Properties

Label 2160.4.a.bv.1.1
Level $2160$
Weight $4$
Character 2160.1
Self dual yes
Analytic conductor $127.444$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(127.444125612\)
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: no (minimal twist has level 1080)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-10.9094\) of defining polynomial
Character \(\chi\) \(=\) 2160.1

$q$-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 \(-0.910755\pi\)
−0.960953 + 0.276713i \(0.910755\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 44.7641 1.22699 0.613495 0.789699i \(-0.289763\pi\)
0.613495 + 0.789699i \(0.289763\pi\)
\(12\) 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 \(-0.761615\pi\)
−0.732433 + 0.680839i \(0.761615\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −152.855 −1.38576 −0.692880 0.721053i \(-0.743659\pi\)
−0.692880 + 0.721053i \(0.743659\pi\)
\(24\) 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 \(-0.345172\pi\)
0.467452 + 0.884019i \(0.345172\pi\)
\(32\) 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 \(-0.453803\pi\)
0.144623 + 0.989487i \(0.453803\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −48.6219 −0.150899 −0.0754493 0.997150i \(-0.524039\pi\)
−0.0754493 + 0.997150i \(0.524039\pi\)
\(48\) 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 \(-0.505787\pi\)
−0.0181795 + 0.999835i \(0.505787\pi\)
\(60\) 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 \(-0.482792\pi\)
0.0540350 + 0.998539i \(0.482792\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −144.370 −0.241317 −0.120659 0.992694i \(-0.538501\pi\)
−0.120659 + 0.992694i \(0.538501\pi\)
\(72\) 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 \(-0.395142\pi\)
0.323496 + 0.946230i \(0.395142\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 165.513 0.218885 0.109442 0.993993i \(-0.465093\pi\)
0.109442 + 0.993993i \(0.465093\pi\)
\(84\) 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 \(-0.496915\pi\)
0.00969149 + 0.999953i \(0.496915\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 359.961 0.325222 0.162611 0.986690i \(-0.448008\pi\)
0.162611 + 0.986690i \(0.448008\pi\)
\(108\) 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 \(-0.710238\pi\)
−0.613499 + 0.789696i \(0.710238\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2388.31 −1.59288 −0.796441 0.604716i \(-0.793287\pi\)
−0.796441 + 0.604716i \(0.793287\pi\)
\(132\) 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 \(-0.577997\pi\)
−0.242590 + 0.970129i \(0.577997\pi\)
\(140\) 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 \(-0.646208\pi\)
−0.443343 + 0.896352i \(0.646208\pi\)
\(152\) 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 \(-0.328494\pi\)
0.513108 + 0.858324i \(0.328494\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3612.36 1.67385 0.836925 0.547318i \(-0.184351\pi\)
0.836925 + 0.547318i \(0.184351\pi\)
\(168\) 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 \(-0.349392\pi\)
0.455691 + 0.890138i \(0.349392\pi\)
\(180\) 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 \(-0.504588\pi\)
−0.0144130 + 0.999896i \(0.504588\pi\)
\(192\) 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 \(-0.307218\pi\)
0.569290 + 0.822137i \(0.307218\pi\)
\(200\) 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 \(-0.250653\pi\)
0.705654 + 0.708557i \(0.250653\pi\)
\(212\) 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 \(-0.565989\pi\)
−0.205830 + 0.978588i \(0.565989\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2676.95 −0.782711 −0.391356 0.920240i \(-0.627994\pi\)
−0.391356 + 0.920240i \(0.627994\pi\)
\(228\) 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 \(-0.253821\pi\)
0.698568 + 0.715543i \(0.253821\pi\)
\(240\) 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 \(-0.455754\pi\)
0.138555 + 0.990355i \(0.455754\pi\)
\(252\) 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 \(-0.448265\pi\)
0.161815 + 0.986821i \(0.448265\pi\)
\(264\) 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 \(-0.403032\pi\)
0.299944 + 0.953957i \(0.403032\pi\)
\(272\) 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 \(-0.377152\pi\)
0.376429 + 0.926446i \(0.377152\pi\)
\(284\) 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 \(-0.672179\pi\)
−0.514923 + 0.857237i \(0.672179\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8862.85 1.61597 0.807984 0.589204i \(-0.200559\pi\)
0.807984 + 0.589204i \(0.200559\pi\)
\(312\) 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 \(-0.733057\pi\)
−0.668484 + 0.743726i \(0.733057\pi\)
\(332\) 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 \(-0.418645\pi\)
0.252811 + 0.967516i \(0.418645\pi\)
\(348\) 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 \(-0.462516\pi\)
0.117487 + 0.993074i \(0.462516\pi\)
\(360\) 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 \(-0.165616\pi\)
0.867671 + 0.497138i \(0.165616\pi\)
\(368\) 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 \(-0.331200\pi\)
0.505794 + 0.862654i \(0.331200\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4064.01 0.542196 0.271098 0.962552i \(-0.412613\pi\)
0.271098 + 0.962552i \(0.412613\pi\)
\(384\) 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 \(-0.445421\pi\)
0.170627 + 0.985336i \(0.445421\pi\)
\(420\) 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 \(-0.584465\pi\)
−0.262253 + 0.964999i \(0.584465\pi\)
\(432\) 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 \(-0.453967\pi\)
0.144113 + 0.989561i \(0.453967\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1899.98 −0.203771 −0.101886 0.994796i \(-0.532488\pi\)
−0.101886 + 0.994796i \(0.532488\pi\)
\(444\) 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 \(-0.450367\pi\)
0.155295 + 0.987868i \(0.450367\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −13827.8 −1.37018 −0.685092 0.728457i \(-0.740238\pi\)
−0.685092 + 0.728457i \(0.740238\pi\)
\(468\) 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 \(-0.472997\pi\)
0.0847313 + 0.996404i \(0.472997\pi\)
\(480\) 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 \(-0.452077\pi\)
0.149986 + 0.988688i \(0.452077\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8699.20 0.799571 0.399786 0.916609i \(-0.369085\pi\)
0.399786 + 0.916609i \(0.369085\pi\)
\(492\) 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 \(-0.678113\pi\)
−0.530813 + 0.847489i \(0.678113\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 15882.5 1.40789 0.703943 0.710256i \(-0.251421\pi\)
0.703943 + 0.710256i \(0.251421\pi\)
\(504\) 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 \(-0.379760\pi\)
0.368825 + 0.929499i \(0.379760\pi\)
\(524\) 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 \(-0.627624\pi\)
−0.390287 + 0.920693i \(0.627624\pi\)
\(548\) 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 \(-0.367154\pi\)
0.405337 + 0.914167i \(0.367154\pi\)
\(564\) 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 \(-0.711683\pi\)
−0.617077 + 0.786903i \(0.711683\pi\)
\(572\) 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 \(-0.523760\pi\)
−0.0745739 + 0.997215i \(0.523760\pi\)
\(588\) 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 \(-0.284290\pi\)
0.626982 + 0.779033i \(0.284290\pi\)
\(600\) 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 \(-0.0801232\pi\)
0.968487 + 0.249065i \(0.0801232\pi\)
\(608\) 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 \(-0.455659\pi\)
0.138851 + 0.990313i \(0.455659\pi\)
\(620\) 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 \(-0.302160\pi\)
0.582283 + 0.812987i \(0.302160\pi\)
\(632\) 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 \(-0.348568\pi\)
0.457996 + 0.888954i \(0.348568\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18764.3 1.14019 0.570093 0.821580i \(-0.306907\pi\)
0.570093 + 0.821580i \(0.306907\pi\)
\(648\) 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 \(-0.434541\pi\)
0.204200 + 0.978929i \(0.434541\pi\)
\(660\) 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 \(-0.488698\pi\)
0.0354994 + 0.999370i \(0.488698\pi\)
\(684\) 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 \(-0.732299\pi\)
−0.666713 + 0.745315i \(0.732299\pi\)
\(692\) 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 \(-0.486549\pi\)
0.0422448 + 0.999107i \(0.486549\pi\)
\(720\) 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 \(-0.220359\pi\)
0.769793 + 0.638293i \(0.220359\pi\)
\(728\) 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 \(-0.677641\pi\)
−0.529554 + 0.848276i \(0.677641\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −9647.90 −0.476375 −0.238188 0.971219i \(-0.576553\pi\)
−0.238188 + 0.971219i \(0.576553\pi\)
\(744\) 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 \(-0.345954\pi\)
0.465280 + 0.885163i \(0.345954\pi\)
\(752\) 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 \(-0.916067\pi\)
−0.965436 + 0.260639i \(0.916067\pi\)
\(788\) 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 \(-0.473877\pi\)
0.0819756 + 0.996634i \(0.473877\pi\)
\(812\) 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 \(-0.644724\pi\)
−0.439160 + 0.898409i \(0.644724\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −10152.0 −0.426866 −0.213433 0.976958i \(-0.568465\pi\)
−0.213433 + 0.976958i \(0.568465\pi\)
\(828\) 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 \(-0.330787\pi\)
0.506911 + 0.861998i \(0.330787\pi\)
\(840\) 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 \(-0.345041\pi\)
0.467817 + 0.883825i \(0.345041\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 9572.78 0.377591 0.188796 0.982016i \(-0.439542\pi\)
0.188796 + 0.982016i \(0.439542\pi\)
\(864\) 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 \(-0.540294\pi\)
−0.126251 + 0.991998i \(0.540294\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −32562.3 −1.23262 −0.616311 0.787503i \(-0.711374\pi\)
−0.616311 + 0.787503i \(0.711374\pi\)
\(888\) 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 \(-0.260432\pi\)
0.683556 + 0.729898i \(0.260432\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 13207.3 0.480328 0.240164 0.970732i \(-0.422799\pi\)
0.240164 + 0.970732i \(0.422799\pi\)
\(912\) 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 \(-0.868161\pi\)
−0.915444 + 0.402445i \(0.868161\pi\)
\(920\) 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 \(-0.831052\pi\)
−0.862420 + 0.506193i \(0.831052\pi\)
\(948\) 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 \(-0.574771\pi\)
−0.232746 + 0.972537i \(0.574771\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 29067.3 0.960675 0.480337 0.877084i \(-0.340514\pi\)
0.480337 + 0.877084i \(0.340514\pi\)
\(972\) 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 \(-0.484721\pi\)
0.0479818 + 0.998848i \(0.484721\pi\)
\(984\) 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 \(-0.454179\pi\)
0.143454 + 0.989657i \(0.454179\pi\)
\(992\) 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 2160.4.a.bv.1.1 4
3.2 odd 2 2160.4.a.bu.1.1 4
4.3 odd 2 1080.4.a.p.1.4 yes 4
12.11 even 2 1080.4.a.o.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.o.1.4 4 12.11 even 2
1080.4.a.p.1.4 yes 4 4.3 odd 2
2160.4.a.bu.1.1 4 3.2 odd 2
2160.4.a.bv.1.1 4 1.1 even 1 trivial