Properties

Label 1080.4.a.p.1.4
Level $1080$
Weight $4$
Character 1080.1
Self dual yes
Analytic conductor $63.722$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - x^{3} - 141x^{2} + 200x + 3500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-10.9094\) of defining polynomial
Character \(\chi\) \(=\) 1080.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.00000 q^{5} +35.5942 q^{7} +O(q^{10})\) \(q+5.00000 q^{5} +35.5942 q^{7} -44.7641 q^{11} +77.8220 q^{13} -120.489 q^{17} +121.319 q^{19} +152.855 q^{23} +25.0000 q^{25} +187.926 q^{29} -161.365 q^{31} +177.971 q^{35} +13.3350 q^{37} +188.065 q^{41} -81.5584 q^{43} +48.6219 q^{47} +923.946 q^{49} -707.373 q^{53} -223.821 q^{55} +16.4774 q^{59} -743.868 q^{61} +389.110 q^{65} -59.2676 q^{67} +144.370 q^{71} +657.273 q^{73} -1593.34 q^{77} -454.297 q^{79} -165.513 q^{83} -602.444 q^{85} +535.743 q^{89} +2770.01 q^{91} +606.594 q^{95} -436.761 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 20 q^{5} + 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 20 q^{5} + 14 q^{7} - 4 q^{11} + 30 q^{13} - 28 q^{17} + 78 q^{19} + 182 q^{23} + 100 q^{25} + 202 q^{29} - 76 q^{31} + 70 q^{35} + 302 q^{37} + 380 q^{41} + 178 q^{43} + 114 q^{47} + 958 q^{49} - 256 q^{53} - 20 q^{55} - 204 q^{59} + 766 q^{61} + 150 q^{65} + 330 q^{67} - 1060 q^{71} + 1442 q^{73} + 216 q^{77} + 742 q^{79} - 768 q^{83} - 140 q^{85} - 400 q^{89} + 3066 q^{91} + 390 q^{95} + 3338 q^{97}+O(q^{100}) \) 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
\(7\) 35.5942 1.92191 0.960953 0.276713i \(-0.0892452\pi\)
0.960953 + 0.276713i \(0.0892452\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −44.7641 −1.22699 −0.613495 0.789699i \(-0.710237\pi\)
−0.613495 + 0.789699i \(0.710237\pi\)
\(12\) 0 0
\(13\) 77.8220 1.66030 0.830152 0.557538i \(-0.188254\pi\)
0.830152 + 0.557538i \(0.188254\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −120.489 −1.71899 −0.859495 0.511145i \(-0.829222\pi\)
−0.859495 + 0.511145i \(0.829222\pi\)
\(18\) 0 0
\(19\) 121.319 1.46487 0.732433 0.680839i \(-0.238385\pi\)
0.732433 + 0.680839i \(0.238385\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 152.855 1.38576 0.692880 0.721053i \(-0.256341\pi\)
0.692880 + 0.721053i \(0.256341\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 187.926 1.20334 0.601672 0.798743i \(-0.294501\pi\)
0.601672 + 0.798743i \(0.294501\pi\)
\(30\) 0 0
\(31\) −161.365 −0.934903 −0.467452 0.884019i \(-0.654828\pi\)
−0.467452 + 0.884019i \(0.654828\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 177.971 0.859502
\(36\) 0 0
\(37\) 13.3350 0.0592501 0.0296251 0.999561i \(-0.490569\pi\)
0.0296251 + 0.999561i \(0.490569\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 188.065 0.716360 0.358180 0.933653i \(-0.383397\pi\)
0.358180 + 0.933653i \(0.383397\pi\)
\(42\) 0 0
\(43\) −81.5584 −0.289245 −0.144623 0.989487i \(-0.546197\pi\)
−0.144623 + 0.989487i \(0.546197\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 48.6219 0.150899 0.0754493 0.997150i \(-0.475961\pi\)
0.0754493 + 0.997150i \(0.475961\pi\)
\(48\) 0 0
\(49\) 923.946 2.69372
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −707.373 −1.83331 −0.916653 0.399685i \(-0.869120\pi\)
−0.916653 + 0.399685i \(0.869120\pi\)
\(54\) 0 0
\(55\) −223.821 −0.548727
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 16.4774 0.0363589 0.0181795 0.999835i \(-0.494213\pi\)
0.0181795 + 0.999835i \(0.494213\pi\)
\(60\) 0 0
\(61\) −743.868 −1.56135 −0.780676 0.624936i \(-0.785125\pi\)
−0.780676 + 0.624936i \(0.785125\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 389.110 0.742510
\(66\) 0 0
\(67\) −59.2676 −0.108070 −0.0540350 0.998539i \(-0.517208\pi\)
−0.0540350 + 0.998539i \(0.517208\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 144.370 0.241317 0.120659 0.992694i \(-0.461499\pi\)
0.120659 + 0.992694i \(0.461499\pi\)
\(72\) 0 0
\(73\) 657.273 1.05381 0.526904 0.849925i \(-0.323353\pi\)
0.526904 + 0.849925i \(0.323353\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1593.34 −2.35816
\(78\) 0 0
\(79\) −454.297 −0.646992 −0.323496 0.946230i \(-0.604858\pi\)
−0.323496 + 0.946230i \(0.604858\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −165.513 −0.218885 −0.109442 0.993993i \(-0.534907\pi\)
−0.109442 + 0.993993i \(0.534907\pi\)
\(84\) 0 0
\(85\) −602.444 −0.768755
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 535.743 0.638074 0.319037 0.947742i \(-0.396641\pi\)
0.319037 + 0.947742i \(0.396641\pi\)
\(90\) 0 0
\(91\) 2770.01 3.19094
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 606.594 0.655108
\(96\) 0 0
\(97\) −436.761 −0.457179 −0.228590 0.973523i \(-0.573411\pi\)
−0.228590 + 0.973523i \(0.573411\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1376.74 1.35635 0.678174 0.734901i \(-0.262771\pi\)
0.678174 + 0.734901i \(0.262771\pi\)
\(102\) 0 0
\(103\) −20.2617 −0.0193830 −0.00969149 0.999953i \(-0.503085\pi\)
−0.00969149 + 0.999953i \(0.503085\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −359.961 −0.325222 −0.162611 0.986690i \(-0.551992\pi\)
−0.162611 + 0.986690i \(0.551992\pi\)
\(108\) 0 0
\(109\) 377.931 0.332103 0.166052 0.986117i \(-0.446898\pi\)
0.166052 + 0.986117i \(0.446898\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −21.3449 −0.0177696 −0.00888478 0.999961i \(-0.502828\pi\)
−0.00888478 + 0.999961i \(0.502828\pi\)
\(114\) 0 0
\(115\) 764.276 0.619731
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −4288.70 −3.30373
\(120\) 0 0
\(121\) 672.826 0.505504
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 1756.10 1.22700 0.613499 0.789696i \(-0.289762\pi\)
0.613499 + 0.789696i \(0.289762\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2388.31 1.59288 0.796441 0.604716i \(-0.206713\pi\)
0.796441 + 0.604716i \(0.206713\pi\)
\(132\) 0 0
\(133\) 4318.24 2.81533
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1124.94 0.701534 0.350767 0.936463i \(-0.385921\pi\)
0.350767 + 0.936463i \(0.385921\pi\)
\(138\) 0 0
\(139\) 795.105 0.485179 0.242590 0.970129i \(-0.422003\pi\)
0.242590 + 0.970129i \(0.422003\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3483.63 −2.03718
\(144\) 0 0
\(145\) 939.630 0.538152
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2933.56 1.61293 0.806466 0.591281i \(-0.201378\pi\)
0.806466 + 0.591281i \(0.201378\pi\)
\(150\) 0 0
\(151\) 1645.26 0.886686 0.443343 0.896352i \(-0.353792\pi\)
0.443343 + 0.896352i \(0.353792\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −806.825 −0.418101
\(156\) 0 0
\(157\) −1001.96 −0.509330 −0.254665 0.967029i \(-0.581965\pi\)
−0.254665 + 0.967029i \(0.581965\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5440.75 2.66330
\(162\) 0 0
\(163\) −2135.60 −1.02622 −0.513108 0.858324i \(-0.671506\pi\)
−0.513108 + 0.858324i \(0.671506\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3612.36 −1.67385 −0.836925 0.547318i \(-0.815649\pi\)
−0.836925 + 0.547318i \(0.815649\pi\)
\(168\) 0 0
\(169\) 3859.26 1.75661
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2684.80 1.17989 0.589947 0.807442i \(-0.299149\pi\)
0.589947 + 0.807442i \(0.299149\pi\)
\(174\) 0 0
\(175\) 889.854 0.384381
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2182.63 −0.911382 −0.455691 0.890138i \(-0.650608\pi\)
−0.455691 + 0.890138i \(0.650608\pi\)
\(180\) 0 0
\(181\) 3117.86 1.28038 0.640189 0.768217i \(-0.278856\pi\)
0.640189 + 0.768217i \(0.278856\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 66.6748 0.0264975
\(186\) 0 0
\(187\) 5393.57 2.10918
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 76.0912 0.0288260 0.0144130 0.999896i \(-0.495412\pi\)
0.0144130 + 0.999896i \(0.495412\pi\)
\(192\) 0 0
\(193\) 533.717 0.199056 0.0995280 0.995035i \(-0.468267\pi\)
0.0995280 + 0.995035i \(0.468267\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2729.47 0.987142 0.493571 0.869706i \(-0.335691\pi\)
0.493571 + 0.869706i \(0.335691\pi\)
\(198\) 0 0
\(199\) −3196.27 −1.13858 −0.569290 0.822137i \(-0.692782\pi\)
−0.569290 + 0.822137i \(0.692782\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6689.07 2.31271
\(204\) 0 0
\(205\) 940.323 0.320366
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −5430.73 −1.79738
\(210\) 0 0
\(211\) −4325.59 −1.41131 −0.705654 0.708557i \(-0.749347\pi\)
−0.705654 + 0.708557i \(0.749347\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −407.792 −0.129354
\(216\) 0 0
\(217\) −5743.65 −1.79680
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −9376.68 −2.85404
\(222\) 0 0
\(223\) 1370.87 0.411660 0.205830 0.978588i \(-0.434011\pi\)
0.205830 + 0.978588i \(0.434011\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2676.95 0.782711 0.391356 0.920240i \(-0.372006\pi\)
0.391356 + 0.920240i \(0.372006\pi\)
\(228\) 0 0
\(229\) −2641.40 −0.762222 −0.381111 0.924529i \(-0.624459\pi\)
−0.381111 + 0.924529i \(0.624459\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1719.93 −0.483589 −0.241795 0.970327i \(-0.577736\pi\)
−0.241795 + 0.970327i \(0.577736\pi\)
\(234\) 0 0
\(235\) 243.109 0.0674839
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −5162.21 −1.39714 −0.698568 0.715543i \(-0.746179\pi\)
−0.698568 + 0.715543i \(0.746179\pi\)
\(240\) 0 0
\(241\) −717.935 −0.191893 −0.0959465 0.995386i \(-0.530588\pi\)
−0.0959465 + 0.995386i \(0.530588\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4619.73 1.20467
\(246\) 0 0
\(247\) 9441.27 2.43212
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1101.95 −0.277110 −0.138555 0.990355i \(-0.544246\pi\)
−0.138555 + 0.990355i \(0.544246\pi\)
\(252\) 0 0
\(253\) −6842.43 −1.70031
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2922.73 0.709396 0.354698 0.934981i \(-0.384584\pi\)
0.354698 + 0.934981i \(0.384584\pi\)
\(258\) 0 0
\(259\) 474.647 0.113873
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1380.33 −0.323631 −0.161815 0.986821i \(-0.551735\pi\)
−0.161815 + 0.986821i \(0.551735\pi\)
\(264\) 0 0
\(265\) −3536.87 −0.819879
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −85.9555 −0.0194825 −0.00974126 0.999953i \(-0.503101\pi\)
−0.00974126 + 0.999953i \(0.503101\pi\)
\(270\) 0 0
\(271\) −2676.24 −0.599889 −0.299944 0.953957i \(-0.596968\pi\)
−0.299944 + 0.953957i \(0.596968\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1119.10 −0.245398
\(276\) 0 0
\(277\) 6135.26 1.33080 0.665400 0.746487i \(-0.268261\pi\)
0.665400 + 0.746487i \(0.268261\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2899.29 −0.615505 −0.307753 0.951466i \(-0.599577\pi\)
−0.307753 + 0.951466i \(0.599577\pi\)
\(282\) 0 0
\(283\) −3584.20 −0.752857 −0.376429 0.926446i \(-0.622848\pi\)
−0.376429 + 0.926446i \(0.622848\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6694.00 1.37678
\(288\) 0 0
\(289\) 9604.55 1.95492
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7592.52 1.51386 0.756928 0.653498i \(-0.226699\pi\)
0.756928 + 0.653498i \(0.226699\pi\)
\(294\) 0 0
\(295\) 82.3871 0.0162602
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 11895.5 2.30078
\(300\) 0 0
\(301\) −2903.01 −0.555902
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3719.34 −0.698258
\(306\) 0 0
\(307\) 5539.62 1.02985 0.514923 0.857237i \(-0.327821\pi\)
0.514923 + 0.857237i \(0.327821\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −8862.85 −1.61597 −0.807984 0.589204i \(-0.799441\pi\)
−0.807984 + 0.589204i \(0.799441\pi\)
\(312\) 0 0
\(313\) 6028.49 1.08866 0.544329 0.838872i \(-0.316784\pi\)
0.544329 + 0.838872i \(0.316784\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10578.3 1.87425 0.937126 0.348991i \(-0.113476\pi\)
0.937126 + 0.348991i \(0.113476\pi\)
\(318\) 0 0
\(319\) −8412.34 −1.47649
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −14617.6 −2.51809
\(324\) 0 0
\(325\) 1945.55 0.332061
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1730.66 0.290013
\(330\) 0 0
\(331\) 8051.25 1.33697 0.668484 0.743726i \(-0.266943\pi\)
0.668484 + 0.743726i \(0.266943\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −296.338 −0.0483304
\(336\) 0 0
\(337\) −3573.37 −0.577608 −0.288804 0.957388i \(-0.593258\pi\)
−0.288804 + 0.957388i \(0.593258\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 7223.36 1.14712
\(342\) 0 0
\(343\) 20678.3 3.25517
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3268.28 −0.505621 −0.252811 0.967516i \(-0.581355\pi\)
−0.252811 + 0.967516i \(0.581355\pi\)
\(348\) 0 0
\(349\) 5831.57 0.894432 0.447216 0.894426i \(-0.352415\pi\)
0.447216 + 0.894426i \(0.352415\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12021.4 −1.81257 −0.906285 0.422667i \(-0.861094\pi\)
−0.906285 + 0.422667i \(0.861094\pi\)
\(354\) 0 0
\(355\) 721.849 0.107920
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1598.32 −0.234975 −0.117487 0.993074i \(-0.537484\pi\)
−0.117487 + 0.993074i \(0.537484\pi\)
\(360\) 0 0
\(361\) 7859.26 1.14583
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3286.37 0.471277
\(366\) 0 0
\(367\) −12200.7 −1.73534 −0.867671 0.497138i \(-0.834384\pi\)
−0.867671 + 0.497138i \(0.834384\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −25178.4 −3.52344
\(372\) 0 0
\(373\) −7613.13 −1.05682 −0.528409 0.848990i \(-0.677211\pi\)
−0.528409 + 0.848990i \(0.677211\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 14624.8 1.99792
\(378\) 0 0
\(379\) −7463.85 −1.01159 −0.505794 0.862654i \(-0.668800\pi\)
−0.505794 + 0.862654i \(0.668800\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4064.01 −0.542196 −0.271098 0.962552i \(-0.587387\pi\)
−0.271098 + 0.962552i \(0.587387\pi\)
\(384\) 0 0
\(385\) −7966.71 −1.05460
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1258.30 0.164006 0.0820028 0.996632i \(-0.473868\pi\)
0.0820028 + 0.996632i \(0.473868\pi\)
\(390\) 0 0
\(391\) −18417.3 −2.38211
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2271.48 −0.289344
\(396\) 0 0
\(397\) −8486.66 −1.07288 −0.536440 0.843938i \(-0.680231\pi\)
−0.536440 + 0.843938i \(0.680231\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −829.986 −0.103360 −0.0516802 0.998664i \(-0.516458\pi\)
−0.0516802 + 0.998664i \(0.516458\pi\)
\(402\) 0 0
\(403\) −12557.7 −1.55222
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −596.928 −0.0726993
\(408\) 0 0
\(409\) −8600.00 −1.03971 −0.519857 0.854254i \(-0.674015\pi\)
−0.519857 + 0.854254i \(0.674015\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 586.500 0.0698784
\(414\) 0 0
\(415\) −827.566 −0.0978882
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2926.84 −0.341254 −0.170627 0.985336i \(-0.554579\pi\)
−0.170627 + 0.985336i \(0.554579\pi\)
\(420\) 0 0
\(421\) −707.503 −0.0819041 −0.0409520 0.999161i \(-0.513039\pi\)
−0.0409520 + 0.999161i \(0.513039\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3012.22 −0.343798
\(426\) 0 0
\(427\) −26477.4 −3.00077
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4693.17 0.524506 0.262253 0.964999i \(-0.415535\pi\)
0.262253 + 0.964999i \(0.415535\pi\)
\(432\) 0 0
\(433\) −279.149 −0.0309816 −0.0154908 0.999880i \(-0.504931\pi\)
−0.0154908 + 0.999880i \(0.504931\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 18544.2 2.02995
\(438\) 0 0
\(439\) −2651.13 −0.288226 −0.144113 0.989561i \(-0.546033\pi\)
−0.144113 + 0.989561i \(0.546033\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1899.98 0.203771 0.101886 0.994796i \(-0.467512\pi\)
0.101886 + 0.994796i \(0.467512\pi\)
\(444\) 0 0
\(445\) 2678.71 0.285355
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6033.52 0.634164 0.317082 0.948398i \(-0.397297\pi\)
0.317082 + 0.948398i \(0.397297\pi\)
\(450\) 0 0
\(451\) −8418.54 −0.878966
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 13850.1 1.42703
\(456\) 0 0
\(457\) −7920.25 −0.810708 −0.405354 0.914160i \(-0.632852\pi\)
−0.405354 + 0.914160i \(0.632852\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3754.65 0.379331 0.189666 0.981849i \(-0.439260\pi\)
0.189666 + 0.981849i \(0.439260\pi\)
\(462\) 0 0
\(463\) −3094.28 −0.310591 −0.155295 0.987868i \(-0.549633\pi\)
−0.155295 + 0.987868i \(0.549633\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 13827.8 1.37018 0.685092 0.728457i \(-0.259762\pi\)
0.685092 + 0.728457i \(0.259762\pi\)
\(468\) 0 0
\(469\) −2109.58 −0.207700
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3650.89 0.354901
\(474\) 0 0
\(475\) 3032.97 0.292973
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1776.55 −0.169463 −0.0847313 0.996404i \(-0.527003\pi\)
−0.0847313 + 0.996404i \(0.527003\pi\)
\(480\) 0 0
\(481\) 1037.75 0.0983732
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2183.81 −0.204457
\(486\) 0 0
\(487\) −3223.84 −0.299971 −0.149986 0.988688i \(-0.547923\pi\)
−0.149986 + 0.988688i \(0.547923\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −8699.20 −0.799571 −0.399786 0.916609i \(-0.630915\pi\)
−0.399786 + 0.916609i \(0.630915\pi\)
\(492\) 0 0
\(493\) −22643.0 −2.06854
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5138.72 0.463789
\(498\) 0 0
\(499\) 11833.7 1.06163 0.530813 0.847489i \(-0.321887\pi\)
0.530813 + 0.847489i \(0.321887\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −15882.5 −1.40789 −0.703943 0.710256i \(-0.748579\pi\)
−0.703943 + 0.710256i \(0.748579\pi\)
\(504\) 0 0
\(505\) 6883.72 0.606577
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −20662.8 −1.79934 −0.899670 0.436570i \(-0.856193\pi\)
−0.899670 + 0.436570i \(0.856193\pi\)
\(510\) 0 0
\(511\) 23395.1 2.02532
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −101.309 −0.00866834
\(516\) 0 0
\(517\) −2176.52 −0.185151
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 13333.8 1.12124 0.560618 0.828075i \(-0.310564\pi\)
0.560618 + 0.828075i \(0.310564\pi\)
\(522\) 0 0
\(523\) −8822.73 −0.737650 −0.368825 0.929499i \(-0.620240\pi\)
−0.368825 + 0.929499i \(0.620240\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 19442.7 1.60709
\(528\) 0 0
\(529\) 11197.7 0.920333
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 14635.6 1.18937
\(534\) 0 0
\(535\) −1799.80 −0.145444
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −41359.6 −3.30517
\(540\) 0 0
\(541\) −16975.6 −1.34905 −0.674527 0.738250i \(-0.735652\pi\)
−0.674527 + 0.738250i \(0.735652\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1889.66 0.148521
\(546\) 0 0
\(547\) 9986.07 0.780573 0.390287 0.920693i \(-0.372376\pi\)
0.390287 + 0.920693i \(0.372376\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 22799.0 1.76274
\(552\) 0 0
\(553\) −16170.3 −1.24346
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −15740.3 −1.19737 −0.598686 0.800984i \(-0.704310\pi\)
−0.598686 + 0.800984i \(0.704310\pi\)
\(558\) 0 0
\(559\) −6347.04 −0.480235
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −10829.5 −0.810674 −0.405337 0.914167i \(-0.632846\pi\)
−0.405337 + 0.914167i \(0.632846\pi\)
\(564\) 0 0
\(565\) −106.725 −0.00794679
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −11858.7 −0.873711 −0.436855 0.899532i \(-0.643908\pi\)
−0.436855 + 0.899532i \(0.643908\pi\)
\(570\) 0 0
\(571\) 16839.3 1.23415 0.617077 0.786903i \(-0.288317\pi\)
0.617077 + 0.786903i \(0.288317\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3821.38 0.277152
\(576\) 0 0
\(577\) 2182.04 0.157434 0.0787169 0.996897i \(-0.474918\pi\)
0.0787169 + 0.996897i \(0.474918\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −5891.31 −0.420676
\(582\) 0 0
\(583\) 31664.9 2.24945
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2121.16 0.149148 0.0745739 0.997215i \(-0.476240\pi\)
0.0745739 + 0.997215i \(0.476240\pi\)
\(588\) 0 0
\(589\) −19576.6 −1.36951
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1602.22 0.110953 0.0554766 0.998460i \(-0.482332\pi\)
0.0554766 + 0.998460i \(0.482332\pi\)
\(594\) 0 0
\(595\) −21443.5 −1.47748
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −18383.4 −1.25396 −0.626982 0.779033i \(-0.715710\pi\)
−0.626982 + 0.779033i \(0.715710\pi\)
\(600\) 0 0
\(601\) −25606.2 −1.73793 −0.868967 0.494870i \(-0.835216\pi\)
−0.868967 + 0.494870i \(0.835216\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3364.13 0.226068
\(606\) 0 0
\(607\) −28967.2 −1.93697 −0.968487 0.249065i \(-0.919877\pi\)
−0.968487 + 0.249065i \(0.919877\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3783.85 0.250537
\(612\) 0 0
\(613\) 9101.21 0.599665 0.299832 0.953992i \(-0.403069\pi\)
0.299832 + 0.953992i \(0.403069\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −13945.3 −0.909913 −0.454957 0.890514i \(-0.650345\pi\)
−0.454957 + 0.890514i \(0.650345\pi\)
\(618\) 0 0
\(619\) −4276.76 −0.277702 −0.138851 0.990313i \(-0.544341\pi\)
−0.138851 + 0.990313i \(0.544341\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 19069.3 1.22632
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1606.71 −0.101850
\(630\) 0 0
\(631\) −18459.0 −1.16457 −0.582283 0.812987i \(-0.697840\pi\)
−0.582283 + 0.812987i \(0.697840\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 8780.50 0.548730
\(636\) 0 0
\(637\) 71903.3 4.47239
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −25102.1 −1.54676 −0.773381 0.633941i \(-0.781436\pi\)
−0.773381 + 0.633941i \(0.781436\pi\)
\(642\) 0 0
\(643\) −14935.1 −0.915991 −0.457996 0.888954i \(-0.651432\pi\)
−0.457996 + 0.888954i \(0.651432\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −18764.3 −1.14019 −0.570093 0.821580i \(-0.693093\pi\)
−0.570093 + 0.821580i \(0.693093\pi\)
\(648\) 0 0
\(649\) −737.597 −0.0446121
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 10200.0 0.611264 0.305632 0.952150i \(-0.401132\pi\)
0.305632 + 0.952150i \(0.401132\pi\)
\(654\) 0 0
\(655\) 11941.5 0.712359
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −6908.98 −0.408400 −0.204200 0.978929i \(-0.565459\pi\)
−0.204200 + 0.978929i \(0.565459\pi\)
\(660\) 0 0
\(661\) −15449.8 −0.909118 −0.454559 0.890717i \(-0.650203\pi\)
−0.454559 + 0.890717i \(0.650203\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 21591.2 1.25906
\(666\) 0 0
\(667\) 28725.5 1.66755
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 33298.6 1.91576
\(672\) 0 0
\(673\) 15782.7 0.903977 0.451989 0.892024i \(-0.350715\pi\)
0.451989 + 0.892024i \(0.350715\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6982.32 0.396384 0.198192 0.980163i \(-0.436493\pi\)
0.198192 + 0.980163i \(0.436493\pi\)
\(678\) 0 0
\(679\) −15546.2 −0.878655
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1267.31 −0.0709988 −0.0354994 0.999370i \(-0.511302\pi\)
−0.0354994 + 0.999370i \(0.511302\pi\)
\(684\) 0 0
\(685\) 5624.71 0.313736
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −55049.2 −3.04384
\(690\) 0 0
\(691\) 24220.7 1.33343 0.666713 0.745315i \(-0.267701\pi\)
0.666713 + 0.745315i \(0.267701\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3975.52 0.216979
\(696\) 0 0
\(697\) −22659.7 −1.23141
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 11822.1 0.636967 0.318483 0.947928i \(-0.396826\pi\)
0.318483 + 0.947928i \(0.396826\pi\)
\(702\) 0 0
\(703\) 1617.78 0.0867935
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 49004.1 2.60677
\(708\) 0 0
\(709\) −15140.6 −0.801997 −0.400998 0.916079i \(-0.631337\pi\)
−0.400998 + 0.916079i \(0.631337\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −24665.5 −1.29555
\(714\) 0 0
\(715\) −17418.2 −0.911052
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1628.91 −0.0844895 −0.0422448 0.999107i \(-0.513451\pi\)
−0.0422448 + 0.999107i \(0.513451\pi\)
\(720\) 0 0
\(721\) −721.200 −0.0372523
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4698.15 0.240669
\(726\) 0 0
\(727\) −30179.1 −1.53959 −0.769793 0.638293i \(-0.779641\pi\)
−0.769793 + 0.638293i \(0.779641\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 9826.87 0.497209
\(732\) 0 0
\(733\) 9233.18 0.465260 0.232630 0.972565i \(-0.425267\pi\)
0.232630 + 0.972565i \(0.425267\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2653.06 0.132601
\(738\) 0 0
\(739\) 21276.8 1.05911 0.529554 0.848276i \(-0.322359\pi\)
0.529554 + 0.848276i \(0.322359\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9647.90 0.476375 0.238188 0.971219i \(-0.423447\pi\)
0.238188 + 0.971219i \(0.423447\pi\)
\(744\) 0 0
\(745\) 14667.8 0.721325
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −12812.5 −0.625045
\(750\) 0 0
\(751\) −19151.6 −0.930560 −0.465280 0.885163i \(-0.654046\pi\)
−0.465280 + 0.885163i \(0.654046\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8226.31 0.396538
\(756\) 0 0
\(757\) 6689.40 0.321176 0.160588 0.987022i \(-0.448661\pi\)
0.160588 + 0.987022i \(0.448661\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 24594.6 1.17156 0.585779 0.810471i \(-0.300789\pi\)
0.585779 + 0.810471i \(0.300789\pi\)
\(762\) 0 0
\(763\) 13452.2 0.638271
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1282.31 0.0603669
\(768\) 0 0
\(769\) −2631.04 −0.123378 −0.0616890 0.998095i \(-0.519649\pi\)
−0.0616890 + 0.998095i \(0.519649\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −15347.0 −0.714091 −0.357045 0.934087i \(-0.616216\pi\)
−0.357045 + 0.934087i \(0.616216\pi\)
\(774\) 0 0
\(775\) −4034.12 −0.186981
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 22815.8 1.04937
\(780\) 0 0
\(781\) −6462.58 −0.296094
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −5009.78 −0.227779
\(786\) 0 0
\(787\) 42630.0 1.93087 0.965436 0.260639i \(-0.0839331\pi\)
0.965436 + 0.260639i \(0.0839331\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −759.755 −0.0341514
\(792\) 0 0
\(793\) −57889.3 −2.59232
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −10002.6 −0.444554 −0.222277 0.974984i \(-0.571349\pi\)
−0.222277 + 0.974984i \(0.571349\pi\)
\(798\) 0 0
\(799\) −5858.39 −0.259393
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −29422.3 −1.29301
\(804\) 0 0
\(805\) 27203.8 1.19106
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 13642.8 0.592898 0.296449 0.955049i \(-0.404198\pi\)
0.296449 + 0.955049i \(0.404198\pi\)
\(810\) 0 0
\(811\) −3786.57 −0.163951 −0.0819756 0.996634i \(-0.526123\pi\)
−0.0819756 + 0.996634i \(0.526123\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −10678.0 −0.458938
\(816\) 0 0
\(817\) −9894.57 −0.423705
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 31609.5 1.34370 0.671851 0.740686i \(-0.265499\pi\)
0.671851 + 0.740686i \(0.265499\pi\)
\(822\) 0 0
\(823\) 20737.3 0.878320 0.439160 0.898409i \(-0.355276\pi\)
0.439160 + 0.898409i \(0.355276\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 10152.0 0.426866 0.213433 0.976958i \(-0.431535\pi\)
0.213433 + 0.976958i \(0.431535\pi\)
\(828\) 0 0
\(829\) −2857.26 −0.119706 −0.0598532 0.998207i \(-0.519063\pi\)
−0.0598532 + 0.998207i \(0.519063\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −111325. −4.63047
\(834\) 0 0
\(835\) −18061.8 −0.748568
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −24638.0 −1.01382 −0.506911 0.861998i \(-0.669213\pi\)
−0.506911 + 0.861998i \(0.669213\pi\)
\(840\) 0 0
\(841\) 10927.2 0.448037
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 19296.3 0.785578
\(846\) 0 0
\(847\) 23948.7 0.971531
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2038.32 0.0821065
\(852\) 0 0
\(853\) 1.13034 4.53719e−5 0 2.26860e−5 1.00000i \(-0.499993\pi\)
2.26860e−5 1.00000i \(0.499993\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 37803.4 1.50681 0.753406 0.657555i \(-0.228409\pi\)
0.753406 + 0.657555i \(0.228409\pi\)
\(858\) 0 0
\(859\) −23555.7 −0.935635 −0.467817 0.883825i \(-0.654959\pi\)
−0.467817 + 0.883825i \(0.654959\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −9572.78 −0.377591 −0.188796 0.982016i \(-0.560458\pi\)
−0.188796 + 0.982016i \(0.560458\pi\)
\(864\) 0 0
\(865\) 13424.0 0.527665
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 20336.2 0.793853
\(870\) 0 0
\(871\) −4612.32 −0.179429
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4449.27 0.171900
\(876\) 0 0
\(877\) 1559.23 0.0600358 0.0300179 0.999549i \(-0.490444\pi\)
0.0300179 + 0.999549i \(0.490444\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −36729.5 −1.40459 −0.702297 0.711884i \(-0.747842\pi\)
−0.702297 + 0.711884i \(0.747842\pi\)
\(882\) 0 0
\(883\) 6625.31 0.252502 0.126251 0.991998i \(-0.459706\pi\)
0.126251 + 0.991998i \(0.459706\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 32562.3 1.23262 0.616311 0.787503i \(-0.288626\pi\)
0.616311 + 0.787503i \(0.288626\pi\)
\(888\) 0 0
\(889\) 62506.9 2.35817
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5898.75 0.221046
\(894\) 0 0
\(895\) −10913.1 −0.407582
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −30324.7 −1.12501
\(900\) 0 0
\(901\) 85230.5 3.15143
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 15589.3 0.572603
\(906\) 0 0
\(907\) −37343.5 −1.36711 −0.683556 0.729898i \(-0.739568\pi\)
−0.683556 + 0.729898i \(0.739568\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −13207.3 −0.480328 −0.240164 0.970732i \(-0.577201\pi\)
−0.240164 + 0.970732i \(0.577201\pi\)
\(912\) 0 0
\(913\) 7409.05 0.268569
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 85009.9 3.06137
\(918\) 0 0
\(919\) 51007.6 1.83089 0.915444 0.402445i \(-0.131839\pi\)
0.915444 + 0.402445i \(0.131839\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 11235.1 0.400660
\(924\) 0 0
\(925\) 333.374 0.0118500
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −25322.8 −0.894311 −0.447155 0.894456i \(-0.647563\pi\)
−0.447155 + 0.894456i \(0.647563\pi\)
\(930\) 0 0
\(931\) 112092. 3.94594
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 26967.9 0.943255
\(936\) 0 0
\(937\) 12667.9 0.441668 0.220834 0.975311i \(-0.429122\pi\)
0.220834 + 0.975311i \(0.429122\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −25676.4 −0.889507 −0.444753 0.895653i \(-0.646709\pi\)
−0.444753 + 0.895653i \(0.646709\pi\)
\(942\) 0 0
\(943\) 28746.6 0.992703
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 50265.9 1.72484 0.862420 0.506193i \(-0.168948\pi\)
0.862420 + 0.506193i \(0.168948\pi\)
\(948\) 0 0
\(949\) 51150.3 1.74964
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 39606.9 1.34627 0.673134 0.739520i \(-0.264948\pi\)
0.673134 + 0.739520i \(0.264948\pi\)
\(954\) 0 0
\(955\) 380.456 0.0128914
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 40041.4 1.34828
\(960\) 0 0
\(961\) −3752.35 −0.125956
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2668.59 0.0890205
\(966\) 0 0
\(967\) 13997.6 0.465493 0.232746 0.972537i \(-0.425229\pi\)
0.232746 + 0.972537i \(0.425229\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −29067.3 −0.960675 −0.480337 0.877084i \(-0.659486\pi\)
−0.480337 + 0.877084i \(0.659486\pi\)
\(972\) 0 0
\(973\) 28301.1 0.932468
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −32811.7 −1.07445 −0.537226 0.843438i \(-0.680528\pi\)
−0.537226 + 0.843438i \(0.680528\pi\)
\(978\) 0 0
\(979\) −23982.0 −0.782911
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −2957.58 −0.0959635 −0.0479818 0.998848i \(-0.515279\pi\)
−0.0479818 + 0.998848i \(0.515279\pi\)
\(984\) 0 0
\(985\) 13647.4 0.441463
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −12466.6 −0.400825
\(990\) 0 0
\(991\) −8950.60 −0.286907 −0.143454 0.989657i \(-0.545821\pi\)
−0.143454 + 0.989657i \(0.545821\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −15981.3 −0.509188
\(996\) 0 0
\(997\) −53211.5 −1.69029 −0.845147 0.534533i \(-0.820487\pi\)
−0.845147 + 0.534533i \(0.820487\pi\)
\(998\) 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.p.1.4 yes 4
3.2 odd 2 1080.4.a.o.1.4 4
4.3 odd 2 2160.4.a.bv.1.1 4
12.11 even 2 2160.4.a.bu.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.o.1.4 4 3.2 odd 2
1080.4.a.p.1.4 yes 4 1.1 even 1 trivial
2160.4.a.bu.1.1 4 12.11 even 2
2160.4.a.bv.1.1 4 4.3 odd 2