Properties

Label 441.4.p.b.80.4
Level $441$
Weight $4$
Character 441.80
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 80.4
Root \(2.23256 - 1.28897i\) of defining polynomial
Character \(\chi\) \(=\) 441.80
Dual form 441.4.p.b.215.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.68205 - 2.70318i) q^{2} +(10.6144 - 18.3846i) q^{4} -71.5195i q^{8} +O(q^{10})\) \(q+(4.68205 - 2.70318i) q^{2} +(10.6144 - 18.3846i) q^{4} -71.5195i q^{8} +(-57.8432 - 33.3958i) q^{11} +(-108.415 - 187.780i) q^{16} -361.099 q^{22} +(108.360 - 62.5618i) q^{23} +(62.5000 - 108.253i) q^{25} +69.7031i q^{29} +(-519.707 - 300.053i) q^{32} +(-5.29150 - 9.16515i) q^{37} +534.442 q^{43} +(-1227.94 + 708.951i) q^{44} +(338.232 - 585.835i) q^{46} -675.795i q^{50} +(-56.7049 - 32.7386i) q^{53} +(188.420 + 326.353i) q^{58} -1509.75 q^{64} +(-370.000 + 640.859i) q^{67} +205.717i q^{71} +(-49.5501 - 28.6078i) q^{74} +(692.000 + 1198.58i) q^{79} +(2502.28 - 1444.69i) q^{86} +(-2388.45 + 4136.91i) q^{88} -2656.22i q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 32 q^{4}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.68205 2.70318i 1.65535 0.955719i 0.680536 0.732714i \(-0.261747\pi\)
0.974817 0.223005i \(-0.0715865\pi\)
\(3\) 0 0
\(4\) 10.6144 18.3846i 1.32680 2.29808i
\(5\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 71.5195i 3.16074i
\(9\) 0 0
\(10\) 0 0
\(11\) −57.8432 33.3958i −1.58549 0.915382i −0.994037 0.109042i \(-0.965222\pi\)
−0.591451 0.806341i \(-0.701445\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −108.415 187.780i −1.69398 2.93407i
\(17\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −361.099 −3.49939
\(23\) 108.360 62.5618i 0.982377 0.567176i 0.0793904 0.996844i \(-0.474703\pi\)
0.902987 + 0.429668i \(0.141369\pi\)
\(24\) 0 0
\(25\) 62.5000 108.253i 0.500000 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 69.7031i 0.446329i 0.974781 + 0.223165i \(0.0716388\pi\)
−0.974781 + 0.223165i \(0.928361\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) −519.707 300.053i −2.87100 1.65757i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.29150 9.16515i −0.0235113 0.0407227i 0.854030 0.520223i \(-0.174151\pi\)
−0.877542 + 0.479500i \(0.840818\pi\)
\(38\) 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 \(-0.103408\pi\)
0.947693 + 0.319183i \(0.103408\pi\)
\(44\) −1227.94 + 708.951i −4.20724 + 2.42905i
\(45\) 0 0
\(46\) 338.232 585.835i 1.08412 1.87775i
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 675.795i 1.91144i
\(51\) 0 0
\(52\) 0 0
\(53\) −56.7049 32.7386i −0.146963 0.0848489i 0.424716 0.905327i \(-0.360374\pi\)
−0.571678 + 0.820478i \(0.693707\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 188.420 + 326.353i 0.426565 + 0.738833i
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1509.75 −2.94873
\(65\) 0 0
\(66\) 0 0
\(67\) −370.000 + 640.859i −0.674667 + 1.16856i 0.301899 + 0.953340i \(0.402379\pi\)
−0.976566 + 0.215218i \(0.930954\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 205.717i 0.343861i 0.985109 + 0.171931i \(0.0550005\pi\)
−0.985109 + 0.171931i \(0.945000\pi\)
\(72\) 0 0
\(73\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) −49.5501 28.6078i −0.0778390 0.0449404i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 692.000 + 1198.58i 0.985520 + 1.70697i 0.639602 + 0.768706i \(0.279099\pi\)
0.345918 + 0.938265i \(0.387568\pi\)
\(80\) 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
\(86\) 2502.28 1444.69i 3.13753 1.81146i
\(87\) 0 0
\(88\) −2388.45 + 4136.91i −2.89329 + 5.01132i
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2656.22i 3.01011i
\(93\) 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
\(100\) −1326.80 2298.08i −1.32680 2.29808i
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −353.994 −0.324367
\(107\) 1916.98 1106.77i 1.73197 0.999955i 0.861261 0.508163i \(-0.169675\pi\)
0.870713 0.491792i \(-0.163658\pi\)
\(108\) 0 0
\(109\) 1137.67 1970.51i 0.999718 1.73156i 0.479312 0.877645i \(-0.340886\pi\)
0.520407 0.853919i \(-0.325780\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2105.12i 1.75251i −0.481849 0.876254i \(-0.660035\pi\)
0.481849 0.876254i \(-0.339965\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1281.47 + 739.856i 1.02570 + 0.592188i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1565.06 + 2710.76i 1.17585 + 2.03663i
\(122\) 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 \(0.253760\pi\)
\(128\) −2911.07 + 1680.71i −2.01019 + 1.16059i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4000.71i 2.57917i
\(135\) 0 0
\(136\) 0 0
\(137\) −1424.90 822.668i −0.888596 0.513031i −0.0151133 0.999886i \(-0.504811\pi\)
−0.873483 + 0.486854i \(0.838144\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 556.091 + 963.177i 0.328635 + 0.569212i
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −224.664 −0.124779
\(149\) −2669.52 + 1541.25i −1.46776 + 0.847409i −0.999348 0.0361037i \(-0.988505\pi\)
−0.468407 + 0.883513i \(0.655172\pi\)
\(150\) 0 0
\(151\) −1124.44 + 1947.59i −0.606000 + 1.04962i 0.385893 + 0.922544i \(0.373893\pi\)
−0.991893 + 0.127079i \(0.959440\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(158\) 6479.95 + 3741.20i 3.26277 + 1.88376i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 890.000 + 1541.53i 0.427670 + 0.740746i 0.996666 0.0815946i \(-0.0260013\pi\)
−0.568996 + 0.822340i \(0.692668\pi\)
\(164\) 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
\(172\) 5672.77 9825.52i 2.51479 4.35575i
\(173\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 14482.4i 6.20257i
\(177\) 0 0
\(178\) 0 0
\(179\) −3917.09 2261.53i −1.63562 0.944328i −0.982315 0.187237i \(-0.940047\pi\)
−0.653310 0.757091i \(-0.726620\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4474.39 7749.87i −1.79270 3.10504i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −435.945 + 251.693i −0.165151 + 0.0953502i −0.580298 0.814405i \(-0.697064\pi\)
0.415146 + 0.909755i \(0.363730\pi\)
\(192\) 0 0
\(193\) −1386.37 + 2401.27i −0.517064 + 0.895581i 0.482740 + 0.875764i \(0.339642\pi\)
−0.999804 + 0.0198172i \(0.993692\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2021.80i 0.731205i −0.930771 0.365603i \(-0.880863\pi\)
0.930771 0.365603i \(-0.119137\pi\)
\(198\) 0 0
\(199\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(200\) −7742.21 4469.97i −2.73728 1.58037i
\(201\) 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 \(-0.406617\pi\)
0.289181 + 0.957274i \(0.406617\pi\)
\(212\) −1203.78 + 695.000i −0.389979 + 0.225155i
\(213\) 0 0
\(214\) 5983.58 10363.9i 1.91135 3.31056i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 12301.3i 3.82180i
\(219\) 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
\(226\) −5690.53 9856.29i −1.67491 2.90102i
\(227\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(228\) 0 0
\(229\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 4985.13 1.41073
\(233\) 6149.85 3550.62i 1.72914 0.998321i 0.835611 0.549322i \(-0.185114\pi\)
0.893532 0.448999i \(-0.148219\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4897.58i 1.32551i 0.748834 + 0.662757i \(0.230614\pi\)
−0.748834 + 0.662757i \(0.769386\pi\)
\(240\) 0 0
\(241\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(242\) 14655.3 + 8461.26i 3.89289 + 2.24756i
\(243\) 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
\(253\) −8357.20 −2.07673
\(254\) 9364.09 5406.36i 2.31321 1.33553i
\(255\) 0 0
\(256\) −3047.51 + 5278.44i −0.744020 + 1.28868i
\(257\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7070.96 4082.42i −1.65785 0.957159i −0.973705 0.227812i \(-0.926843\pi\)
−0.684144 0.729347i \(-0.739824\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 7854.64 + 13604.6i 1.79029 + 3.10088i
\(269\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(270\) 0 0
\(271\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −8895.29 −1.96126
\(275\) −7230.40 + 4174.47i −1.58549 + 0.915382i
\(276\) 0 0
\(277\) −3655.00 + 6330.65i −0.792807 + 1.37318i 0.131415 + 0.991327i \(0.458048\pi\)
−0.924222 + 0.381855i \(0.875285\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2841.56i 0.603250i 0.953427 + 0.301625i \(0.0975291\pi\)
−0.953427 + 0.301625i \(0.902471\pi\)
\(282\) 0 0
\(283\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(284\) 3782.04 + 2183.56i 0.790220 + 0.456234i
\(285\) 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
\(296\) −655.487 + 378.445i −0.128714 + 0.0743131i
\(297\) 0 0
\(298\) −8332.54 + 14432.4i −1.61977 + 2.80552i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 12158.3i 2.31666i
\(303\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 29380.6 5.23034
\(317\) −9705.58 + 5603.52i −1.71962 + 0.992823i −0.800014 + 0.599982i \(0.795175\pi\)
−0.919606 + 0.392841i \(0.871492\pi\)
\(318\) 0 0
\(319\) 2327.79 4031.85i 0.408562 0.707650i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 8334.04 + 4811.66i 1.41589 + 0.817464i
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2553.15 4422.19i −0.423969 0.734336i 0.572354 0.820006i \(-0.306030\pi\)
−0.996324 + 0.0856702i \(0.972697\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 11916.5 1.92621 0.963103 0.269135i \(-0.0867376\pi\)
0.963103 + 0.269135i \(0.0867376\pi\)
\(338\) 10286.5 5938.89i 1.65535 0.955719i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 38223.0i 5.99083i
\(345\) 0 0
\(346\) 0 0
\(347\) 4997.21 + 2885.14i 0.773096 + 0.446347i 0.833978 0.551798i \(-0.186058\pi\)
−0.0608819 + 0.998145i \(0.519391\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 20041.0 + 34712.1i 3.03463 + 5.25613i
\(353\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −24453.3 −3.61005
\(359\) −11654.0 + 6728.46i −1.71330 + 0.989177i −0.783287 + 0.621660i \(0.786459\pi\)
−0.930017 + 0.367517i \(0.880208\pi\)
\(360\) 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 \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(368\) −23495.8 13565.3i −3.32826 1.92157i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6985.00 12098.4i −0.969624 1.67944i −0.696643 0.717418i \(-0.745324\pi\)
−0.272980 0.962020i \(-0.588009\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −8704.52 −1.17974 −0.589870 0.807498i \(-0.700821\pi\)
−0.589870 + 0.807498i \(0.700821\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1360.74 + 2356.88i −0.182256 + 0.315676i
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14990.5i 1.97667i
\(387\) 0 0
\(388\) 0 0
\(389\) 391.349 + 225.946i 0.0510082 + 0.0294496i 0.525287 0.850925i \(-0.323958\pi\)
−0.474279 + 0.880375i \(0.657291\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) −5465.30 9466.17i −0.698827 1.21040i
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −27103.8 −3.38797
\(401\) 10764.5 6214.88i 1.34053 0.773956i 0.353646 0.935379i \(-0.384942\pi\)
0.986885 + 0.161423i \(0.0516085\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 706.855i 0.0860873i
\(408\) 0 0
\(409\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(410\) 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 \(0.844753\pi\)
\(422\) 8299.65 4791.80i 0.957395 0.552752i
\(423\) 0 0
\(424\) −2341.45 + 4055.51i −0.268186 + 0.464511i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 46990.6i 5.30695i
\(429\) 0 0
\(430\) 0 0
\(431\) 14879.0 + 8590.39i 1.66287 + 0.960057i 0.971336 + 0.237712i \(0.0763975\pi\)
0.691533 + 0.722345i \(0.256936\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −24151.4 41831.4i −2.65285 4.59487i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12353.2 + 7132.14i −1.32488 + 0.764917i −0.984502 0.175373i \(-0.943887\pi\)
−0.340373 + 0.940290i \(0.610553\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 11421.0i 1.20043i 0.799841 + 0.600213i \(0.204917\pi\)
−0.799841 + 0.600213i \(0.795083\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −38702.0 22344.6i −4.02740 2.32522i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8910.89 15434.1i −0.912109 1.57982i −0.811079 0.584936i \(-0.801120\pi\)
−0.101030 0.994883i \(-0.532214\pi\)
\(458\) 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 \(0.639229\pi\)
\(464\) 13088.9 7556.87i 1.30956 0.756075i
\(465\) 0 0
\(466\) 19195.9 33248.3i 1.90823 3.30515i
\(467\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −30913.8 17848.1i −3.00511 1.73500i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 13239.0 + 22930.7i 1.26682 + 2.19420i
\(479\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 66448.4 6.24045
\(485\) 0 0
\(486\) 0 0
\(487\) −1648.30 + 2854.94i −0.153371 + 0.265647i −0.932465 0.361261i \(-0.882346\pi\)
0.779094 + 0.626908i \(0.215680\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 19811.9i 1.82097i 0.413540 + 0.910486i \(0.364292\pi\)
−0.413540 + 0.910486i \(0.635708\pi\)
\(492\) 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 0.754022 + 0.656850i \(0.228111\pi\)
0.191838 + 0.981427i \(0.438555\pi\)
\(500\) 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
\(506\) −39128.8 + 22591.0i −3.43772 + 1.98477i
\(507\) 0 0
\(508\) 21228.8 36769.3i 1.85408 3.21136i
\(509\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 6060.53i 0.523125i
\(513\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(522\) 0 0
\(523\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −44142.1 −3.65910
\(527\) 0 0
\(528\) 0 0
\(529\) 1744.47 3021.51i 0.143377 0.248336i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 45833.9 + 26462.2i 3.69351 + 2.13245i
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7939.00 13750.8i −0.630914 1.09277i −0.987365 0.158461i \(-0.949347\pi\)
0.356452 0.934314i \(-0.383986\pi\)
\(542\) 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 \(0.330644\pi\)
\(548\) −30248.9 + 17464.2i −2.35797 + 1.36138i
\(549\) 0 0
\(550\) −22568.7 + 39090.1i −1.74970 + 3.03056i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 39520.5i 3.03080i
\(555\) 0 0
\(556\) 0 0
\(557\) 22638.7 + 13070.5i 1.72214 + 0.994280i 0.914473 + 0.404646i \(0.132605\pi\)
0.807671 + 0.589634i \(0.200728\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 7681.25 + 13304.3i 0.576538 + 0.998593i
\(563\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 14712.8 1.08686
\(569\) 18679.9 10784.9i 1.37628 0.794596i 0.384572 0.923095i \(-0.374349\pi\)
0.991710 + 0.128499i \(0.0410159\pi\)
\(570\) 0 0
\(571\) −3394.00 + 5878.58i −0.248747 + 0.430842i −0.963178 0.268863i \(-0.913352\pi\)
0.714431 + 0.699705i \(0.246685\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 15640.5i 1.13435i
\(576\) 0 0
\(577\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(578\) 23002.9 + 13280.7i 1.65535 + 0.955719i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 2186.66 + 3787.41i 0.155338 + 0.269054i
\(584\) 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
\(592\) −1147.36 + 1987.28i −0.0796555 + 0.137967i
\(593\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 65437.5i 4.49736i
\(597\) 0 0
\(598\) 0 0
\(599\) 5507.54 + 3179.78i 0.375680 + 0.216899i 0.675937 0.736960i \(-0.263739\pi\)
−0.300257 + 0.953858i \(0.597073\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 23870.6 + 41345.0i 1.60808 + 2.78527i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 7505.00 12999.0i 0.494493 0.856487i −0.505487 0.862834i \(-0.668687\pi\)
0.999980 + 0.00634752i \(0.00202049\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23368.2i 1.52474i −0.647139 0.762372i \(-0.724035\pi\)
0.647139 0.762372i \(-0.275965\pi\)
\(618\) 0 0
\(619\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(620\) 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 \(0.190488\pi\)
\(632\) 85721.7 49491.5i 5.39530 3.11498i
\(633\) 0 0
\(634\) −30294.6 + 52471.9i −1.89772 + 3.28695i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 25169.8i 1.56188i
\(639\) 0 0
\(640\) 0 0
\(641\) −13681.5 7899.04i −0.843040 0.486729i 0.0152566 0.999884i \(-0.495143\pi\)
−0.858296 + 0.513154i \(0.828477\pi\)
\(642\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 37787.2 2.26972
\(653\) 4594.75 2652.78i 0.275354 0.158976i −0.355964 0.934500i \(-0.615847\pi\)
0.631318 + 0.775524i \(0.282514\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 25166.1i 1.48761i −0.668399 0.743803i \(-0.733020\pi\)
0.668399 0.743803i \(-0.266980\pi\)
\(660\) 0 0
\(661\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(662\) −23907.9 13803.3i −1.40364 0.810391i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4360.76 + 7553.05i 0.253147 + 0.438464i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −9609.37 −0.550392 −0.275196 0.961388i \(-0.588743\pi\)
−0.275196 + 0.961388i \(0.588743\pi\)
\(674\) 55793.4 32212.4i 3.18855 1.84091i
\(675\) 0 0
\(676\) 23319.8 40391.1i 1.32680 2.29808i
\(677\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 14308.6 + 8261.08i 0.801616 + 0.462813i 0.844036 0.536286i \(-0.180173\pi\)
−0.0424197 + 0.999100i \(0.513507\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −57941.5 100358.i −3.21076 5.56119i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 31196.2 1.70633
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 29047.8i 1.56508i −0.622602 0.782539i \(-0.713924\pi\)
0.622602 0.782539i \(-0.286076\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 87328.8 + 50419.3i 4.67518 + 2.69922i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 17805.9 + 30840.7i 0.943180 + 1.63364i 0.759354 + 0.650677i \(0.225515\pi\)
0.183826 + 0.982959i \(0.441152\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −83154.9 + 48009.5i −4.34028 + 2.50586i
\(717\) 0 0
\(718\) −36376.5 + 63005.9i −1.89075 + 3.27487i
\(719\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 37082.2i 1.91144i
\(723\) 0 0
\(724\) 0 0
\(725\) 7545.59 + 4356.45i 0.386532 + 0.223165i
\(726\) 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 \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −75087.5 −3.76055
\(737\) 42804.0 24712.9i 2.13935 1.23516i
\(738\) 0 0
\(739\) 12662.0 21931.2i 0.630283 1.09168i −0.357211 0.934024i \(-0.616272\pi\)
0.987494 0.157658i \(-0.0503945\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 40237.0i 1.98675i 0.114933 + 0.993373i \(0.463335\pi\)
−0.114933 + 0.993373i \(0.536665\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −65408.2 37763.4i −3.21014 1.85338i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 20544.3 + 35583.7i 0.998230 + 1.72898i 0.550620 + 0.834756i \(0.314391\pi\)
0.447610 + 0.894229i \(0.352275\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −22848.7 −1.09703 −0.548514 0.836141i \(-0.684806\pi\)
−0.548514 + 0.836141i \(0.684806\pi\)
\(758\) −40755.0 + 23529.9i −1.95289 + 1.12750i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 10686.3i 0.506041i
\(765\) 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
\(772\) 29431.0 + 50976.0i 1.37208 + 2.37651i
\(773\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 2443.09 0.112582
\(779\) 0 0
\(780\) 0 0
\(781\) 6870.08 11899.3i 0.314764 0.545188i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(788\) −37170.1 21460.2i −1.68037 0.970161i
\(789\) 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
\(800\) −64963.4 + 37506.6i −2.87100 + 1.65757i
\(801\) 0 0
\(802\) 33599.9 58196.7i 1.47937 2.56234i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 39335.6 + 22710.4i 1.70948 + 0.986967i 0.935199 + 0.354123i \(0.115221\pi\)
0.774279 + 0.632845i \(0.218113\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 1910.76 + 3309.53i 0.0822752 + 0.142505i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 15741.4 9088.31i 0.669159 0.386339i −0.126599 0.991954i \(-0.540406\pi\)
0.795758 + 0.605615i \(0.207073\pi\)
\(822\) 0 0
\(823\) 23120.0 40045.0i 0.979238 1.69609i 0.314062 0.949403i \(-0.398310\pi\)
0.665176 0.746687i \(-0.268357\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12090.1i 0.508359i 0.967157 + 0.254179i \(0.0818054\pi\)
−0.967157 + 0.254179i \(0.918195\pi\)
\(828\) 0 0
\(829\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(830\) 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
\(841\) 19530.5 0.800790
\(842\) −71457.4 + 41256.0i −2.92469 + 1.68857i
\(843\) 0 0
\(844\) 18815.6 32589.6i 0.767370 1.32912i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 14197.4i 0.574931i
\(849\) 0 0
\(850\) 0 0
\(851\) −1146.78 662.092i −0.0461939 0.0266701i
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −79155.4 137101.i −3.16060 5.47432i
\(857\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(858\) 0 0
\(859\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 92885.6 3.67018
\(863\) 16109.7 9300.93i 0.635435 0.366868i −0.147419 0.989074i \(-0.547097\pi\)
0.782854 + 0.622206i \(0.213763\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 92439.5i 3.60851i
\(870\) 0 0
\(871\) 0 0
\(872\) −140930. 81365.8i −5.47303 3.15985i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3275.00 + 5672.47i 0.126099 + 0.218410i 0.922162 0.386804i \(-0.126421\pi\)
−0.796063 + 0.605214i \(0.793088\pi\)
\(878\) 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 \(-0.194150\pi\)
0.819681 + 0.572820i \(0.194150\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −38558.9 + 66786.0i −1.46209 + 2.53242i
\(887\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(888\) 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
\(898\) 30873.1 + 53473.7i 1.14727 + 1.98713i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −150557. −5.53923
\(905\) 0 0
\(906\) 0 0
\(907\) −7125.01 + 12340.9i −0.260840 + 0.451788i −0.966465 0.256797i \(-0.917333\pi\)
0.705625 + 0.708585i \(0.250666\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 54982.7i 1.99962i 0.0193736 + 0.999812i \(0.493833\pi\)
−0.0193736 + 0.999812i \(0.506167\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −83442.4 48175.5i −3.01973 1.74344i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −25650.6 44428.1i −0.920711 1.59472i −0.798317 0.602238i \(-0.794276\pi\)
−0.122395 0.992482i \(-0.539057\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1322.88 −0.0470226
\(926\) −39516.5 + 22814.9i −1.40237 + 0.809657i
\(927\) 0 0
\(928\) 20914.6 36225.2i 0.739824 1.28141i
\(929\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 150750.i 5.29828i
\(933\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −192987. −6.63270
\(947\) −10398.7 + 6003.70i −0.356824 + 0.206013i −0.667687 0.744442i \(-0.732715\pi\)
0.310862 + 0.950455i \(0.399382\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 15373.4i 0.522553i 0.965264 + 0.261276i \(0.0841434\pi\)
−0.965264 + 0.261276i \(0.915857\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 90040.2 + 51984.8i 3.04614 + 1.75869i
\(957\) 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 \(0.167127\pi\)
\(968\) 193872. 111932.i 6.43727 3.71656i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 17822.6i 0.586319i
\(975\) 0 0
\(976\) 0 0
\(977\) 6568.41 + 3792.27i 0.215089 + 0.124182i 0.603674 0.797231i \(-0.293703\pi\)
−0.388585 + 0.921413i \(0.627036\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 53555.1 + 92760.1i 1.74034 + 3.01435i
\(983\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 57912.3 33435.7i 1.86198 1.07502i
\(990\) 0 0
\(991\) −12077.9 + 20919.5i −0.387150 + 0.670564i −0.992065 0.125727i \(-0.959874\pi\)
0.604915 + 0.796290i \(0.293207\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(998\) 98728.6 + 57001.0i 3.13146 + 1.80795i
\(999\) 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.80.4 8
3.2 odd 2 inner 441.4.p.b.80.1 8
7.2 even 3 inner 441.4.p.b.215.1 8
7.3 odd 6 63.4.c.a.62.4 yes 4
7.4 even 3 63.4.c.a.62.4 yes 4
7.5 odd 6 inner 441.4.p.b.215.1 8
7.6 odd 2 CM 441.4.p.b.80.4 8
21.2 odd 6 inner 441.4.p.b.215.4 8
21.5 even 6 inner 441.4.p.b.215.4 8
21.11 odd 6 63.4.c.a.62.1 4
21.17 even 6 63.4.c.a.62.1 4
21.20 even 2 inner 441.4.p.b.80.1 8
28.3 even 6 1008.4.k.a.881.3 4
28.11 odd 6 1008.4.k.a.881.3 4
84.11 even 6 1008.4.k.a.881.4 4
84.59 odd 6 1008.4.k.a.881.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.c.a.62.1 4 21.11 odd 6
63.4.c.a.62.1 4 21.17 even 6
63.4.c.a.62.4 yes 4 7.3 odd 6
63.4.c.a.62.4 yes 4 7.4 even 3
441.4.p.b.80.1 8 3.2 odd 2 inner
441.4.p.b.80.1 8 21.20 even 2 inner
441.4.p.b.80.4 8 1.1 even 1 trivial
441.4.p.b.80.4 8 7.6 odd 2 CM
441.4.p.b.215.1 8 7.2 even 3 inner
441.4.p.b.215.1 8 7.5 odd 6 inner
441.4.p.b.215.4 8 21.2 odd 6 inner
441.4.p.b.215.4 8 21.5 even 6 inner
1008.4.k.a.881.3 4 28.3 even 6
1008.4.k.a.881.3 4 28.11 odd 6
1008.4.k.a.881.4 4 84.11 even 6
1008.4.k.a.881.4 4 84.59 odd 6