Properties

Label 1080.4.a.n.1.2
Level $1080$
Weight $4$
Character 1080.1
Self dual yes
Analytic conductor $63.722$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1080.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.7220628062\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.47977.1
Defining polynomial: \( x^{3} - x^{2} - 60x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(8.58548\) of defining polynomial
Character \(\chi\) \(=\) 1080.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.00000 q^{5} +9.46549 q^{7} +O(q^{10})\) \(q+5.00000 q^{5} +9.46549 q^{7} +49.0474 q^{11} +55.4438 q^{13} -27.9783 q^{17} +26.3965 q^{19} +9.53451 q^{23} +25.0000 q^{25} +218.379 q^{29} -158.961 q^{31} +47.3274 q^{35} +189.655 q^{37} -246.146 q^{41} +40.2327 q^{43} -213.965 q^{47} -253.405 q^{49} -283.310 q^{53} +245.237 q^{55} -92.0000 q^{59} +370.172 q^{61} +277.219 q^{65} +1087.53 q^{67} +333.681 q^{71} -606.791 q^{73} +464.257 q^{77} -44.3751 q^{79} -107.732 q^{83} -139.892 q^{85} +257.819 q^{89} +524.803 q^{91} +131.982 q^{95} +1312.05 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 15 q^{5} + 9 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 15 q^{5} + 9 q^{7} + 18 q^{11} - 21 q^{13} + 84 q^{17} + 21 q^{19} + 48 q^{23} + 75 q^{25} - 36 q^{29} + 324 q^{31} + 45 q^{35} + 33 q^{37} + 114 q^{41} + 282 q^{43} + 282 q^{47} + 228 q^{49} + 222 q^{53} + 90 q^{55} - 276 q^{59} + 303 q^{61} - 105 q^{65} + 1035 q^{67} + 510 q^{71} + 447 q^{73} - 1578 q^{77} + 777 q^{79} + 78 q^{83} + 420 q^{85} - 324 q^{89} + 1995 q^{91} + 105 q^{95} + 1191 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 9.46549 0.511088 0.255544 0.966797i \(-0.417745\pi\)
0.255544 + 0.966797i \(0.417745\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 49.0474 1.34439 0.672197 0.740372i \(-0.265351\pi\)
0.672197 + 0.740372i \(0.265351\pi\)
\(12\) 0 0
\(13\) 55.4438 1.18287 0.591437 0.806351i \(-0.298561\pi\)
0.591437 + 0.806351i \(0.298561\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −27.9783 −0.399162 −0.199581 0.979881i \(-0.563958\pi\)
−0.199581 + 0.979881i \(0.563958\pi\)
\(18\) 0 0
\(19\) 26.3965 0.318724 0.159362 0.987220i \(-0.449056\pi\)
0.159362 + 0.987220i \(0.449056\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 9.53451 0.0864384 0.0432192 0.999066i \(-0.486239\pi\)
0.0432192 + 0.999066i \(0.486239\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 218.379 1.39834 0.699171 0.714954i \(-0.253552\pi\)
0.699171 + 0.714954i \(0.253552\pi\)
\(30\) 0 0
\(31\) −158.961 −0.920974 −0.460487 0.887666i \(-0.652325\pi\)
−0.460487 + 0.887666i \(0.652325\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 47.3274 0.228566
\(36\) 0 0
\(37\) 189.655 0.842678 0.421339 0.906903i \(-0.361560\pi\)
0.421339 + 0.906903i \(0.361560\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −246.146 −0.937599 −0.468800 0.883304i \(-0.655313\pi\)
−0.468800 + 0.883304i \(0.655313\pi\)
\(42\) 0 0
\(43\) 40.2327 0.142684 0.0713422 0.997452i \(-0.477272\pi\)
0.0713422 + 0.997452i \(0.477272\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −213.965 −0.664042 −0.332021 0.943272i \(-0.607730\pi\)
−0.332021 + 0.943272i \(0.607730\pi\)
\(48\) 0 0
\(49\) −253.405 −0.738789
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −283.310 −0.734257 −0.367128 0.930170i \(-0.619659\pi\)
−0.367128 + 0.930170i \(0.619659\pi\)
\(54\) 0 0
\(55\) 245.237 0.601232
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −92.0000 −0.203006 −0.101503 0.994835i \(-0.532365\pi\)
−0.101503 + 0.994835i \(0.532365\pi\)
\(60\) 0 0
\(61\) 370.172 0.776978 0.388489 0.921453i \(-0.372997\pi\)
0.388489 + 0.921453i \(0.372997\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 277.219 0.528997
\(66\) 0 0
\(67\) 1087.53 1.98302 0.991510 0.130029i \(-0.0415071\pi\)
0.991510 + 0.130029i \(0.0415071\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 333.681 0.557755 0.278877 0.960327i \(-0.410038\pi\)
0.278877 + 0.960327i \(0.410038\pi\)
\(72\) 0 0
\(73\) −606.791 −0.972871 −0.486435 0.873717i \(-0.661703\pi\)
−0.486435 + 0.873717i \(0.661703\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 464.257 0.687104
\(78\) 0 0
\(79\) −44.3751 −0.0631973 −0.0315986 0.999501i \(-0.510060\pi\)
−0.0315986 + 0.999501i \(0.510060\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −107.732 −0.142471 −0.0712355 0.997460i \(-0.522694\pi\)
−0.0712355 + 0.997460i \(0.522694\pi\)
\(84\) 0 0
\(85\) −139.892 −0.178510
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 257.819 0.307065 0.153532 0.988144i \(-0.450935\pi\)
0.153532 + 0.988144i \(0.450935\pi\)
\(90\) 0 0
\(91\) 524.803 0.604553
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 131.982 0.142538
\(96\) 0 0
\(97\) 1312.05 1.37339 0.686693 0.726947i \(-0.259062\pi\)
0.686693 + 0.726947i \(0.259062\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −63.3293 −0.0623911 −0.0311955 0.999513i \(-0.509931\pi\)
−0.0311955 + 0.999513i \(0.509931\pi\)
\(102\) 0 0
\(103\) 1219.85 1.16695 0.583474 0.812132i \(-0.301693\pi\)
0.583474 + 0.812132i \(0.301693\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −42.3947 −0.0383033 −0.0191517 0.999817i \(-0.506097\pi\)
−0.0191517 + 0.999817i \(0.506097\pi\)
\(108\) 0 0
\(109\) 266.741 0.234396 0.117198 0.993109i \(-0.462609\pi\)
0.117198 + 0.993109i \(0.462609\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1871.60 −1.55810 −0.779049 0.626964i \(-0.784297\pi\)
−0.779049 + 0.626964i \(0.784297\pi\)
\(114\) 0 0
\(115\) 47.6726 0.0386564
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −264.829 −0.204007
\(120\) 0 0
\(121\) 1074.64 0.807397
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −1232.11 −0.860881 −0.430441 0.902619i \(-0.641642\pi\)
−0.430441 + 0.902619i \(0.641642\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1209.17 0.806458 0.403229 0.915099i \(-0.367888\pi\)
0.403229 + 0.915099i \(0.367888\pi\)
\(132\) 0 0
\(133\) 249.855 0.162896
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2629.66 −1.63990 −0.819952 0.572432i \(-0.806000\pi\)
−0.819952 + 0.572432i \(0.806000\pi\)
\(138\) 0 0
\(139\) 2215.06 1.35165 0.675823 0.737064i \(-0.263788\pi\)
0.675823 + 0.737064i \(0.263788\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2719.37 1.59025
\(144\) 0 0
\(145\) 1091.89 0.625358
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2446.94 −1.34538 −0.672689 0.739925i \(-0.734861\pi\)
−0.672689 + 0.739925i \(0.734861\pi\)
\(150\) 0 0
\(151\) 119.365 0.0643298 0.0321649 0.999483i \(-0.489760\pi\)
0.0321649 + 0.999483i \(0.489760\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −794.804 −0.411872
\(156\) 0 0
\(157\) 2212.86 1.12487 0.562437 0.826840i \(-0.309864\pi\)
0.562437 + 0.826840i \(0.309864\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 90.2488 0.0441777
\(162\) 0 0
\(163\) 3480.54 1.67250 0.836249 0.548349i \(-0.184744\pi\)
0.836249 + 0.548349i \(0.184744\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2477.15 1.14783 0.573916 0.818914i \(-0.305424\pi\)
0.573916 + 0.818914i \(0.305424\pi\)
\(168\) 0 0
\(169\) 877.018 0.399189
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −26.7446 −0.0117535 −0.00587676 0.999983i \(-0.501871\pi\)
−0.00587676 + 0.999983i \(0.501871\pi\)
\(174\) 0 0
\(175\) 236.637 0.102218
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1400.75 −0.584901 −0.292451 0.956281i \(-0.594471\pi\)
−0.292451 + 0.956281i \(0.594471\pi\)
\(180\) 0 0
\(181\) 3492.48 1.43422 0.717111 0.696959i \(-0.245464\pi\)
0.717111 + 0.696959i \(0.245464\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 948.275 0.376857
\(186\) 0 0
\(187\) −1372.26 −0.536631
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2728.16 1.03352 0.516762 0.856129i \(-0.327137\pi\)
0.516762 + 0.856129i \(0.327137\pi\)
\(192\) 0 0
\(193\) 2439.82 0.909959 0.454980 0.890502i \(-0.349647\pi\)
0.454980 + 0.890502i \(0.349647\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 765.906 0.276998 0.138499 0.990363i \(-0.455772\pi\)
0.138499 + 0.990363i \(0.455772\pi\)
\(198\) 0 0
\(199\) 754.031 0.268602 0.134301 0.990941i \(-0.457121\pi\)
0.134301 + 0.990941i \(0.457121\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2067.06 0.714676
\(204\) 0 0
\(205\) −1230.73 −0.419307
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1294.68 0.428491
\(210\) 0 0
\(211\) −1024.68 −0.334320 −0.167160 0.985930i \(-0.553460\pi\)
−0.167160 + 0.985930i \(0.553460\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 201.163 0.0638104
\(216\) 0 0
\(217\) −1504.64 −0.470699
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1551.23 −0.472157
\(222\) 0 0
\(223\) −1938.61 −0.582147 −0.291074 0.956701i \(-0.594012\pi\)
−0.291074 + 0.956701i \(0.594012\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5876.07 1.71810 0.859050 0.511892i \(-0.171055\pi\)
0.859050 + 0.511892i \(0.171055\pi\)
\(228\) 0 0
\(229\) 5061.06 1.46046 0.730228 0.683204i \(-0.239414\pi\)
0.730228 + 0.683204i \(0.239414\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1241.58 −0.349093 −0.174547 0.984649i \(-0.555846\pi\)
−0.174547 + 0.984649i \(0.555846\pi\)
\(234\) 0 0
\(235\) −1069.82 −0.296968
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3141.68 −0.850286 −0.425143 0.905126i \(-0.639776\pi\)
−0.425143 + 0.905126i \(0.639776\pi\)
\(240\) 0 0
\(241\) −2682.15 −0.716898 −0.358449 0.933549i \(-0.616694\pi\)
−0.358449 + 0.933549i \(0.616694\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1267.02 −0.330396
\(246\) 0 0
\(247\) 1463.52 0.377010
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4159.18 −1.04592 −0.522959 0.852358i \(-0.675172\pi\)
−0.522959 + 0.852358i \(0.675172\pi\)
\(252\) 0 0
\(253\) 467.643 0.116207
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2820.31 0.684537 0.342268 0.939602i \(-0.388805\pi\)
0.342268 + 0.939602i \(0.388805\pi\)
\(258\) 0 0
\(259\) 1795.18 0.430683
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −27.1884 −0.00637457 −0.00318728 0.999995i \(-0.501015\pi\)
−0.00318728 + 0.999995i \(0.501015\pi\)
\(264\) 0 0
\(265\) −1416.55 −0.328370
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −4955.54 −1.12321 −0.561607 0.827404i \(-0.689817\pi\)
−0.561607 + 0.827404i \(0.689817\pi\)
\(270\) 0 0
\(271\) 6619.08 1.48369 0.741846 0.670570i \(-0.233950\pi\)
0.741846 + 0.670570i \(0.233950\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1226.18 0.268879
\(276\) 0 0
\(277\) −2999.18 −0.650554 −0.325277 0.945619i \(-0.605457\pi\)
−0.325277 + 0.945619i \(0.605457\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3387.06 −0.719058 −0.359529 0.933134i \(-0.617063\pi\)
−0.359529 + 0.933134i \(0.617063\pi\)
\(282\) 0 0
\(283\) 79.4525 0.0166889 0.00834446 0.999965i \(-0.497344\pi\)
0.00834446 + 0.999965i \(0.497344\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2329.89 −0.479196
\(288\) 0 0
\(289\) −4130.21 −0.840670
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −6045.37 −1.20537 −0.602686 0.797978i \(-0.705903\pi\)
−0.602686 + 0.797978i \(0.705903\pi\)
\(294\) 0 0
\(295\) −460.000 −0.0907872
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 528.630 0.102246
\(300\) 0 0
\(301\) 380.822 0.0729243
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1850.86 0.347475
\(306\) 0 0
\(307\) −1529.87 −0.284412 −0.142206 0.989837i \(-0.545420\pi\)
−0.142206 + 0.989837i \(0.545420\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6583.78 1.20042 0.600212 0.799841i \(-0.295083\pi\)
0.600212 + 0.799841i \(0.295083\pi\)
\(312\) 0 0
\(313\) 110.352 0.0199281 0.00996403 0.999950i \(-0.496828\pi\)
0.00996403 + 0.999950i \(0.496828\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4003.48 −0.709331 −0.354666 0.934993i \(-0.615405\pi\)
−0.354666 + 0.934993i \(0.615405\pi\)
\(318\) 0 0
\(319\) 10710.9 1.87992
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −738.529 −0.127222
\(324\) 0 0
\(325\) 1386.10 0.236575
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2025.28 −0.339384
\(330\) 0 0
\(331\) −3066.52 −0.509219 −0.254609 0.967044i \(-0.581947\pi\)
−0.254609 + 0.967044i \(0.581947\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5437.63 0.886834
\(336\) 0 0
\(337\) 10714.9 1.73199 0.865993 0.500057i \(-0.166687\pi\)
0.865993 + 0.500057i \(0.166687\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −7796.61 −1.23815
\(342\) 0 0
\(343\) −5645.26 −0.888674
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9008.73 1.39370 0.696850 0.717217i \(-0.254584\pi\)
0.696850 + 0.717217i \(0.254584\pi\)
\(348\) 0 0
\(349\) −9529.14 −1.46156 −0.730778 0.682615i \(-0.760843\pi\)
−0.730778 + 0.682615i \(0.760843\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −982.433 −0.148129 −0.0740646 0.997253i \(-0.523597\pi\)
−0.0740646 + 0.997253i \(0.523597\pi\)
\(354\) 0 0
\(355\) 1668.40 0.249436
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 9887.90 1.45366 0.726829 0.686818i \(-0.240993\pi\)
0.726829 + 0.686818i \(0.240993\pi\)
\(360\) 0 0
\(361\) −6162.23 −0.898415
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3033.96 −0.435081
\(366\) 0 0
\(367\) −3348.54 −0.476274 −0.238137 0.971232i \(-0.576537\pi\)
−0.238137 + 0.971232i \(0.576537\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2681.67 −0.375270
\(372\) 0 0
\(373\) 9572.04 1.32874 0.664372 0.747402i \(-0.268699\pi\)
0.664372 + 0.747402i \(0.268699\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12107.8 1.65406
\(378\) 0 0
\(379\) −9976.30 −1.35211 −0.676053 0.736853i \(-0.736311\pi\)
−0.676053 + 0.736853i \(0.736311\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −7619.84 −1.01659 −0.508297 0.861182i \(-0.669725\pi\)
−0.508297 + 0.861182i \(0.669725\pi\)
\(384\) 0 0
\(385\) 2321.29 0.307282
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −7802.78 −1.01701 −0.508504 0.861059i \(-0.669801\pi\)
−0.508504 + 0.861059i \(0.669801\pi\)
\(390\) 0 0
\(391\) −266.760 −0.0345029
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −221.875 −0.0282627
\(396\) 0 0
\(397\) −8073.51 −1.02065 −0.510325 0.859982i \(-0.670475\pi\)
−0.510325 + 0.859982i \(0.670475\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6306.80 −0.785403 −0.392702 0.919666i \(-0.628459\pi\)
−0.392702 + 0.919666i \(0.628459\pi\)
\(402\) 0 0
\(403\) −8813.39 −1.08940
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9302.08 1.13289
\(408\) 0 0
\(409\) −15362.3 −1.85726 −0.928629 0.371010i \(-0.879012\pi\)
−0.928629 + 0.371010i \(0.879012\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −870.825 −0.103754
\(414\) 0 0
\(415\) −538.659 −0.0637150
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 13791.3 1.60799 0.803994 0.594637i \(-0.202704\pi\)
0.803994 + 0.594637i \(0.202704\pi\)
\(420\) 0 0
\(421\) −9093.23 −1.05268 −0.526339 0.850275i \(-0.676436\pi\)
−0.526339 + 0.850275i \(0.676436\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −699.459 −0.0798323
\(426\) 0 0
\(427\) 3503.86 0.397104
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 400.347 0.0447425 0.0223713 0.999750i \(-0.492878\pi\)
0.0223713 + 0.999750i \(0.492878\pi\)
\(432\) 0 0
\(433\) −10063.9 −1.11695 −0.558474 0.829522i \(-0.688613\pi\)
−0.558474 + 0.829522i \(0.688613\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 251.677 0.0275500
\(438\) 0 0
\(439\) −11684.8 −1.27035 −0.635176 0.772368i \(-0.719072\pi\)
−0.635176 + 0.772368i \(0.719072\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3698.68 −0.396681 −0.198341 0.980133i \(-0.563555\pi\)
−0.198341 + 0.980133i \(0.563555\pi\)
\(444\) 0 0
\(445\) 1289.09 0.137323
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −11318.1 −1.18961 −0.594804 0.803871i \(-0.702770\pi\)
−0.594804 + 0.803871i \(0.702770\pi\)
\(450\) 0 0
\(451\) −12072.8 −1.26050
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2624.01 0.270364
\(456\) 0 0
\(457\) −2436.98 −0.249446 −0.124723 0.992192i \(-0.539804\pi\)
−0.124723 + 0.992192i \(0.539804\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8575.54 0.866384 0.433192 0.901302i \(-0.357387\pi\)
0.433192 + 0.901302i \(0.357387\pi\)
\(462\) 0 0
\(463\) −4711.30 −0.472900 −0.236450 0.971644i \(-0.575984\pi\)
−0.236450 + 0.971644i \(0.575984\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11075.4 1.09745 0.548726 0.836002i \(-0.315113\pi\)
0.548726 + 0.836002i \(0.315113\pi\)
\(468\) 0 0
\(469\) 10294.0 1.01350
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1973.31 0.191824
\(474\) 0 0
\(475\) 659.911 0.0637449
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −17455.1 −1.66502 −0.832510 0.554010i \(-0.813097\pi\)
−0.832510 + 0.554010i \(0.813097\pi\)
\(480\) 0 0
\(481\) 10515.2 0.996781
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6560.25 0.614197
\(486\) 0 0
\(487\) 8285.77 0.770974 0.385487 0.922713i \(-0.374034\pi\)
0.385487 + 0.922713i \(0.374034\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −7055.06 −0.648453 −0.324226 0.945980i \(-0.605104\pi\)
−0.324226 + 0.945980i \(0.605104\pi\)
\(492\) 0 0
\(493\) −6109.88 −0.558165
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3158.45 0.285062
\(498\) 0 0
\(499\) 2778.49 0.249263 0.124632 0.992203i \(-0.460225\pi\)
0.124632 + 0.992203i \(0.460225\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −5445.52 −0.482711 −0.241356 0.970437i \(-0.577592\pi\)
−0.241356 + 0.970437i \(0.577592\pi\)
\(504\) 0 0
\(505\) −316.646 −0.0279021
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −16567.4 −1.44270 −0.721352 0.692569i \(-0.756479\pi\)
−0.721352 + 0.692569i \(0.756479\pi\)
\(510\) 0 0
\(511\) −5743.58 −0.497223
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6099.26 0.521875
\(516\) 0 0
\(517\) −10494.4 −0.892734
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3805.03 0.319964 0.159982 0.987120i \(-0.448856\pi\)
0.159982 + 0.987120i \(0.448856\pi\)
\(522\) 0 0
\(523\) −7489.80 −0.626207 −0.313104 0.949719i \(-0.601369\pi\)
−0.313104 + 0.949719i \(0.601369\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4447.46 0.367617
\(528\) 0 0
\(529\) −12076.1 −0.992528
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −13647.3 −1.10906
\(534\) 0 0
\(535\) −211.974 −0.0171298
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −12428.8 −0.993224
\(540\) 0 0
\(541\) −2063.73 −0.164005 −0.0820026 0.996632i \(-0.526132\pi\)
−0.0820026 + 0.996632i \(0.526132\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1333.71 0.104825
\(546\) 0 0
\(547\) −12954.6 −1.01261 −0.506307 0.862353i \(-0.668990\pi\)
−0.506307 + 0.862353i \(0.668990\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5764.43 0.445686
\(552\) 0 0
\(553\) −420.032 −0.0322994
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14340.8 1.09091 0.545457 0.838139i \(-0.316356\pi\)
0.545457 + 0.838139i \(0.316356\pi\)
\(558\) 0 0
\(559\) 2230.65 0.168777
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12827.3 0.960222 0.480111 0.877208i \(-0.340596\pi\)
0.480111 + 0.877208i \(0.340596\pi\)
\(564\) 0 0
\(565\) −9357.98 −0.696802
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −15102.9 −1.11274 −0.556370 0.830935i \(-0.687806\pi\)
−0.556370 + 0.830935i \(0.687806\pi\)
\(570\) 0 0
\(571\) 22553.6 1.65295 0.826477 0.562970i \(-0.190342\pi\)
0.826477 + 0.562970i \(0.190342\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 238.363 0.0172877
\(576\) 0 0
\(577\) 5007.38 0.361283 0.180641 0.983549i \(-0.442183\pi\)
0.180641 + 0.983549i \(0.442183\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1019.73 −0.0728153
\(582\) 0 0
\(583\) −13895.6 −0.987131
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 24975.4 1.75612 0.878062 0.478547i \(-0.158837\pi\)
0.878062 + 0.478547i \(0.158837\pi\)
\(588\) 0 0
\(589\) −4196.00 −0.293537
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 19247.2 1.33287 0.666433 0.745565i \(-0.267820\pi\)
0.666433 + 0.745565i \(0.267820\pi\)
\(594\) 0 0
\(595\) −1324.14 −0.0912346
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6000.84 −0.409328 −0.204664 0.978832i \(-0.565610\pi\)
−0.204664 + 0.978832i \(0.565610\pi\)
\(600\) 0 0
\(601\) −8576.08 −0.582073 −0.291036 0.956712i \(-0.594000\pi\)
−0.291036 + 0.956712i \(0.594000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5373.22 0.361079
\(606\) 0 0
\(607\) −2960.22 −0.197943 −0.0989715 0.995090i \(-0.531555\pi\)
−0.0989715 + 0.995090i \(0.531555\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −11863.0 −0.785477
\(612\) 0 0
\(613\) 6483.88 0.427212 0.213606 0.976920i \(-0.431479\pi\)
0.213606 + 0.976920i \(0.431479\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1169.84 0.0763304 0.0381652 0.999271i \(-0.487849\pi\)
0.0381652 + 0.999271i \(0.487849\pi\)
\(618\) 0 0
\(619\) 2930.97 0.190316 0.0951580 0.995462i \(-0.469664\pi\)
0.0951580 + 0.995462i \(0.469664\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2440.38 0.156937
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −5306.23 −0.336365
\(630\) 0 0
\(631\) −29359.9 −1.85230 −0.926150 0.377156i \(-0.876902\pi\)
−0.926150 + 0.377156i \(0.876902\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6160.54 −0.384998
\(636\) 0 0
\(637\) −14049.7 −0.873894
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 14661.8 0.903443 0.451721 0.892159i \(-0.350810\pi\)
0.451721 + 0.892159i \(0.350810\pi\)
\(642\) 0 0
\(643\) 11141.3 0.683312 0.341656 0.939825i \(-0.389012\pi\)
0.341656 + 0.939825i \(0.389012\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −17044.5 −1.03569 −0.517844 0.855475i \(-0.673265\pi\)
−0.517844 + 0.855475i \(0.673265\pi\)
\(648\) 0 0
\(649\) −4512.36 −0.272921
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −19960.2 −1.19618 −0.598089 0.801430i \(-0.704073\pi\)
−0.598089 + 0.801430i \(0.704073\pi\)
\(654\) 0 0
\(655\) 6045.87 0.360659
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −13422.1 −0.793399 −0.396700 0.917948i \(-0.629845\pi\)
−0.396700 + 0.917948i \(0.629845\pi\)
\(660\) 0 0
\(661\) 13349.5 0.785531 0.392765 0.919639i \(-0.371518\pi\)
0.392765 + 0.919639i \(0.371518\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1249.28 0.0728494
\(666\) 0 0
\(667\) 2082.14 0.120871
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 18156.0 1.04456
\(672\) 0 0
\(673\) 6927.58 0.396788 0.198394 0.980122i \(-0.436427\pi\)
0.198394 + 0.980122i \(0.436427\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6607.86 0.375126 0.187563 0.982253i \(-0.439941\pi\)
0.187563 + 0.982253i \(0.439941\pi\)
\(678\) 0 0
\(679\) 12419.2 0.701922
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −15422.7 −0.864031 −0.432015 0.901866i \(-0.642197\pi\)
−0.432015 + 0.901866i \(0.642197\pi\)
\(684\) 0 0
\(685\) −13148.3 −0.733388
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −15707.8 −0.868533
\(690\) 0 0
\(691\) −16254.6 −0.894870 −0.447435 0.894317i \(-0.647662\pi\)
−0.447435 + 0.894317i \(0.647662\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 11075.3 0.604475
\(696\) 0 0
\(697\) 6886.76 0.374254
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −10389.8 −0.559796 −0.279898 0.960030i \(-0.590301\pi\)
−0.279898 + 0.960030i \(0.590301\pi\)
\(702\) 0 0
\(703\) 5006.22 0.268582
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −599.442 −0.0318873
\(708\) 0 0
\(709\) 7530.22 0.398877 0.199438 0.979910i \(-0.436088\pi\)
0.199438 + 0.979910i \(0.436088\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1515.61 −0.0796075
\(714\) 0 0
\(715\) 13596.9 0.711181
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −33888.4 −1.75775 −0.878875 0.477051i \(-0.841706\pi\)
−0.878875 + 0.477051i \(0.841706\pi\)
\(720\) 0 0
\(721\) 11546.5 0.596413
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5459.47 0.279669
\(726\) 0 0
\(727\) −13863.3 −0.707239 −0.353619 0.935389i \(-0.615049\pi\)
−0.353619 + 0.935389i \(0.615049\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1125.64 −0.0569541
\(732\) 0 0
\(733\) 24497.7 1.23444 0.617219 0.786792i \(-0.288259\pi\)
0.617219 + 0.786792i \(0.288259\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 53340.3 2.66596
\(738\) 0 0
\(739\) 19029.1 0.947221 0.473611 0.880734i \(-0.342950\pi\)
0.473611 + 0.880734i \(0.342950\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5639.67 −0.278465 −0.139233 0.990260i \(-0.544464\pi\)
−0.139233 + 0.990260i \(0.544464\pi\)
\(744\) 0 0
\(745\) −12234.7 −0.601672
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −401.287 −0.0195764
\(750\) 0 0
\(751\) −14566.0 −0.707753 −0.353876 0.935292i \(-0.615137\pi\)
−0.353876 + 0.935292i \(0.615137\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 596.825 0.0287691
\(756\) 0 0
\(757\) −2958.83 −0.142061 −0.0710307 0.997474i \(-0.522629\pi\)
−0.0710307 + 0.997474i \(0.522629\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −281.640 −0.0134158 −0.00670791 0.999978i \(-0.502135\pi\)
−0.00670791 + 0.999978i \(0.502135\pi\)
\(762\) 0 0
\(763\) 2524.84 0.119797
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5100.83 −0.240131
\(768\) 0 0
\(769\) 29799.2 1.39738 0.698690 0.715425i \(-0.253767\pi\)
0.698690 + 0.715425i \(0.253767\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −8377.18 −0.389788 −0.194894 0.980824i \(-0.562436\pi\)
−0.194894 + 0.980824i \(0.562436\pi\)
\(774\) 0 0
\(775\) −3974.02 −0.184195
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6497.39 −0.298836
\(780\) 0 0
\(781\) 16366.2 0.749843
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 11064.3 0.503059
\(786\) 0 0
\(787\) −17368.8 −0.786697 −0.393348 0.919389i \(-0.628683\pi\)
−0.393348 + 0.919389i \(0.628683\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −17715.6 −0.796325
\(792\) 0 0
\(793\) 20523.7 0.919066
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6003.86 0.266835 0.133418 0.991060i \(-0.457405\pi\)
0.133418 + 0.991060i \(0.457405\pi\)
\(798\) 0 0
\(799\) 5986.38 0.265060
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −29761.5 −1.30792
\(804\) 0 0
\(805\) 451.244 0.0197568
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 42392.2 1.84231 0.921157 0.389191i \(-0.127246\pi\)
0.921157 + 0.389191i \(0.127246\pi\)
\(810\) 0 0
\(811\) −7148.82 −0.309530 −0.154765 0.987951i \(-0.549462\pi\)
−0.154765 + 0.987951i \(0.549462\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 17402.7 0.747964
\(816\) 0 0
\(817\) 1062.00 0.0454770
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −25700.4 −1.09251 −0.546254 0.837619i \(-0.683947\pi\)
−0.546254 + 0.837619i \(0.683947\pi\)
\(822\) 0 0
\(823\) −19064.7 −0.807476 −0.403738 0.914875i \(-0.632289\pi\)
−0.403738 + 0.914875i \(0.632289\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 41947.6 1.76380 0.881900 0.471437i \(-0.156264\pi\)
0.881900 + 0.471437i \(0.156264\pi\)
\(828\) 0 0
\(829\) −31403.8 −1.31568 −0.657841 0.753157i \(-0.728530\pi\)
−0.657841 + 0.753157i \(0.728530\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 7089.84 0.294896
\(834\) 0 0
\(835\) 12385.8 0.513326
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −25495.3 −1.04910 −0.524550 0.851380i \(-0.675766\pi\)
−0.524550 + 0.851380i \(0.675766\pi\)
\(840\) 0 0
\(841\) 23300.3 0.955362
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4385.09 0.178523
\(846\) 0 0
\(847\) 10172.0 0.412651
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1808.27 0.0728398
\(852\) 0 0
\(853\) −13537.1 −0.543377 −0.271689 0.962385i \(-0.587582\pi\)
−0.271689 + 0.962385i \(0.587582\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 15337.3 0.611332 0.305666 0.952139i \(-0.401121\pi\)
0.305666 + 0.952139i \(0.401121\pi\)
\(858\) 0 0
\(859\) 19720.1 0.783286 0.391643 0.920117i \(-0.371907\pi\)
0.391643 + 0.920117i \(0.371907\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −21464.2 −0.846641 −0.423321 0.905980i \(-0.639136\pi\)
−0.423321 + 0.905980i \(0.639136\pi\)
\(864\) 0 0
\(865\) −133.723 −0.00525633
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2176.48 −0.0849621
\(870\) 0 0
\(871\) 60296.6 2.34566
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1183.19 0.0457131
\(876\) 0 0
\(877\) 3671.90 0.141381 0.0706905 0.997498i \(-0.477480\pi\)
0.0706905 + 0.997498i \(0.477480\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 48518.9 1.85544 0.927720 0.373276i \(-0.121766\pi\)
0.927720 + 0.373276i \(0.121766\pi\)
\(882\) 0 0
\(883\) 13413.5 0.511210 0.255605 0.966781i \(-0.417725\pi\)
0.255605 + 0.966781i \(0.417725\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 50035.0 1.89404 0.947018 0.321180i \(-0.104079\pi\)
0.947018 + 0.321180i \(0.104079\pi\)
\(888\) 0 0
\(889\) −11662.5 −0.439986
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −5647.91 −0.211646
\(894\) 0 0
\(895\) −7003.77 −0.261576
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −34713.7 −1.28784
\(900\) 0 0
\(901\) 7926.54 0.293087
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 17462.4 0.641403
\(906\) 0 0
\(907\) 35905.1 1.31446 0.657228 0.753692i \(-0.271729\pi\)
0.657228 + 0.753692i \(0.271729\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 36938.6 1.34339 0.671697 0.740826i \(-0.265566\pi\)
0.671697 + 0.740826i \(0.265566\pi\)
\(912\) 0 0
\(913\) −5283.96 −0.191537
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 11445.4 0.412171
\(918\) 0 0
\(919\) 35867.0 1.28742 0.643712 0.765268i \(-0.277394\pi\)
0.643712 + 0.765268i \(0.277394\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 18500.5 0.659753
\(924\) 0 0
\(925\) 4741.37 0.168536
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 11391.2 0.402297 0.201148 0.979561i \(-0.435533\pi\)
0.201148 + 0.979561i \(0.435533\pi\)
\(930\) 0 0
\(931\) −6688.98 −0.235470
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −6861.32 −0.239988
\(936\) 0 0
\(937\) 30928.7 1.07833 0.539166 0.842199i \(-0.318740\pi\)
0.539166 + 0.842199i \(0.318740\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −36684.3 −1.27086 −0.635428 0.772160i \(-0.719176\pi\)
−0.635428 + 0.772160i \(0.719176\pi\)
\(942\) 0 0
\(943\) −2346.88 −0.0810446
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −39211.7 −1.34552 −0.672761 0.739860i \(-0.734892\pi\)
−0.672761 + 0.739860i \(0.734892\pi\)
\(948\) 0 0
\(949\) −33642.8 −1.15078
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −30476.1 −1.03591 −0.517953 0.855409i \(-0.673306\pi\)
−0.517953 + 0.855409i \(0.673306\pi\)
\(954\) 0 0
\(955\) 13640.8 0.462206
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −24891.0 −0.838136
\(960\) 0 0
\(961\) −4522.48 −0.151807
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 12199.1 0.406946
\(966\) 0 0
\(967\) 48866.8 1.62508 0.812538 0.582908i \(-0.198085\pi\)
0.812538 + 0.582908i \(0.198085\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −54060.7 −1.78670 −0.893352 0.449358i \(-0.851653\pi\)
−0.893352 + 0.449358i \(0.851653\pi\)
\(972\) 0 0
\(973\) 20966.6 0.690811
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 27233.7 0.891795 0.445898 0.895084i \(-0.352885\pi\)
0.445898 + 0.895084i \(0.352885\pi\)
\(978\) 0 0
\(979\) 12645.3 0.412816
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −949.700 −0.0308146 −0.0154073 0.999881i \(-0.504904\pi\)
−0.0154073 + 0.999881i \(0.504904\pi\)
\(984\) 0 0
\(985\) 3829.53 0.123877
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 383.599 0.0123334
\(990\) 0 0
\(991\) 37146.4 1.19071 0.595356 0.803462i \(-0.297011\pi\)
0.595356 + 0.803462i \(0.297011\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3770.15 0.120123
\(996\) 0 0
\(997\) 45360.4 1.44090 0.720451 0.693506i \(-0.243935\pi\)
0.720451 + 0.693506i \(0.243935\pi\)
\(998\) 0 0
\(999\) 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 1080.4.a.n.1.2 yes 3
3.2 odd 2 1080.4.a.h.1.2 3
4.3 odd 2 2160.4.a.bn.1.2 3
12.11 even 2 2160.4.a.bf.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.h.1.2 3 3.2 odd 2
1080.4.a.n.1.2 yes 3 1.1 even 1 trivial
2160.4.a.bf.1.2 3 12.11 even 2
2160.4.a.bn.1.2 3 4.3 odd 2