167.36 + 1828.68i 0.509710 + 0.294281i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 11769.5i 1.84467i
\(345\) 0 0
\(346\) 0 0
\(347\) −10018.7 + 5784.28i −1.54994 + 0.894860i −0.551798 + 0.833978i \(0.686058\pi\)
−0.998145 + 0.0608819i \(0.980609\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2713.99 4700.76i 0.410955 0.711794i
\(353\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 2623.31 0.387280
\(359\) 1728.70 + 998.065i 0.254143 + 0.146729i 0.621660 0.783287i \(-0.286459\pi\)
−0.367517 + 0.930017i \(0.619792\pi\)
\(360\) 0 0
\(361\) −3429.50 5940.07i −0.500000 0.866025i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(368\) 813.492 469.670i 0.115234 0.0665305i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6985.00 + 12098.4i −0.969624 + 1.67944i −0.272980 + 0.962020i \(0.588009\pi\)
−0.696643 + 0.717418i \(0.745324\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8704.52 1.17974 0.589870 0.807498i \(-0.299179\pi\)
0.589870 + 0.807498i \(0.299179\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −4374.26 7576.43i −0.585881 1.01477i
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4615.81i 0.608649i
\(387\) 0 0
\(388\) 0 0
\(389\) −13283.0 + 7668.95i −1.73130 + 0.999566i −0.850925 + 0.525287i \(0.823958\pi\)
−0.880375 + 0.474279i \(0.842709\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 4284.30 7420.62i 0.547817 0.948846i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −646.243 −0.0807804
\(401\) −8807.35 5084.92i −1.09680 0.633239i −0.161423 0.986885i \(-0.551608\pi\)
−0.935379 + 0.353646i \(0.884942\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 310.876i 0.0378612i
\(408\) 0 0
\(409\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −15262.0 −1.76680 −0.883402 0.468616i \(-0.844753\pi\)
−0.883402 + 0.468616i \(0.844753\pi\)
\(422\) −2555.59 1475.47i −0.294797 0.170201i
\(423\) 0 0
\(424\) 8466.45 + 14664.3i 0.969734 + 1.67963i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 109.402i 0.0123555i
\(429\) 0 0
\(430\) 0 0
\(431\) −4336.40 + 2503.62i −0.484633 + 0.279803i −0.722345 0.691533i \(-0.756936\pi\)
0.237712 + 0.971336i \(0.423602\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −5948.62 + 10303.3i −0.653411 + 1.13174i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10402.5 6005.90i −1.11566 0.644129i −0.175373 0.984502i \(-0.556113\pi\)
−0.940290 + 0.340373i \(0.889447\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 15219.6i 1.59968i 0.600213 + 0.799841i \(0.295083\pi\)
−0.600213 + 0.799841i \(0.704917\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −5241.89 + 3026.41i −0.545482 + 0.314934i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8910.89 15434.1i 0.912109 1.57982i 0.101030 0.994883i \(-0.467786\pi\)
0.811079 0.584936i \(-0.198880\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −8440.00 −0.847171 −0.423585 0.905856i \(-0.639229\pi\)
−0.423585 + 0.905856i \(0.639229\pi\)
\(464\) −1363.17 787.028i −0.136387 0.0787432i
\(465\) 0 0
\(466\) −342.937 593.984i −0.0340906 0.0590467i
\(467\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 13595.9 7849.61i 1.32165 0.763056i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 4605.96 7977.75i 0.440735 0.763376i
\(479\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 2447.64 0.229868
\(485\) 0 0
\(486\) 0 0
\(487\) 1648.30 + 2854.94i 0.153371 + 0.265647i 0.932465 0.361261i \(-0.117654\pi\)
−0.779094 + 0.626908i \(0.784320\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8998.48i 0.827079i −0.910486 0.413540i \(-0.864292\pi\)
0.910486 0.413540i \(-0.135708\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −10543.3 + 18261.6i −0.945859 + 1.63828i −0.191838 + 0.981427i \(0.561445\pi\)
−0.754022 + 0.656850i \(0.771889\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 7694.52 + 4442.44i 0.676015 + 0.390297i
\(507\) 0 0
\(508\) −5228.76 9056.47i −0.456670 0.790976i
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1866.13i 0.161079i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(522\) 0 0
\(523\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −4111.89 −0.340850
\(527\) 0 0
\(528\) 0 0
\(529\) 10422.5 + 18052.4i 0.856623 + 1.48371i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −14113.0 + 8148.13i −1.13729 + 0.656615i
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7939.00 + 13750.8i −0.630914 + 1.09277i 0.356452 + 0.934314i \(0.383986\pi\)
−0.987365 + 0.158461i \(0.949347\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 12980.0 1.01460 0.507299 0.861770i \(-0.330644\pi\)
0.507299 + 0.861770i \(0.330644\pi\)
\(548\) 12465.6 + 7197.04i 0.971725 + 0.561026i
\(549\) 0 0
\(550\) −3056.29 5293.65i −0.236947 0.410404i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 12169.0i 0.933233i
\(555\) 0 0
\(556\) 0 0
\(557\) 2431.79 1403.99i 0.184988 0.106803i −0.404646 0.914473i \(-0.632605\pi\)
0.589634 + 0.807671i \(0.299272\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −7476.25 + 12949.3i −0.561151 + 0.971942i
\(563\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 25957.2 1.91750
\(569\) 14273.0 + 8240.54i 1.05159 + 0.607138i 0.923095 0.384572i \(-0.125651\pi\)
0.128499 + 0.991710i \(0.458984\pi\)
\(570\) 0 0
\(571\) −3394.00 5878.58i −0.248747 0.430842i 0.714431 0.699705i \(-0.246685\pi\)
−0.963178 + 0.268863i \(0.913352\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 22711.5i 1.64719i
\(576\) 0 0
\(577\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(578\) 7082.96 4089.35i 0.509710 0.294281i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 11293.3 19560.6i 0.802268 1.38957i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 27.3568 + 47.3834i 0.00189925 + 0.00328960i
\(593\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10098.4i 0.694040i
\(597\) 0 0
\(598\) 0 0
\(599\) 24787.7 14311.2i 1.69082 0.976194i 0.736960 0.675937i \(-0.236261\pi\)
0.953858 0.300257i \(-0.0970725\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 5879.45 10183.5i 0.396078 0.686028i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 7505.00 + 12999.0i 0.494493 + 0.856487i 0.999980 0.00634752i \(-0.00202049\pi\)
−0.505487 + 0.862834i \(0.668687\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 19836.0i 1.29428i 0.762372 + 0.647139i \(0.224035\pi\)
−0.762372 + 0.647139i \(0.775965\pi\)
\(618\) 0 0
\(619\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −7812.50 + 13531.6i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 26192.0 1.65244 0.826218 0.563351i \(-0.190488\pi\)
0.826218 + 0.563351i \(0.190488\pi\)
\(632\) −26395.1 15239.2i −1.66130 0.959151i
\(633\) 0 0
\(634\) 1123.65 + 1946.22i 0.0703877 + 0.121915i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 14888.4i 0.923885i
\(639\) 0 0
\(640\) 0 0
\(641\) 24554.8 14176.7i 1.51304 0.873553i 0.513154 0.858296i \(-0.328477\pi\)
0.999884 0.0152566i \(-0.00485652\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −9307.19 −0.559045
\(653\) 28534.6 + 16474.5i 1.71002 + 0.987282i 0.934500 + 0.355964i \(0.115847\pi\)
0.775524 + 0.631318i \(0.217486\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 22614.9i 1.33680i −0.743803 0.668399i \(-0.766980\pi\)
0.743803 0.668399i \(-0.233020\pi\)
\(660\) 0 0
\(661\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(662\) 7361.64 4250.24i 0.432203 0.249532i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 27659.2 47907.2i 1.60565 2.78107i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 9609.37 0.550392 0.275196 0.961388i \(-0.411257\pi\)
0.275196 + 0.961388i \(0.411257\pi\)
\(674\) −17179.7 9918.70i −0.981806 0.566846i
\(675\) 0 0
\(676\) −5743.79 9948.53i −0.326797 0.566030i
\(677\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 27406.2 15823.0i 1.53539 0.886456i 0.536286 0.844036i \(-0.319827\pi\)
0.999100 0.0424197i \(-0.0135066\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 1381.52 2392.86i 0.0765551 0.132597i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −19258.2 −1.05336
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 23110.9i 1.24520i −0.782539 0.622602i \(-0.786076\pi\)
0.782539 0.622602i \(-0.213924\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 6773.22 3910.52i 0.362607 0.209351i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −17805.9 + 30840.7i −0.943180 + 1.63364i −0.183826 + 0.982959i \(0.558848\pi\)
−0.759354 + 0.650677i \(0.774485\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −7135.78 4119.84i −0.372453 0.215036i
\(717\) 0 0
\(718\) 1661.48 + 2877.77i 0.0863593 + 0.149579i
\(719\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 11418.2i 0.588562i
\(723\) 0 0
\(724\) 0 0
\(725\) −32959.0 + 19028.9i −1.68837 + 0.974781i
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 33573.5 1.68143
\(737\) 18825.2 + 10868.7i 0.940889 + 0.543223i
\(738\) 0 0
\(739\) 12662.0 + 21931.2i 0.630283 + 1.09168i 0.987494 + 0.157658i \(0.0503945\pi\)
−0.357211 + 0.934024i \(0.616272\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4655.40i 0.229865i 0.993373 + 0.114933i \(0.0366652\pi\)
−0.993373 + 0.114933i \(0.963335\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −20140.2 + 11628.0i −0.988453 + 0.570684i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −20544.3 + 35583.7i −0.998230 + 1.72898i −0.447610 + 0.894229i \(0.647725\pi\)
−0.550620 + 0.834756i \(0.685609\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 22848.7 1.09703 0.548514 0.836141i \(-0.315194\pi\)
0.548514 + 0.836141i \(0.315194\pi\)
\(758\) 12549.1 + 7245.23i 0.601325 + 0.347175i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 27478.6i 1.30123i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7249.01 12555.7i 0.337950 0.585347i
\(773\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −25533.1 −1.17661
\(779\) 0 0
\(780\) 0 0
\(781\) −17312.1 29985.4i −0.793182 1.37383i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(788\) −23307.8 + 13456.8i −1.05369 + 0.608347i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −20003.3 11548.9i −0.884028 0.510394i
\(801\) 0 0
\(802\) −8464.90 14661.6i −0.372701 0.645537i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 6413.48 3702.82i 0.278722 0.160920i −0.354123 0.935199i \(-0.615221\pi\)
0.632845 + 0.774279i \(0.281887\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −258.758 + 448.182i −0.0111418 + 0.0192982i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 37581.5 + 21697.7i 1.59757 + 0.922357i 0.991954 + 0.126599i \(0.0404061\pi\)
0.605615 + 0.795758i \(0.292927\pi\)
\(822\) 0 0
\(823\) 23120.0 + 40045.0i 0.979238 + 1.69609i 0.665176 + 0.746687i \(0.268357\pi\)
0.314062 + 0.949403i \(0.398310\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 46002.9i 1.93431i 0.254179 + 0.967157i \(0.418195\pi\)
−0.254179 + 0.967157i \(0.581805\pi\)
\(828\) 0 0
\(829\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −68308.5 −2.80079
\(842\) −22002.9 12703.4i −0.900557 0.519937i
\(843\) 0 0
\(844\) 4634.39 + 8026.99i 0.189007 + 0.327370i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 3975.22i 0.160978i
\(849\) 0 0
\(850\) 0 0
\(851\) −1665.24 + 961.425i −0.0670782 + 0.0387276i
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 230.384 399.037i 0.00919902 0.0159332i
\(857\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(858\) 0 0
\(859\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −8335.58 −0.329363
\(863\) −40849.5 23584.5i −1.61128 0.930273i −0.989074 0.147419i \(-0.952903\pi\)
−0.622206 0.782854i \(-0.713763\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 40655.0i 1.58703i
\(870\) 0 0
\(871\) 0 0
\(872\) −43394.5 + 25053.8i −1.68523 + 0.972969i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3275.00 5672.47i 0.126099 0.218410i −0.796063 0.605214i \(-0.793088\pi\)
0.922162 + 0.386804i \(0.126421\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −43014.6 −1.63936 −0.819681 0.572820i \(-0.805850\pi\)
−0.819681 + 0.572820i \(0.805850\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −9998.06 17317.1i −0.379110 0.656637i
\(887\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −12668.1 + 21941.7i −0.470756 + 0.815373i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −25492.7 −0.937913
\(905\) 0 0
\(906\) 0 0
\(907\) 7125.01 + 12340.9i 0.260840 + 0.451788i 0.966465 0.256797i \(-0.0826671\pi\)
−0.705625 + 0.708585i \(0.749334\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1065.41i 0.0387473i −0.999812 0.0193736i \(-0.993833\pi\)
0.999812 0.0193736i \(-0.00616720\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 25693.3 14834.0i 0.929822 0.536833i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 25650.6 44428.1i 0.920711 1.59472i 0.122395 0.992482i \(-0.460943\pi\)
0.798317 0.602238i \(-0.205724\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1322.88 0.0470226
\(926\) −12167.8 7025.06i −0.431811 0.249306i
\(927\) 0 0
\(928\) −28129.6 48722.0i −0.995044 1.72347i
\(929\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 2154.29i 0.0757149i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 26134.6 0.898211
\(947\) −49393.3 28517.3i −1.69490 0.978549i −0.950455 0.310862i \(-0.899382\pi\)
−0.744442 0.667687i \(-0.767285\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 56795.7i 1.93053i 0.261276 + 0.965264i \(0.415857\pi\)
−0.261276 + 0.965264i \(0.584143\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −25057.7 + 14467.1i −0.847724 + 0.489434i
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −14895.5 + 25799.8i −0.500000 + 0.866025i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 52040.0 1.73060 0.865302 0.501251i \(-0.167127\pi\)
0.865302 + 0.501251i \(0.167127\pi\)
\(968\) 8927.62 + 5154.36i 0.296430 + 0.171144i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 5487.88i 0.180537i
\(975\) 0 0
\(976\) 0 0
\(977\) −52484.1 + 30301.7i −1.71864 + 0.992259i −0.797231 + 0.603674i \(0.793703\pi\)
−0.921413 + 0.388585i \(0.872964\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 7489.91 12972.9i 0.243394 0.421570i
\(983\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 84094.5 + 48552.0i 2.70379 + 1.56103i
\(990\) 0 0
\(991\) 12077.9 + 20919.5i 0.387150 + 0.670564i 0.992065 0.125727i \(-0.0401262\pi\)
−0.604915 + 0.796290i \(0.706793\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(998\) −30400.1 + 17551.5i −0.964227 + 0.556697i
\(999\) 0 0
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 441.4.p.b.215.3 8
3.2 odd 2 inner 441.4.p.b.215.2 8
7.2 even 3 63.4.c.a.62.2 4
7.3 odd 6 inner 441.4.p.b.80.2 8
7.4 even 3 inner 441.4.p.b.80.2 8
7.5 odd 6 63.4.c.a.62.2 4
7.6 odd 2 CM 441.4.p.b.215.3 8
21.2 odd 6 63.4.c.a.62.3 yes 4
21.5 even 6 63.4.c.a.62.3 yes 4
21.11 odd 6 inner 441.4.p.b.80.3 8
21.17 even 6 inner 441.4.p.b.80.3 8
21.20 even 2 inner 441.4.p.b.215.2 8
28.19 even 6 1008.4.k.a.881.2 4
28.23 odd 6 1008.4.k.a.881.2 4
84.23 even 6 1008.4.k.a.881.1 4
84.47 odd 6 1008.4.k.a.881.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.c.a.62.2 4 7.2 even 3
63.4.c.a.62.2 4 7.5 odd 6
63.4.c.a.62.3 yes 4 21.2 odd 6
63.4.c.a.62.3 yes 4 21.5 even 6
441.4.p.b.80.2 8 7.3 odd 6 inner
441.4.p.b.80.2 8 7.4 even 3 inner
441.4.p.b.80.3 8 21.11 odd 6 inner
441.4.p.b.80.3 8 21.17 even 6 inner
441.4.p.b.215.2 8 3.2 odd 2 inner
441.4.p.b.215.2 8 21.20 even 2 inner
441.4.p.b.215.3 8 1.1 even 1 trivial
441.4.p.b.215.3 8 7.6 odd 2 CM
1008.4.k.a.881.1 4 84.23 even 6
1008.4.k.a.881.1 4 84.47 odd 6
1008.4.k.a.881.2 4 28.19 even 6
1008.4.k.a.881.2 4 28.23 odd 6