Properties

Label 2160.4.a.bs.1.1
Level $2160$
Weight $4$
Character 2160.1
Self dual yes
Analytic conductor $127.444$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1257.1
Defining polynomial: \( x^{3} - x^{2} - 8x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 1080)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.72396\) of defining polynomial
Character \(\chi\) \(=\) 2160.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.00000 q^{5} -24.0794 q^{7} +O(q^{10})\) \(q+5.00000 q^{5} -24.0794 q^{7} -2.95982 q^{11} -22.1759 q^{13} -76.0223 q^{17} +72.1187 q^{19} +176.003 q^{23} +25.0000 q^{25} -42.5134 q^{29} +327.010 q^{31} -120.397 q^{35} +182.089 q^{37} -154.624 q^{41} -173.531 q^{43} +338.059 q^{47} +236.820 q^{49} +26.5490 q^{53} -14.7991 q^{55} -391.670 q^{59} -191.191 q^{61} -110.879 q^{65} +507.583 q^{67} -576.547 q^{71} -390.440 q^{73} +71.2709 q^{77} -1220.43 q^{79} +247.332 q^{83} -380.111 q^{85} -1500.81 q^{89} +533.983 q^{91} +360.594 q^{95} +959.331 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 15 q^{5} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 15 q^{5} + 10 q^{7} - 28 q^{11} - 78 q^{13} + 11 q^{17} + 71 q^{19} + 25 q^{23} + 75 q^{25} - 118 q^{29} + 107 q^{31} + 50 q^{35} - 410 q^{37} - 592 q^{41} - 52 q^{43} + 580 q^{47} + 479 q^{49} - 169 q^{53} - 140 q^{55} - 234 q^{59} - 673 q^{61} - 390 q^{65} - 386 q^{67} - 16 q^{71} - 892 q^{73} - 1800 q^{77} - 1263 q^{79} + 1815 q^{83} + 55 q^{85} - 1800 q^{89} - 1284 q^{91} + 355 q^{95} - 840 q^{97}+O(q^{100}) \) 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
\(7\) −24.0794 −1.30017 −0.650084 0.759862i \(-0.725266\pi\)
−0.650084 + 0.759862i \(0.725266\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.95982 −0.0811292 −0.0405646 0.999177i \(-0.512916\pi\)
−0.0405646 + 0.999177i \(0.512916\pi\)
\(12\) 0 0
\(13\) −22.1759 −0.473114 −0.236557 0.971618i \(-0.576019\pi\)
−0.236557 + 0.971618i \(0.576019\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −76.0223 −1.08459 −0.542297 0.840187i \(-0.682445\pi\)
−0.542297 + 0.840187i \(0.682445\pi\)
\(18\) 0 0
\(19\) 72.1187 0.870798 0.435399 0.900238i \(-0.356607\pi\)
0.435399 + 0.900238i \(0.356607\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 176.003 1.59562 0.797810 0.602909i \(-0.205992\pi\)
0.797810 + 0.602909i \(0.205992\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −42.5134 −0.272226 −0.136113 0.990693i \(-0.543461\pi\)
−0.136113 + 0.990693i \(0.543461\pi\)
\(30\) 0 0
\(31\) 327.010 1.89460 0.947301 0.320345i \(-0.103799\pi\)
0.947301 + 0.320345i \(0.103799\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −120.397 −0.581453
\(36\) 0 0
\(37\) 182.089 0.809061 0.404530 0.914525i \(-0.367435\pi\)
0.404530 + 0.914525i \(0.367435\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −154.624 −0.588981 −0.294491 0.955654i \(-0.595150\pi\)
−0.294491 + 0.955654i \(0.595150\pi\)
\(42\) 0 0
\(43\) −173.531 −0.615424 −0.307712 0.951479i \(-0.599563\pi\)
−0.307712 + 0.951479i \(0.599563\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 338.059 1.04917 0.524585 0.851358i \(-0.324221\pi\)
0.524585 + 0.851358i \(0.324221\pi\)
\(48\) 0 0
\(49\) 236.820 0.690436
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 26.5490 0.0688074 0.0344037 0.999408i \(-0.489047\pi\)
0.0344037 + 0.999408i \(0.489047\pi\)
\(54\) 0 0
\(55\) −14.7991 −0.0362821
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −391.670 −0.864256 −0.432128 0.901812i \(-0.642237\pi\)
−0.432128 + 0.901812i \(0.642237\pi\)
\(60\) 0 0
\(61\) −191.191 −0.401303 −0.200652 0.979663i \(-0.564306\pi\)
−0.200652 + 0.979663i \(0.564306\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −110.879 −0.211583
\(66\) 0 0
\(67\) 507.583 0.925540 0.462770 0.886478i \(-0.346856\pi\)
0.462770 + 0.886478i \(0.346856\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −576.547 −0.963712 −0.481856 0.876251i \(-0.660037\pi\)
−0.481856 + 0.876251i \(0.660037\pi\)
\(72\) 0 0
\(73\) −390.440 −0.625994 −0.312997 0.949754i \(-0.601333\pi\)
−0.312997 + 0.949754i \(0.601333\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 71.2709 0.105482
\(78\) 0 0
\(79\) −1220.43 −1.73808 −0.869042 0.494739i \(-0.835264\pi\)
−0.869042 + 0.494739i \(0.835264\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 247.332 0.327088 0.163544 0.986536i \(-0.447708\pi\)
0.163544 + 0.986536i \(0.447708\pi\)
\(84\) 0 0
\(85\) −380.111 −0.485045
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1500.81 −1.78748 −0.893740 0.448586i \(-0.851928\pi\)
−0.893740 + 0.448586i \(0.851928\pi\)
\(90\) 0 0
\(91\) 533.983 0.615128
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 360.594 0.389433
\(96\) 0 0
\(97\) 959.331 1.00418 0.502089 0.864816i \(-0.332565\pi\)
0.502089 + 0.864816i \(0.332565\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −587.011 −0.578314 −0.289157 0.957282i \(-0.593375\pi\)
−0.289157 + 0.957282i \(0.593375\pi\)
\(102\) 0 0
\(103\) −1359.95 −1.30097 −0.650483 0.759521i \(-0.725433\pi\)
−0.650483 + 0.759521i \(0.725433\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1041.85 0.941306 0.470653 0.882318i \(-0.344018\pi\)
0.470653 + 0.882318i \(0.344018\pi\)
\(108\) 0 0
\(109\) −457.732 −0.402227 −0.201114 0.979568i \(-0.564456\pi\)
−0.201114 + 0.979568i \(0.564456\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −467.527 −0.389214 −0.194607 0.980881i \(-0.562343\pi\)
−0.194607 + 0.980881i \(0.562343\pi\)
\(114\) 0 0
\(115\) 880.017 0.713583
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1830.57 1.41015
\(120\) 0 0
\(121\) −1322.24 −0.993418
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −2368.02 −1.65455 −0.827274 0.561799i \(-0.810109\pi\)
−0.827274 + 0.561799i \(0.810109\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2532.97 1.68937 0.844683 0.535267i \(-0.179789\pi\)
0.844683 + 0.535267i \(0.179789\pi\)
\(132\) 0 0
\(133\) −1736.58 −1.13218
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 424.096 0.264474 0.132237 0.991218i \(-0.457784\pi\)
0.132237 + 0.991218i \(0.457784\pi\)
\(138\) 0 0
\(139\) −1461.14 −0.891596 −0.445798 0.895134i \(-0.647080\pi\)
−0.445798 + 0.895134i \(0.647080\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 65.6368 0.0383834
\(144\) 0 0
\(145\) −212.567 −0.121743
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −497.826 −0.273715 −0.136857 0.990591i \(-0.543700\pi\)
−0.136857 + 0.990591i \(0.543700\pi\)
\(150\) 0 0
\(151\) −708.375 −0.381767 −0.190883 0.981613i \(-0.561135\pi\)
−0.190883 + 0.981613i \(0.561135\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1635.05 0.847292
\(156\) 0 0
\(157\) 351.998 0.178933 0.0894666 0.995990i \(-0.471484\pi\)
0.0894666 + 0.995990i \(0.471484\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4238.06 −2.07457
\(162\) 0 0
\(163\) 791.353 0.380267 0.190133 0.981758i \(-0.439108\pi\)
0.190133 + 0.981758i \(0.439108\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2756.06 1.27707 0.638535 0.769593i \(-0.279541\pi\)
0.638535 + 0.769593i \(0.279541\pi\)
\(168\) 0 0
\(169\) −1705.23 −0.776163
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3691.62 −1.62236 −0.811182 0.584794i \(-0.801175\pi\)
−0.811182 + 0.584794i \(0.801175\pi\)
\(174\) 0 0
\(175\) −601.986 −0.260034
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2036.15 −0.850217 −0.425109 0.905142i \(-0.639764\pi\)
−0.425109 + 0.905142i \(0.639764\pi\)
\(180\) 0 0
\(181\) −3268.74 −1.34234 −0.671170 0.741304i \(-0.734208\pi\)
−0.671170 + 0.741304i \(0.734208\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 910.445 0.361823
\(186\) 0 0
\(187\) 225.013 0.0879922
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2192.48 −0.830589 −0.415294 0.909687i \(-0.636321\pi\)
−0.415294 + 0.909687i \(0.636321\pi\)
\(192\) 0 0
\(193\) 3326.76 1.24075 0.620376 0.784304i \(-0.286980\pi\)
0.620376 + 0.784304i \(0.286980\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1818.42 0.657648 0.328824 0.944391i \(-0.393348\pi\)
0.328824 + 0.944391i \(0.393348\pi\)
\(198\) 0 0
\(199\) 1191.04 0.424275 0.212137 0.977240i \(-0.431958\pi\)
0.212137 + 0.977240i \(0.431958\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1023.70 0.353939
\(204\) 0 0
\(205\) −773.121 −0.263400
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −213.459 −0.0706471
\(210\) 0 0
\(211\) −2929.49 −0.955803 −0.477902 0.878413i \(-0.658602\pi\)
−0.477902 + 0.878413i \(0.658602\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −867.656 −0.275226
\(216\) 0 0
\(217\) −7874.21 −2.46330
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1685.86 0.513137
\(222\) 0 0
\(223\) −1479.80 −0.444370 −0.222185 0.975004i \(-0.571319\pi\)
−0.222185 + 0.975004i \(0.571319\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1543.24 0.451228 0.225614 0.974217i \(-0.427561\pi\)
0.225614 + 0.974217i \(0.427561\pi\)
\(228\) 0 0
\(229\) 3163.97 0.913018 0.456509 0.889719i \(-0.349100\pi\)
0.456509 + 0.889719i \(0.349100\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1466.95 0.412459 0.206229 0.978504i \(-0.433881\pi\)
0.206229 + 0.978504i \(0.433881\pi\)
\(234\) 0 0
\(235\) 1690.29 0.469203
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1346.42 −0.364405 −0.182202 0.983261i \(-0.558323\pi\)
−0.182202 + 0.983261i \(0.558323\pi\)
\(240\) 0 0
\(241\) −47.7311 −0.0127578 −0.00637890 0.999980i \(-0.502030\pi\)
−0.00637890 + 0.999980i \(0.502030\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1184.10 0.308772
\(246\) 0 0
\(247\) −1599.30 −0.411987
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5074.18 1.27601 0.638007 0.770031i \(-0.279759\pi\)
0.638007 + 0.770031i \(0.279759\pi\)
\(252\) 0 0
\(253\) −520.939 −0.129451
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3088.03 −0.749518 −0.374759 0.927122i \(-0.622275\pi\)
−0.374759 + 0.927122i \(0.622275\pi\)
\(258\) 0 0
\(259\) −4384.60 −1.05191
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4777.33 1.12009 0.560043 0.828463i \(-0.310784\pi\)
0.560043 + 0.828463i \(0.310784\pi\)
\(264\) 0 0
\(265\) 132.745 0.0307716
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7972.82 −1.80710 −0.903552 0.428478i \(-0.859050\pi\)
−0.903552 + 0.428478i \(0.859050\pi\)
\(270\) 0 0
\(271\) −3478.08 −0.779625 −0.389813 0.920894i \(-0.627460\pi\)
−0.389813 + 0.920894i \(0.627460\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −73.9956 −0.0162258
\(276\) 0 0
\(277\) −7166.62 −1.55451 −0.777257 0.629184i \(-0.783389\pi\)
−0.777257 + 0.629184i \(0.783389\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8157.14 1.73172 0.865862 0.500284i \(-0.166771\pi\)
0.865862 + 0.500284i \(0.166771\pi\)
\(282\) 0 0
\(283\) −4435.42 −0.931654 −0.465827 0.884876i \(-0.654243\pi\)
−0.465827 + 0.884876i \(0.654243\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3723.26 0.765774
\(288\) 0 0
\(289\) 866.384 0.176345
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1811.46 0.361183 0.180592 0.983558i \(-0.442199\pi\)
0.180592 + 0.983558i \(0.442199\pi\)
\(294\) 0 0
\(295\) −1958.35 −0.386507
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3903.03 −0.754911
\(300\) 0 0
\(301\) 4178.53 0.800155
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −955.955 −0.179468
\(306\) 0 0
\(307\) 9369.97 1.74193 0.870965 0.491345i \(-0.163495\pi\)
0.870965 + 0.491345i \(0.163495\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −824.869 −0.150399 −0.0751994 0.997169i \(-0.523959\pi\)
−0.0751994 + 0.997169i \(0.523959\pi\)
\(312\) 0 0
\(313\) 3447.76 0.622616 0.311308 0.950309i \(-0.399233\pi\)
0.311308 + 0.950309i \(0.399233\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3822.49 0.677264 0.338632 0.940919i \(-0.390036\pi\)
0.338632 + 0.940919i \(0.390036\pi\)
\(318\) 0 0
\(319\) 125.832 0.0220854
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5482.63 −0.944463
\(324\) 0 0
\(325\) −554.397 −0.0946229
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8140.27 −1.36410
\(330\) 0 0
\(331\) 297.908 0.0494698 0.0247349 0.999694i \(-0.492126\pi\)
0.0247349 + 0.999694i \(0.492126\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2537.92 0.413914
\(336\) 0 0
\(337\) 738.773 0.119417 0.0597085 0.998216i \(-0.480983\pi\)
0.0597085 + 0.998216i \(0.480983\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −967.891 −0.153707
\(342\) 0 0
\(343\) 2556.77 0.402485
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3379.76 −0.522868 −0.261434 0.965221i \(-0.584195\pi\)
−0.261434 + 0.965221i \(0.584195\pi\)
\(348\) 0 0
\(349\) −2622.01 −0.402157 −0.201078 0.979575i \(-0.564445\pi\)
−0.201078 + 0.979575i \(0.564445\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3052.95 −0.460318 −0.230159 0.973153i \(-0.573925\pi\)
−0.230159 + 0.973153i \(0.573925\pi\)
\(354\) 0 0
\(355\) −2882.73 −0.430985
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1613.36 −0.237187 −0.118593 0.992943i \(-0.537838\pi\)
−0.118593 + 0.992943i \(0.537838\pi\)
\(360\) 0 0
\(361\) −1657.89 −0.241710
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1952.20 −0.279953
\(366\) 0 0
\(367\) −5387.79 −0.766323 −0.383161 0.923681i \(-0.625165\pi\)
−0.383161 + 0.923681i \(0.625165\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −639.286 −0.0894611
\(372\) 0 0
\(373\) −13742.0 −1.90759 −0.953796 0.300455i \(-0.902861\pi\)
−0.953796 + 0.300455i \(0.902861\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 942.773 0.128794
\(378\) 0 0
\(379\) −6648.79 −0.901123 −0.450561 0.892745i \(-0.648776\pi\)
−0.450561 + 0.892745i \(0.648776\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −9086.53 −1.21227 −0.606136 0.795361i \(-0.707281\pi\)
−0.606136 + 0.795361i \(0.707281\pi\)
\(384\) 0 0
\(385\) 356.355 0.0471728
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3098.40 0.403844 0.201922 0.979402i \(-0.435281\pi\)
0.201922 + 0.979402i \(0.435281\pi\)
\(390\) 0 0
\(391\) −13380.2 −1.73060
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −6102.13 −0.777294
\(396\) 0 0
\(397\) −13686.5 −1.73024 −0.865122 0.501562i \(-0.832759\pi\)
−0.865122 + 0.501562i \(0.832759\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3734.83 −0.465108 −0.232554 0.972583i \(-0.574708\pi\)
−0.232554 + 0.972583i \(0.574708\pi\)
\(402\) 0 0
\(403\) −7251.73 −0.896363
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −538.952 −0.0656384
\(408\) 0 0
\(409\) 3672.62 0.444009 0.222004 0.975046i \(-0.428740\pi\)
0.222004 + 0.975046i \(0.428740\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 9431.20 1.12368
\(414\) 0 0
\(415\) 1236.66 0.146278
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −9111.68 −1.06237 −0.531187 0.847255i \(-0.678254\pi\)
−0.531187 + 0.847255i \(0.678254\pi\)
\(420\) 0 0
\(421\) 13640.0 1.57903 0.789514 0.613733i \(-0.210333\pi\)
0.789514 + 0.613733i \(0.210333\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1900.56 −0.216919
\(426\) 0 0
\(427\) 4603.77 0.521762
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −9252.97 −1.03411 −0.517053 0.855953i \(-0.672971\pi\)
−0.517053 + 0.855953i \(0.672971\pi\)
\(432\) 0 0
\(433\) −8990.85 −0.997858 −0.498929 0.866643i \(-0.666273\pi\)
−0.498929 + 0.866643i \(0.666273\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 12693.1 1.38946
\(438\) 0 0
\(439\) −10409.3 −1.13169 −0.565843 0.824513i \(-0.691449\pi\)
−0.565843 + 0.824513i \(0.691449\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3083.63 0.330717 0.165359 0.986233i \(-0.447122\pi\)
0.165359 + 0.986233i \(0.447122\pi\)
\(444\) 0 0
\(445\) −7504.05 −0.799385
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 16734.4 1.75890 0.879448 0.475996i \(-0.157912\pi\)
0.879448 + 0.475996i \(0.157912\pi\)
\(450\) 0 0
\(451\) 457.660 0.0477836
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2669.92 0.275094
\(456\) 0 0
\(457\) −19008.4 −1.94568 −0.972841 0.231473i \(-0.925646\pi\)
−0.972841 + 0.231473i \(0.925646\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8624.58 0.871338 0.435669 0.900107i \(-0.356512\pi\)
0.435669 + 0.900107i \(0.356512\pi\)
\(462\) 0 0
\(463\) −8116.40 −0.814689 −0.407345 0.913275i \(-0.633545\pi\)
−0.407345 + 0.913275i \(0.633545\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3645.17 0.361196 0.180598 0.983557i \(-0.442197\pi\)
0.180598 + 0.983557i \(0.442197\pi\)
\(468\) 0 0
\(469\) −12222.3 −1.20336
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 513.622 0.0499289
\(474\) 0 0
\(475\) 1802.97 0.174160
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −6118.27 −0.583613 −0.291807 0.956477i \(-0.594256\pi\)
−0.291807 + 0.956477i \(0.594256\pi\)
\(480\) 0 0
\(481\) −4037.99 −0.382778
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4796.66 0.449082
\(486\) 0 0
\(487\) 20400.0 1.89818 0.949091 0.315004i \(-0.102006\pi\)
0.949091 + 0.315004i \(0.102006\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4429.08 0.407091 0.203545 0.979066i \(-0.434754\pi\)
0.203545 + 0.979066i \(0.434754\pi\)
\(492\) 0 0
\(493\) 3231.96 0.295254
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 13882.9 1.25299
\(498\) 0 0
\(499\) −17665.2 −1.58477 −0.792386 0.610020i \(-0.791161\pi\)
−0.792386 + 0.610020i \(0.791161\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −1518.17 −0.134577 −0.0672883 0.997734i \(-0.521435\pi\)
−0.0672883 + 0.997734i \(0.521435\pi\)
\(504\) 0 0
\(505\) −2935.05 −0.258630
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8679.43 0.755813 0.377906 0.925844i \(-0.376644\pi\)
0.377906 + 0.925844i \(0.376644\pi\)
\(510\) 0 0
\(511\) 9401.58 0.813897
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6799.73 −0.581809
\(516\) 0 0
\(517\) −1000.59 −0.0851182
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −5080.54 −0.427222 −0.213611 0.976919i \(-0.568522\pi\)
−0.213611 + 0.976919i \(0.568522\pi\)
\(522\) 0 0
\(523\) −13889.3 −1.16125 −0.580626 0.814170i \(-0.697192\pi\)
−0.580626 + 0.814170i \(0.697192\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −24860.0 −2.05487
\(528\) 0 0
\(529\) 18810.2 1.54600
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3428.93 0.278656
\(534\) 0 0
\(535\) 5209.26 0.420965
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −700.944 −0.0560145
\(540\) 0 0
\(541\) −20787.4 −1.65198 −0.825989 0.563686i \(-0.809383\pi\)
−0.825989 + 0.563686i \(0.809383\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2288.66 −0.179882
\(546\) 0 0
\(547\) 2015.14 0.157516 0.0787578 0.996894i \(-0.474905\pi\)
0.0787578 + 0.996894i \(0.474905\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3066.01 −0.237054
\(552\) 0 0
\(553\) 29387.2 2.25980
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16460.8 −1.25218 −0.626092 0.779750i \(-0.715346\pi\)
−0.626092 + 0.779750i \(0.715346\pi\)
\(558\) 0 0
\(559\) 3848.21 0.291166
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 968.373 0.0724903 0.0362452 0.999343i \(-0.488460\pi\)
0.0362452 + 0.999343i \(0.488460\pi\)
\(564\) 0 0
\(565\) −2337.63 −0.174062
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3305.91 0.243569 0.121785 0.992557i \(-0.461138\pi\)
0.121785 + 0.992557i \(0.461138\pi\)
\(570\) 0 0
\(571\) −22291.2 −1.63373 −0.816863 0.576831i \(-0.804289\pi\)
−0.816863 + 0.576831i \(0.804289\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4400.09 0.319124
\(576\) 0 0
\(577\) 12887.3 0.929820 0.464910 0.885358i \(-0.346087\pi\)
0.464910 + 0.885358i \(0.346087\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −5955.63 −0.425269
\(582\) 0 0
\(583\) −78.5805 −0.00558228
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8042.37 0.565493 0.282746 0.959195i \(-0.408755\pi\)
0.282746 + 0.959195i \(0.408755\pi\)
\(588\) 0 0
\(589\) 23583.5 1.64982
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4779.03 0.330946 0.165473 0.986214i \(-0.447085\pi\)
0.165473 + 0.986214i \(0.447085\pi\)
\(594\) 0 0
\(595\) 9152.87 0.630640
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4890.20 0.333569 0.166785 0.985993i \(-0.446662\pi\)
0.166785 + 0.985993i \(0.446662\pi\)
\(600\) 0 0
\(601\) −2732.86 −0.185484 −0.0927418 0.995690i \(-0.529563\pi\)
−0.0927418 + 0.995690i \(0.529563\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −6611.20 −0.444270
\(606\) 0 0
\(607\) 18006.9 1.20408 0.602041 0.798465i \(-0.294355\pi\)
0.602041 + 0.798465i \(0.294355\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −7496.76 −0.496377
\(612\) 0 0
\(613\) −9779.27 −0.644341 −0.322170 0.946682i \(-0.604412\pi\)
−0.322170 + 0.946682i \(0.604412\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −18024.5 −1.17608 −0.588040 0.808832i \(-0.700100\pi\)
−0.588040 + 0.808832i \(0.700100\pi\)
\(618\) 0 0
\(619\) −2990.51 −0.194182 −0.0970911 0.995275i \(-0.530954\pi\)
−0.0970911 + 0.995275i \(0.530954\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 36138.7 2.32402
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −13842.8 −0.877503
\(630\) 0 0
\(631\) 28203.6 1.77935 0.889673 0.456599i \(-0.150932\pi\)
0.889673 + 0.456599i \(0.150932\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −11840.1 −0.739936
\(636\) 0 0
\(637\) −5251.69 −0.326655
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 27172.9 1.67436 0.837180 0.546928i \(-0.184203\pi\)
0.837180 + 0.546928i \(0.184203\pi\)
\(642\) 0 0
\(643\) 278.480 0.0170796 0.00853979 0.999964i \(-0.497282\pi\)
0.00853979 + 0.999964i \(0.497282\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −22641.1 −1.37576 −0.687878 0.725826i \(-0.741458\pi\)
−0.687878 + 0.725826i \(0.741458\pi\)
\(648\) 0 0
\(649\) 1159.28 0.0701164
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 23064.5 1.38221 0.691104 0.722755i \(-0.257125\pi\)
0.691104 + 0.722755i \(0.257125\pi\)
\(654\) 0 0
\(655\) 12664.9 0.755507
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 22595.0 1.33563 0.667813 0.744329i \(-0.267231\pi\)
0.667813 + 0.744329i \(0.267231\pi\)
\(660\) 0 0
\(661\) −14627.9 −0.860756 −0.430378 0.902649i \(-0.641620\pi\)
−0.430378 + 0.902649i \(0.641620\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8682.89 −0.506328
\(666\) 0 0
\(667\) −7482.50 −0.434368
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 565.892 0.0325574
\(672\) 0 0
\(673\) 22246.9 1.27423 0.637113 0.770770i \(-0.280128\pi\)
0.637113 + 0.770770i \(0.280128\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −13502.1 −0.766513 −0.383256 0.923642i \(-0.625197\pi\)
−0.383256 + 0.923642i \(0.625197\pi\)
\(678\) 0 0
\(679\) −23100.2 −1.30560
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −25582.5 −1.43322 −0.716609 0.697475i \(-0.754307\pi\)
−0.716609 + 0.697475i \(0.754307\pi\)
\(684\) 0 0
\(685\) 2120.48 0.118276
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −588.749 −0.0325538
\(690\) 0 0
\(691\) 12434.6 0.684565 0.342283 0.939597i \(-0.388800\pi\)
0.342283 + 0.939597i \(0.388800\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7305.68 −0.398734
\(696\) 0 0
\(697\) 11754.9 0.638806
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 13773.1 0.742088 0.371044 0.928615i \(-0.379000\pi\)
0.371044 + 0.928615i \(0.379000\pi\)
\(702\) 0 0
\(703\) 13132.0 0.704529
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 14134.9 0.751906
\(708\) 0 0
\(709\) −11950.6 −0.633023 −0.316511 0.948589i \(-0.602512\pi\)
−0.316511 + 0.948589i \(0.602512\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 57554.8 3.02306
\(714\) 0 0
\(715\) 328.184 0.0171656
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −23596.2 −1.22391 −0.611953 0.790894i \(-0.709616\pi\)
−0.611953 + 0.790894i \(0.709616\pi\)
\(720\) 0 0
\(721\) 32746.7 1.69147
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1062.83 −0.0544451
\(726\) 0 0
\(727\) 14414.5 0.735358 0.367679 0.929953i \(-0.380152\pi\)
0.367679 + 0.929953i \(0.380152\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 13192.2 0.667486
\(732\) 0 0
\(733\) 28989.8 1.46079 0.730397 0.683023i \(-0.239335\pi\)
0.730397 + 0.683023i \(0.239335\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1502.36 −0.0750883
\(738\) 0 0
\(739\) −5035.96 −0.250678 −0.125339 0.992114i \(-0.540002\pi\)
−0.125339 + 0.992114i \(0.540002\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 36268.2 1.79078 0.895390 0.445282i \(-0.146897\pi\)
0.895390 + 0.445282i \(0.146897\pi\)
\(744\) 0 0
\(745\) −2489.13 −0.122409
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −25087.2 −1.22386
\(750\) 0 0
\(751\) −6876.86 −0.334141 −0.167071 0.985945i \(-0.553431\pi\)
−0.167071 + 0.985945i \(0.553431\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −3541.88 −0.170731
\(756\) 0 0
\(757\) −32381.0 −1.55470 −0.777350 0.629068i \(-0.783437\pi\)
−0.777350 + 0.629068i \(0.783437\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9253.37 0.440781 0.220391 0.975412i \(-0.429267\pi\)
0.220391 + 0.975412i \(0.429267\pi\)
\(762\) 0 0
\(763\) 11021.9 0.522963
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8685.64 0.408892
\(768\) 0 0
\(769\) 31587.9 1.48126 0.740630 0.671913i \(-0.234527\pi\)
0.740630 + 0.671913i \(0.234527\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −38886.4 −1.80937 −0.904687 0.426077i \(-0.859895\pi\)
−0.904687 + 0.426077i \(0.859895\pi\)
\(774\) 0 0
\(775\) 8175.24 0.378920
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −11151.3 −0.512884
\(780\) 0 0
\(781\) 1706.48 0.0781851
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1759.99 0.0800214
\(786\) 0 0
\(787\) 8792.97 0.398266 0.199133 0.979972i \(-0.436187\pi\)
0.199133 + 0.979972i \(0.436187\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 11257.8 0.506044
\(792\) 0 0
\(793\) 4239.83 0.189862
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 21136.5 0.939391 0.469695 0.882829i \(-0.344364\pi\)
0.469695 + 0.882829i \(0.344364\pi\)
\(798\) 0 0
\(799\) −25700.0 −1.13792
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1155.63 0.0507864
\(804\) 0 0
\(805\) −21190.3 −0.927777
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −31974.0 −1.38955 −0.694776 0.719227i \(-0.744496\pi\)
−0.694776 + 0.719227i \(0.744496\pi\)
\(810\) 0 0
\(811\) 15694.2 0.679528 0.339764 0.940511i \(-0.389653\pi\)
0.339764 + 0.940511i \(0.389653\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3956.76 0.170061
\(816\) 0 0
\(817\) −12514.8 −0.535910
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 156.635 0.00665848 0.00332924 0.999994i \(-0.498940\pi\)
0.00332924 + 0.999994i \(0.498940\pi\)
\(822\) 0 0
\(823\) −13428.1 −0.568743 −0.284372 0.958714i \(-0.591785\pi\)
−0.284372 + 0.958714i \(0.591785\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −33717.9 −1.41776 −0.708880 0.705329i \(-0.750799\pi\)
−0.708880 + 0.705329i \(0.750799\pi\)
\(828\) 0 0
\(829\) 38733.5 1.62276 0.811380 0.584518i \(-0.198717\pi\)
0.811380 + 0.584518i \(0.198717\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −18003.6 −0.748843
\(834\) 0 0
\(835\) 13780.3 0.571123
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −14106.0 −0.580443 −0.290221 0.956960i \(-0.593729\pi\)
−0.290221 + 0.956960i \(0.593729\pi\)
\(840\) 0 0
\(841\) −22581.6 −0.925893
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −8526.15 −0.347111
\(846\) 0 0
\(847\) 31838.8 1.29161
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 32048.3 1.29095
\(852\) 0 0
\(853\) −24614.3 −0.988015 −0.494007 0.869458i \(-0.664468\pi\)
−0.494007 + 0.869458i \(0.664468\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −21660.2 −0.863360 −0.431680 0.902027i \(-0.642079\pi\)
−0.431680 + 0.902027i \(0.642079\pi\)
\(858\) 0 0
\(859\) 26600.0 1.05655 0.528277 0.849072i \(-0.322838\pi\)
0.528277 + 0.849072i \(0.322838\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 47381.0 1.86891 0.934454 0.356083i \(-0.115888\pi\)
0.934454 + 0.356083i \(0.115888\pi\)
\(864\) 0 0
\(865\) −18458.1 −0.725543
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 3612.24 0.141009
\(870\) 0 0
\(871\) −11256.1 −0.437886
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3009.93 −0.116291
\(876\) 0 0
\(877\) −31597.5 −1.21662 −0.608308 0.793701i \(-0.708151\pi\)
−0.608308 + 0.793701i \(0.708151\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −33206.9 −1.26989 −0.634943 0.772559i \(-0.718976\pi\)
−0.634943 + 0.772559i \(0.718976\pi\)
\(882\) 0 0
\(883\) 12479.4 0.475612 0.237806 0.971313i \(-0.423572\pi\)
0.237806 + 0.971313i \(0.423572\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −7331.93 −0.277545 −0.138772 0.990324i \(-0.544316\pi\)
−0.138772 + 0.990324i \(0.544316\pi\)
\(888\) 0 0
\(889\) 57020.5 2.15119
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 24380.4 0.913615
\(894\) 0 0
\(895\) −10180.7 −0.380229
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −13902.3 −0.515759
\(900\) 0 0
\(901\) −2018.32 −0.0746281
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −16343.7 −0.600312
\(906\) 0 0
\(907\) 4489.18 0.164345 0.0821725 0.996618i \(-0.473814\pi\)
0.0821725 + 0.996618i \(0.473814\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 25710.9 0.935059 0.467529 0.883978i \(-0.345144\pi\)
0.467529 + 0.883978i \(0.345144\pi\)
\(912\) 0 0
\(913\) −732.061 −0.0265363
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −60992.6 −2.19646
\(918\) 0 0
\(919\) −24637.6 −0.884353 −0.442176 0.896928i \(-0.645793\pi\)
−0.442176 + 0.896928i \(0.645793\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 12785.4 0.455946
\(924\) 0 0
\(925\) 4552.23 0.161812
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 6833.00 0.241317 0.120658 0.992694i \(-0.461499\pi\)
0.120658 + 0.992694i \(0.461499\pi\)
\(930\) 0 0
\(931\) 17079.1 0.601231
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1125.06 0.0393513
\(936\) 0 0
\(937\) −25120.0 −0.875810 −0.437905 0.899021i \(-0.644279\pi\)
−0.437905 + 0.899021i \(0.644279\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −54122.3 −1.87496 −0.937480 0.348040i \(-0.886847\pi\)
−0.937480 + 0.348040i \(0.886847\pi\)
\(942\) 0 0
\(943\) −27214.4 −0.939790
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 47670.7 1.63579 0.817894 0.575370i \(-0.195142\pi\)
0.817894 + 0.575370i \(0.195142\pi\)
\(948\) 0 0
\(949\) 8658.36 0.296167
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −11963.3 −0.406642 −0.203321 0.979112i \(-0.565174\pi\)
−0.203321 + 0.979112i \(0.565174\pi\)
\(954\) 0 0
\(955\) −10962.4 −0.371451
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −10212.0 −0.343861
\(960\) 0 0
\(961\) 77144.3 2.58952
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 16633.8 0.554881
\(966\) 0 0
\(967\) 36546.2 1.21535 0.607676 0.794185i \(-0.292102\pi\)
0.607676 + 0.794185i \(0.292102\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −785.854 −0.0259725 −0.0129862 0.999916i \(-0.504134\pi\)
−0.0129862 + 0.999916i \(0.504134\pi\)
\(972\) 0 0
\(973\) 35183.3 1.15922
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 16159.0 0.529144 0.264572 0.964366i \(-0.414769\pi\)
0.264572 + 0.964366i \(0.414769\pi\)
\(978\) 0 0
\(979\) 4442.14 0.145017
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 19453.9 0.631213 0.315607 0.948890i \(-0.397792\pi\)
0.315607 + 0.948890i \(0.397792\pi\)
\(984\) 0 0
\(985\) 9092.08 0.294109
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −30542.1 −0.981983
\(990\) 0 0
\(991\) −25557.5 −0.819234 −0.409617 0.912258i \(-0.634338\pi\)
−0.409617 + 0.912258i \(0.634338\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 5955.20 0.189741
\(996\) 0 0
\(997\) −10058.6 −0.319519 −0.159760 0.987156i \(-0.551072\pi\)
−0.159760 + 0.987156i \(0.551072\pi\)
\(998\) 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.bs.1.1 3
3.2 odd 2 2160.4.a.bk.1.1 3
4.3 odd 2 1080.4.a.j.1.3 yes 3
12.11 even 2 1080.4.a.d.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.d.1.3 3 12.11 even 2
1080.4.a.j.1.3 yes 3 4.3 odd 2
2160.4.a.bk.1.1 3 3.2 odd 2
2160.4.a.bs.1.1 3 1.1 even 1 trivial