Properties

Label 2160.4.a.bf.1.1
Level $2160$
Weight $4$
Character 2160.1
Self dual yes
Analytic conductor $127.444$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.47977.1
Defining polynomial: \( x^{3} - x^{2} - 60x - 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1080)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-6.83575\) of defining polynomial
Character \(\chi\) \(=\) 2160.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.00000 q^{5} -23.9261 q^{7} +O(q^{10})\) \(q-5.00000 q^{5} -23.9261 q^{7} -57.9406 q^{11} -8.16237 q^{13} -50.0884 q^{17} -69.7782 q^{19} -4.92607 q^{23} +25.0000 q^{25} +79.4277 q^{29} -260.294 q^{31} +119.630 q^{35} -223.836 q^{37} -337.939 q^{41} -326.511 q^{43} -89.6851 q^{47} +229.457 q^{49} -543.672 q^{53} +289.703 q^{55} -92.0000 q^{59} +159.129 q^{61} +40.8119 q^{65} +910.561 q^{67} +293.232 q^{71} +142.022 q^{73} +1386.29 q^{77} -1106.50 q^{79} -813.367 q^{83} +250.442 q^{85} +956.666 q^{89} +195.294 q^{91} +348.891 q^{95} +106.363 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 15 q^{5} - 9 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 15 q^{5} - 9 q^{7} + 18 q^{11} - 21 q^{13} - 84 q^{17} - 21 q^{19} + 48 q^{23} + 75 q^{25} + 36 q^{29} - 324 q^{31} + 45 q^{35} + 33 q^{37} - 114 q^{41} - 282 q^{43} + 282 q^{47} + 228 q^{49} - 222 q^{53} - 90 q^{55} - 276 q^{59} + 303 q^{61} + 105 q^{65} - 1035 q^{67} + 510 q^{71} + 447 q^{73} + 1578 q^{77} - 777 q^{79} + 78 q^{83} + 420 q^{85} + 324 q^{89} - 1995 q^{91} + 105 q^{95} + 1191 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −23.9261 −1.29189 −0.645943 0.763386i \(-0.723536\pi\)
−0.645943 + 0.763386i \(0.723536\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −57.9406 −1.58816 −0.794079 0.607814i \(-0.792046\pi\)
−0.794079 + 0.607814i \(0.792046\pi\)
\(12\) 0 0
\(13\) −8.16237 −0.174141 −0.0870706 0.996202i \(-0.527751\pi\)
−0.0870706 + 0.996202i \(0.527751\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −50.0884 −0.714602 −0.357301 0.933989i \(-0.616303\pi\)
−0.357301 + 0.933989i \(0.616303\pi\)
\(18\) 0 0
\(19\) −69.7782 −0.842538 −0.421269 0.906936i \(-0.638415\pi\)
−0.421269 + 0.906936i \(0.638415\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.92607 −0.0446590 −0.0223295 0.999751i \(-0.507108\pi\)
−0.0223295 + 0.999751i \(0.507108\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 79.4277 0.508598 0.254299 0.967126i \(-0.418155\pi\)
0.254299 + 0.967126i \(0.418155\pi\)
\(30\) 0 0
\(31\) −260.294 −1.50807 −0.754036 0.656833i \(-0.771896\pi\)
−0.754036 + 0.656833i \(0.771896\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 119.630 0.577749
\(36\) 0 0
\(37\) −223.836 −0.994553 −0.497276 0.867592i \(-0.665666\pi\)
−0.497276 + 0.867592i \(0.665666\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −337.939 −1.28725 −0.643625 0.765341i \(-0.722570\pi\)
−0.643625 + 0.765341i \(0.722570\pi\)
\(42\) 0 0
\(43\) −326.511 −1.15797 −0.578983 0.815340i \(-0.696550\pi\)
−0.578983 + 0.815340i \(0.696550\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −89.6851 −0.278339 −0.139169 0.990269i \(-0.544443\pi\)
−0.139169 + 0.990269i \(0.544443\pi\)
\(48\) 0 0
\(49\) 229.457 0.668970
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −543.672 −1.40904 −0.704520 0.709684i \(-0.748838\pi\)
−0.704520 + 0.709684i \(0.748838\pi\)
\(54\) 0 0
\(55\) 289.703 0.710246
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −92.0000 −0.203006 −0.101503 0.994835i \(-0.532365\pi\)
−0.101503 + 0.994835i \(0.532365\pi\)
\(60\) 0 0
\(61\) 159.129 0.334006 0.167003 0.985956i \(-0.446591\pi\)
0.167003 + 0.985956i \(0.446591\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 40.8119 0.0778783
\(66\) 0 0
\(67\) 910.561 1.66034 0.830169 0.557511i \(-0.188244\pi\)
0.830169 + 0.557511i \(0.188244\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 293.232 0.490144 0.245072 0.969505i \(-0.421188\pi\)
0.245072 + 0.969505i \(0.421188\pi\)
\(72\) 0 0
\(73\) 142.022 0.227705 0.113853 0.993498i \(-0.463681\pi\)
0.113853 + 0.993498i \(0.463681\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1386.29 2.05172
\(78\) 0 0
\(79\) −1106.50 −1.57584 −0.787920 0.615777i \(-0.788842\pi\)
−0.787920 + 0.615777i \(0.788842\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −813.367 −1.07565 −0.537823 0.843058i \(-0.680753\pi\)
−0.537823 + 0.843058i \(0.680753\pi\)
\(84\) 0 0
\(85\) 250.442 0.319580
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 956.666 1.13940 0.569699 0.821853i \(-0.307060\pi\)
0.569699 + 0.821853i \(0.307060\pi\)
\(90\) 0 0
\(91\) 195.294 0.224971
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 348.891 0.376794
\(96\) 0 0
\(97\) 106.363 0.111335 0.0556677 0.998449i \(-0.482271\pi\)
0.0556677 + 0.998449i \(0.482271\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −703.082 −0.692666 −0.346333 0.938112i \(-0.612573\pi\)
−0.346333 + 0.938112i \(0.612573\pi\)
\(102\) 0 0
\(103\) −968.360 −0.926363 −0.463181 0.886263i \(-0.653292\pi\)
−0.463181 + 0.886263i \(0.653292\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −366.393 −0.331033 −0.165517 0.986207i \(-0.552929\pi\)
−0.165517 + 0.986207i \(0.552929\pi\)
\(108\) 0 0
\(109\) 723.614 0.635869 0.317934 0.948113i \(-0.397011\pi\)
0.317934 + 0.948113i \(0.397011\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 978.279 0.814413 0.407207 0.913336i \(-0.366503\pi\)
0.407207 + 0.913336i \(0.366503\pi\)
\(114\) 0 0
\(115\) 24.6303 0.0199721
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1198.42 0.923184
\(120\) 0 0
\(121\) 2026.11 1.52225
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −2821.70 −1.97154 −0.985770 0.168099i \(-0.946237\pi\)
−0.985770 + 0.168099i \(0.946237\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2131.82 1.42182 0.710908 0.703285i \(-0.248284\pi\)
0.710908 + 0.703285i \(0.248284\pi\)
\(132\) 0 0
\(133\) 1669.52 1.08846
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −816.808 −0.509377 −0.254688 0.967023i \(-0.581973\pi\)
−0.254688 + 0.967023i \(0.581973\pi\)
\(138\) 0 0
\(139\) −1876.80 −1.14524 −0.572620 0.819821i \(-0.694073\pi\)
−0.572620 + 0.819821i \(0.694073\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 472.933 0.276564
\(144\) 0 0
\(145\) −397.139 −0.227452
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2542.71 1.39803 0.699016 0.715106i \(-0.253621\pi\)
0.699016 + 0.715106i \(0.253621\pi\)
\(150\) 0 0
\(151\) −333.443 −0.179703 −0.0898516 0.995955i \(-0.528639\pi\)
−0.0898516 + 0.995955i \(0.528639\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1301.47 0.674430
\(156\) 0 0
\(157\) 3650.02 1.85543 0.927716 0.373286i \(-0.121769\pi\)
0.927716 + 0.373286i \(0.121769\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 117.861 0.0576943
\(162\) 0 0
\(163\) −745.580 −0.358272 −0.179136 0.983824i \(-0.557330\pi\)
−0.179136 + 0.983824i \(0.557330\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3321.73 1.53918 0.769591 0.638537i \(-0.220460\pi\)
0.769591 + 0.638537i \(0.220460\pi\)
\(168\) 0 0
\(169\) −2130.38 −0.969675
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1038.58 0.456426 0.228213 0.973611i \(-0.426712\pi\)
0.228213 + 0.973611i \(0.426712\pi\)
\(174\) 0 0
\(175\) −598.152 −0.258377
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −250.181 −0.104466 −0.0522329 0.998635i \(-0.516634\pi\)
−0.0522329 + 0.998635i \(0.516634\pi\)
\(180\) 0 0
\(181\) −1171.61 −0.481132 −0.240566 0.970633i \(-0.577333\pi\)
−0.240566 + 0.970633i \(0.577333\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1119.18 0.444778
\(186\) 0 0
\(187\) 2902.15 1.13490
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1091.59 0.413533 0.206766 0.978390i \(-0.433706\pi\)
0.206766 + 0.978390i \(0.433706\pi\)
\(192\) 0 0
\(193\) −171.475 −0.0639537 −0.0319768 0.999489i \(-0.510180\pi\)
−0.0319768 + 0.999489i \(0.510180\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2368.70 0.856665 0.428332 0.903621i \(-0.359101\pi\)
0.428332 + 0.903621i \(0.359101\pi\)
\(198\) 0 0
\(199\) −3373.82 −1.20183 −0.600914 0.799314i \(-0.705197\pi\)
−0.600914 + 0.799314i \(0.705197\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1900.39 −0.657051
\(204\) 0 0
\(205\) 1689.70 0.575676
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4042.99 1.33808
\(210\) 0 0
\(211\) −5148.57 −1.67982 −0.839910 0.542726i \(-0.817392\pi\)
−0.839910 + 0.542726i \(0.817392\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1632.56 0.517858
\(216\) 0 0
\(217\) 6227.82 1.94826
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 408.841 0.124442
\(222\) 0 0
\(223\) 3604.51 1.08240 0.541201 0.840893i \(-0.317970\pi\)
0.541201 + 0.840893i \(0.317970\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3324.89 −0.972162 −0.486081 0.873914i \(-0.661574\pi\)
−0.486081 + 0.873914i \(0.661574\pi\)
\(228\) 0 0
\(229\) −2809.94 −0.810856 −0.405428 0.914127i \(-0.632877\pi\)
−0.405428 + 0.914127i \(0.632877\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4200.85 −1.18115 −0.590573 0.806984i \(-0.701098\pi\)
−0.590573 + 0.806984i \(0.701098\pi\)
\(234\) 0 0
\(235\) 448.426 0.124477
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5787.06 1.56625 0.783125 0.621865i \(-0.213625\pi\)
0.783125 + 0.621865i \(0.213625\pi\)
\(240\) 0 0
\(241\) −2971.77 −0.794309 −0.397154 0.917752i \(-0.630002\pi\)
−0.397154 + 0.917752i \(0.630002\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1147.28 −0.299172
\(246\) 0 0
\(247\) 569.556 0.146721
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4035.62 −1.01484 −0.507422 0.861698i \(-0.669401\pi\)
−0.507422 + 0.861698i \(0.669401\pi\)
\(252\) 0 0
\(253\) 285.419 0.0709255
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2852.67 −0.692392 −0.346196 0.938162i \(-0.612527\pi\)
−0.346196 + 0.938162i \(0.612527\pi\)
\(258\) 0 0
\(259\) 5355.52 1.28485
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2910.63 0.682423 0.341212 0.939987i \(-0.389163\pi\)
0.341212 + 0.939987i \(0.389163\pi\)
\(264\) 0 0
\(265\) 2718.36 0.630142
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 476.299 0.107957 0.0539785 0.998542i \(-0.482810\pi\)
0.0539785 + 0.998542i \(0.482810\pi\)
\(270\) 0 0
\(271\) 4186.19 0.938350 0.469175 0.883105i \(-0.344551\pi\)
0.469175 + 0.883105i \(0.344551\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1448.51 −0.317632
\(276\) 0 0
\(277\) 4993.37 1.08311 0.541556 0.840664i \(-0.317835\pi\)
0.541556 + 0.840664i \(0.317835\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4480.61 0.951212 0.475606 0.879658i \(-0.342229\pi\)
0.475606 + 0.879658i \(0.342229\pi\)
\(282\) 0 0
\(283\) −8117.91 −1.70516 −0.852579 0.522599i \(-0.824963\pi\)
−0.852579 + 0.522599i \(0.824963\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8085.56 1.66298
\(288\) 0 0
\(289\) −2404.15 −0.489344
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5999.46 1.19622 0.598109 0.801414i \(-0.295919\pi\)
0.598109 + 0.801414i \(0.295919\pi\)
\(294\) 0 0
\(295\) 460.000 0.0907872
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 40.2084 0.00777696
\(300\) 0 0
\(301\) 7812.14 1.49596
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −795.644 −0.149372
\(306\) 0 0
\(307\) 1058.34 0.196751 0.0983753 0.995149i \(-0.468635\pi\)
0.0983753 + 0.995149i \(0.468635\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3120.17 −0.568902 −0.284451 0.958691i \(-0.591811\pi\)
−0.284451 + 0.958691i \(0.591811\pi\)
\(312\) 0 0
\(313\) −4425.91 −0.799257 −0.399628 0.916677i \(-0.630861\pi\)
−0.399628 + 0.916677i \(0.630861\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9032.06 −1.60029 −0.800144 0.599808i \(-0.795243\pi\)
−0.800144 + 0.599808i \(0.795243\pi\)
\(318\) 0 0
\(319\) −4602.09 −0.807735
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3495.08 0.602079
\(324\) 0 0
\(325\) −204.059 −0.0348282
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2145.81 0.359582
\(330\) 0 0
\(331\) −3153.85 −0.523720 −0.261860 0.965106i \(-0.584336\pi\)
−0.261860 + 0.965106i \(0.584336\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4552.80 −0.742526
\(336\) 0 0
\(337\) −3438.53 −0.555811 −0.277906 0.960608i \(-0.589640\pi\)
−0.277906 + 0.960608i \(0.589640\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 15081.6 2.39506
\(342\) 0 0
\(343\) 2716.64 0.427653
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11108.0 1.71848 0.859238 0.511576i \(-0.170938\pi\)
0.859238 + 0.511576i \(0.170938\pi\)
\(348\) 0 0
\(349\) 1687.51 0.258826 0.129413 0.991591i \(-0.458691\pi\)
0.129413 + 0.991591i \(0.458691\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12048.9 1.81671 0.908356 0.418199i \(-0.137338\pi\)
0.908356 + 0.418199i \(0.137338\pi\)
\(354\) 0 0
\(355\) −1466.16 −0.219199
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −11017.0 −1.61965 −0.809827 0.586669i \(-0.800439\pi\)
−0.809827 + 0.586669i \(0.800439\pi\)
\(360\) 0 0
\(361\) −1990.00 −0.290130
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −710.112 −0.101833
\(366\) 0 0
\(367\) 10621.2 1.51069 0.755343 0.655329i \(-0.227470\pi\)
0.755343 + 0.655329i \(0.227470\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 13007.9 1.82032
\(372\) 0 0
\(373\) 24.9392 0.00346194 0.00173097 0.999999i \(-0.499449\pi\)
0.00173097 + 0.999999i \(0.499449\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −648.319 −0.0885679
\(378\) 0 0
\(379\) −4755.57 −0.644531 −0.322265 0.946649i \(-0.604444\pi\)
−0.322265 + 0.946649i \(0.604444\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1229.91 0.164088 0.0820440 0.996629i \(-0.473855\pi\)
0.0820440 + 0.996629i \(0.473855\pi\)
\(384\) 0 0
\(385\) −6931.45 −0.917557
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 10616.3 1.38372 0.691862 0.722030i \(-0.256791\pi\)
0.691862 + 0.722030i \(0.256791\pi\)
\(390\) 0 0
\(391\) 246.739 0.0319134
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5532.52 0.704738
\(396\) 0 0
\(397\) −13086.4 −1.65437 −0.827187 0.561927i \(-0.810060\pi\)
−0.827187 + 0.561927i \(0.810060\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5253.36 −0.654215 −0.327107 0.944987i \(-0.606074\pi\)
−0.327107 + 0.944987i \(0.606074\pi\)
\(402\) 0 0
\(403\) 2124.62 0.262618
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12969.2 1.57951
\(408\) 0 0
\(409\) 4778.69 0.577729 0.288865 0.957370i \(-0.406722\pi\)
0.288865 + 0.957370i \(0.406722\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2201.20 0.262261
\(414\) 0 0
\(415\) 4066.84 0.481044
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −7770.14 −0.905958 −0.452979 0.891521i \(-0.649639\pi\)
−0.452979 + 0.891521i \(0.649639\pi\)
\(420\) 0 0
\(421\) −14440.5 −1.67170 −0.835852 0.548955i \(-0.815026\pi\)
−0.835852 + 0.548955i \(0.815026\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1252.21 −0.142920
\(426\) 0 0
\(427\) −3807.32 −0.431497
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −13416.1 −1.49937 −0.749686 0.661794i \(-0.769795\pi\)
−0.749686 + 0.661794i \(0.769795\pi\)
\(432\) 0 0
\(433\) −187.211 −0.0207778 −0.0103889 0.999946i \(-0.503307\pi\)
−0.0103889 + 0.999946i \(0.503307\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 343.732 0.0376269
\(438\) 0 0
\(439\) −2171.47 −0.236079 −0.118039 0.993009i \(-0.537661\pi\)
−0.118039 + 0.993009i \(0.537661\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 11129.8 1.19366 0.596830 0.802368i \(-0.296427\pi\)
0.596830 + 0.802368i \(0.296427\pi\)
\(444\) 0 0
\(445\) −4783.33 −0.509554
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −15602.0 −1.63988 −0.819938 0.572452i \(-0.805992\pi\)
−0.819938 + 0.572452i \(0.805992\pi\)
\(450\) 0 0
\(451\) 19580.4 2.04436
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −976.468 −0.100610
\(456\) 0 0
\(457\) −12065.3 −1.23499 −0.617495 0.786575i \(-0.711852\pi\)
−0.617495 + 0.786575i \(0.711852\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9703.61 0.980352 0.490176 0.871623i \(-0.336933\pi\)
0.490176 + 0.871623i \(0.336933\pi\)
\(462\) 0 0
\(463\) 9787.17 0.982394 0.491197 0.871048i \(-0.336559\pi\)
0.491197 + 0.871048i \(0.336559\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9051.97 0.896949 0.448474 0.893796i \(-0.351968\pi\)
0.448474 + 0.893796i \(0.351968\pi\)
\(468\) 0 0
\(469\) −21786.1 −2.14497
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 18918.3 1.83903
\(474\) 0 0
\(475\) −1744.46 −0.168508
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 8492.58 0.810096 0.405048 0.914295i \(-0.367255\pi\)
0.405048 + 0.914295i \(0.367255\pi\)
\(480\) 0 0
\(481\) 1827.04 0.173193
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −531.816 −0.0497907
\(486\) 0 0
\(487\) 7190.96 0.669104 0.334552 0.942377i \(-0.391415\pi\)
0.334552 + 0.942377i \(0.391415\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 20304.9 1.86629 0.933146 0.359499i \(-0.117052\pi\)
0.933146 + 0.359499i \(0.117052\pi\)
\(492\) 0 0
\(493\) −3978.41 −0.363445
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7015.88 −0.633210
\(498\) 0 0
\(499\) −5079.64 −0.455703 −0.227852 0.973696i \(-0.573170\pi\)
−0.227852 + 0.973696i \(0.573170\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1329.81 0.117880 0.0589398 0.998262i \(-0.481228\pi\)
0.0589398 + 0.998262i \(0.481228\pi\)
\(504\) 0 0
\(505\) 3515.41 0.309770
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6501.31 0.566140 0.283070 0.959099i \(-0.408647\pi\)
0.283070 + 0.959099i \(0.408647\pi\)
\(510\) 0 0
\(511\) −3398.04 −0.294169
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4841.80 0.414282
\(516\) 0 0
\(517\) 5196.41 0.442046
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −5846.09 −0.491597 −0.245798 0.969321i \(-0.579050\pi\)
−0.245798 + 0.969321i \(0.579050\pi\)
\(522\) 0 0
\(523\) 2544.28 0.212722 0.106361 0.994328i \(-0.466080\pi\)
0.106361 + 0.994328i \(0.466080\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 13037.7 1.07767
\(528\) 0 0
\(529\) −12142.7 −0.998006
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2758.39 0.224163
\(534\) 0 0
\(535\) 1831.97 0.148043
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −13294.9 −1.06243
\(540\) 0 0
\(541\) 19350.2 1.53776 0.768880 0.639393i \(-0.220814\pi\)
0.768880 + 0.639393i \(0.220814\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3618.07 −0.284369
\(546\) 0 0
\(547\) 4957.23 0.387488 0.193744 0.981052i \(-0.437937\pi\)
0.193744 + 0.981052i \(0.437937\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −5542.32 −0.428513
\(552\) 0 0
\(553\) 26474.3 2.03581
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1559.47 −0.118630 −0.0593148 0.998239i \(-0.518892\pi\)
−0.0593148 + 0.998239i \(0.518892\pi\)
\(558\) 0 0
\(559\) 2665.11 0.201650
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −7533.60 −0.563949 −0.281974 0.959422i \(-0.590989\pi\)
−0.281974 + 0.959422i \(0.590989\pi\)
\(564\) 0 0
\(565\) −4891.39 −0.364217
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 19257.9 1.41886 0.709432 0.704774i \(-0.248952\pi\)
0.709432 + 0.704774i \(0.248952\pi\)
\(570\) 0 0
\(571\) 26102.2 1.91304 0.956518 0.291675i \(-0.0942125\pi\)
0.956518 + 0.291675i \(0.0942125\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −123.152 −0.00893179
\(576\) 0 0
\(577\) −12553.3 −0.905721 −0.452860 0.891581i \(-0.649596\pi\)
−0.452860 + 0.891581i \(0.649596\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 19460.7 1.38961
\(582\) 0 0
\(583\) 31500.7 2.23778
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −16778.4 −1.17976 −0.589881 0.807490i \(-0.700825\pi\)
−0.589881 + 0.807490i \(0.700825\pi\)
\(588\) 0 0
\(589\) 18162.9 1.27061
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −21693.4 −1.50226 −0.751132 0.660152i \(-0.770492\pi\)
−0.751132 + 0.660152i \(0.770492\pi\)
\(594\) 0 0
\(595\) −5992.10 −0.412861
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −22980.4 −1.56753 −0.783767 0.621055i \(-0.786704\pi\)
−0.783767 + 0.621055i \(0.786704\pi\)
\(600\) 0 0
\(601\) −10048.9 −0.682038 −0.341019 0.940056i \(-0.610772\pi\)
−0.341019 + 0.940056i \(0.610772\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −10130.6 −0.680770
\(606\) 0 0
\(607\) −24514.6 −1.63923 −0.819617 0.572911i \(-0.805814\pi\)
−0.819617 + 0.572911i \(0.805814\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 732.043 0.0484702
\(612\) 0 0
\(613\) 1845.67 0.121609 0.0608043 0.998150i \(-0.480633\pi\)
0.0608043 + 0.998150i \(0.480633\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −9365.42 −0.611082 −0.305541 0.952179i \(-0.598837\pi\)
−0.305541 + 0.952179i \(0.598837\pi\)
\(618\) 0 0
\(619\) 19517.2 1.26731 0.633653 0.773618i \(-0.281555\pi\)
0.633653 + 0.773618i \(0.281555\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −22889.3 −1.47197
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 11211.6 0.710709
\(630\) 0 0
\(631\) −22314.1 −1.40778 −0.703892 0.710307i \(-0.748556\pi\)
−0.703892 + 0.710307i \(0.748556\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 14108.5 0.881700
\(636\) 0 0
\(637\) −1872.91 −0.116495
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −20150.8 −1.24167 −0.620833 0.783943i \(-0.713205\pi\)
−0.620833 + 0.783943i \(0.713205\pi\)
\(642\) 0 0
\(643\) −5345.25 −0.327832 −0.163916 0.986474i \(-0.552413\pi\)
−0.163916 + 0.986474i \(0.552413\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13701.4 0.832549 0.416274 0.909239i \(-0.363336\pi\)
0.416274 + 0.909239i \(0.363336\pi\)
\(648\) 0 0
\(649\) 5330.53 0.322406
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 713.119 0.0427358 0.0213679 0.999772i \(-0.493198\pi\)
0.0213679 + 0.999772i \(0.493198\pi\)
\(654\) 0 0
\(655\) −10659.1 −0.635856
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −26491.2 −1.56593 −0.782967 0.622063i \(-0.786295\pi\)
−0.782967 + 0.622063i \(0.786295\pi\)
\(660\) 0 0
\(661\) 645.746 0.0379979 0.0189989 0.999820i \(-0.493952\pi\)
0.0189989 + 0.999820i \(0.493952\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8347.59 −0.486775
\(666\) 0 0
\(667\) −391.266 −0.0227135
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −9220.01 −0.530454
\(672\) 0 0
\(673\) −6602.56 −0.378172 −0.189086 0.981961i \(-0.560553\pi\)
−0.189086 + 0.981961i \(0.560553\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4968.68 0.282070 0.141035 0.990005i \(-0.454957\pi\)
0.141035 + 0.990005i \(0.454957\pi\)
\(678\) 0 0
\(679\) −2544.85 −0.143833
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −24694.2 −1.38345 −0.691727 0.722159i \(-0.743150\pi\)
−0.691727 + 0.722159i \(0.743150\pi\)
\(684\) 0 0
\(685\) 4084.04 0.227800
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4437.66 0.245372
\(690\) 0 0
\(691\) 19259.2 1.06028 0.530141 0.847910i \(-0.322139\pi\)
0.530141 + 0.847910i \(0.322139\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9384.02 0.512167
\(696\) 0 0
\(697\) 16926.8 0.919871
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −31090.7 −1.67515 −0.837575 0.546323i \(-0.816027\pi\)
−0.837575 + 0.546323i \(0.816027\pi\)
\(702\) 0 0
\(703\) 15618.9 0.837948
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 16822.0 0.894845
\(708\) 0 0
\(709\) −20832.0 −1.10347 −0.551736 0.834019i \(-0.686034\pi\)
−0.551736 + 0.834019i \(0.686034\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1282.23 0.0673490
\(714\) 0 0
\(715\) −2364.66 −0.123683
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −21328.1 −1.10626 −0.553132 0.833093i \(-0.686568\pi\)
−0.553132 + 0.833093i \(0.686568\pi\)
\(720\) 0 0
\(721\) 23169.1 1.19676
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1985.69 0.101720
\(726\) 0 0
\(727\) −4121.56 −0.210262 −0.105131 0.994458i \(-0.533526\pi\)
−0.105131 + 0.994458i \(0.533526\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 16354.5 0.827485
\(732\) 0 0
\(733\) −11102.2 −0.559439 −0.279720 0.960082i \(-0.590241\pi\)
−0.279720 + 0.960082i \(0.590241\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −52758.4 −2.63688
\(738\) 0 0
\(739\) 13749.5 0.684414 0.342207 0.939625i \(-0.388826\pi\)
0.342207 + 0.939625i \(0.388826\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 13962.9 0.689433 0.344717 0.938707i \(-0.387975\pi\)
0.344717 + 0.938707i \(0.387975\pi\)
\(744\) 0 0
\(745\) −12713.5 −0.625219
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 8766.35 0.427657
\(750\) 0 0
\(751\) −32801.6 −1.59380 −0.796902 0.604108i \(-0.793530\pi\)
−0.796902 + 0.604108i \(0.793530\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1667.21 0.0803657
\(756\) 0 0
\(757\) −14664.9 −0.704102 −0.352051 0.935981i \(-0.614516\pi\)
−0.352051 + 0.935981i \(0.614516\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −3304.79 −0.157422 −0.0787112 0.996897i \(-0.525081\pi\)
−0.0787112 + 0.996897i \(0.525081\pi\)
\(762\) 0 0
\(763\) −17313.2 −0.821470
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 750.938 0.0353518
\(768\) 0 0
\(769\) −33085.5 −1.55149 −0.775743 0.631049i \(-0.782625\pi\)
−0.775743 + 0.631049i \(0.782625\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 27880.8 1.29729 0.648643 0.761093i \(-0.275337\pi\)
0.648643 + 0.761093i \(0.275337\pi\)
\(774\) 0 0
\(775\) −6507.36 −0.301614
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 23580.8 1.08456
\(780\) 0 0
\(781\) −16990.0 −0.778426
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −18250.1 −0.829775
\(786\) 0 0
\(787\) −17813.9 −0.806856 −0.403428 0.915011i \(-0.632181\pi\)
−0.403428 + 0.915011i \(0.632181\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −23406.4 −1.05213
\(792\) 0 0
\(793\) −1298.87 −0.0581641
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 8005.10 0.355778 0.177889 0.984051i \(-0.443073\pi\)
0.177889 + 0.984051i \(0.443073\pi\)
\(798\) 0 0
\(799\) 4492.19 0.198901
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −8228.86 −0.361632
\(804\) 0 0
\(805\) −589.307 −0.0258017
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 10140.7 0.440700 0.220350 0.975421i \(-0.429280\pi\)
0.220350 + 0.975421i \(0.429280\pi\)
\(810\) 0 0
\(811\) 14662.7 0.634865 0.317433 0.948281i \(-0.397179\pi\)
0.317433 + 0.948281i \(0.397179\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3727.90 0.160224
\(816\) 0 0
\(817\) 22783.4 0.975630
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −36216.1 −1.53953 −0.769763 0.638330i \(-0.779626\pi\)
−0.769763 + 0.638330i \(0.779626\pi\)
\(822\) 0 0
\(823\) 9695.36 0.410643 0.205321 0.978695i \(-0.434176\pi\)
0.205321 + 0.978695i \(0.434176\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −21471.9 −0.902844 −0.451422 0.892311i \(-0.649083\pi\)
−0.451422 + 0.892311i \(0.649083\pi\)
\(828\) 0 0
\(829\) 10447.9 0.437723 0.218861 0.975756i \(-0.429766\pi\)
0.218861 + 0.975756i \(0.429766\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −11493.1 −0.478047
\(834\) 0 0
\(835\) −16608.7 −0.688343
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 4917.61 0.202354 0.101177 0.994868i \(-0.467739\pi\)
0.101177 + 0.994868i \(0.467739\pi\)
\(840\) 0 0
\(841\) −18080.2 −0.741328
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 10651.9 0.433652
\(846\) 0 0
\(847\) −48476.9 −1.96657
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1102.63 0.0444157
\(852\) 0 0
\(853\) 45031.3 1.80755 0.903777 0.428004i \(-0.140783\pi\)
0.903777 + 0.428004i \(0.140783\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −14826.6 −0.590976 −0.295488 0.955346i \(-0.595482\pi\)
−0.295488 + 0.955346i \(0.595482\pi\)
\(858\) 0 0
\(859\) −43287.9 −1.71940 −0.859700 0.510799i \(-0.829350\pi\)
−0.859700 + 0.510799i \(0.829350\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −40830.8 −1.61054 −0.805270 0.592908i \(-0.797980\pi\)
−0.805270 + 0.592908i \(0.797980\pi\)
\(864\) 0 0
\(865\) −5192.89 −0.204120
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 64111.5 2.50268
\(870\) 0 0
\(871\) −7432.34 −0.289133
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2990.76 0.115550
\(876\) 0 0
\(877\) 2322.20 0.0894127 0.0447064 0.999000i \(-0.485765\pi\)
0.0447064 + 0.999000i \(0.485765\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 10407.8 0.398010 0.199005 0.979998i \(-0.436229\pi\)
0.199005 + 0.979998i \(0.436229\pi\)
\(882\) 0 0
\(883\) 14346.9 0.546784 0.273392 0.961903i \(-0.411854\pi\)
0.273392 + 0.961903i \(0.411854\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 22740.9 0.860838 0.430419 0.902629i \(-0.358366\pi\)
0.430419 + 0.902629i \(0.358366\pi\)
\(888\) 0 0
\(889\) 67512.2 2.54701
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6258.07 0.234511
\(894\) 0 0
\(895\) 1250.90 0.0467186
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −20674.6 −0.767003
\(900\) 0 0
\(901\) 27231.7 1.00690
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 5858.04 0.215169
\(906\) 0 0
\(907\) 14222.6 0.520678 0.260339 0.965517i \(-0.416166\pi\)
0.260339 + 0.965517i \(0.416166\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −6825.69 −0.248238 −0.124119 0.992267i \(-0.539610\pi\)
−0.124119 + 0.992267i \(0.539610\pi\)
\(912\) 0 0
\(913\) 47127.0 1.70830
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −51006.1 −1.83683
\(918\) 0 0
\(919\) −18851.0 −0.676647 −0.338323 0.941030i \(-0.609860\pi\)
−0.338323 + 0.941030i \(0.609860\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −2393.47 −0.0853542
\(924\) 0 0
\(925\) −5595.91 −0.198911
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −7109.70 −0.251089 −0.125544 0.992088i \(-0.540068\pi\)
−0.125544 + 0.992088i \(0.540068\pi\)
\(930\) 0 0
\(931\) −16011.1 −0.563633
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −14510.8 −0.507543
\(936\) 0 0
\(937\) −14107.6 −0.491864 −0.245932 0.969287i \(-0.579094\pi\)
−0.245932 + 0.969287i \(0.579094\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 49409.8 1.71170 0.855852 0.517221i \(-0.173034\pi\)
0.855852 + 0.517221i \(0.173034\pi\)
\(942\) 0 0
\(943\) 1664.71 0.0574872
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 24070.5 0.825963 0.412982 0.910739i \(-0.364487\pi\)
0.412982 + 0.910739i \(0.364487\pi\)
\(948\) 0 0
\(949\) −1159.24 −0.0396528
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −55644.1 −1.89138 −0.945692 0.325064i \(-0.894614\pi\)
−0.945692 + 0.325064i \(0.894614\pi\)
\(954\) 0 0
\(955\) −5457.95 −0.184937
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 19543.0 0.658057
\(960\) 0 0
\(961\) 37962.1 1.27428
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 857.377 0.0286010
\(966\) 0 0
\(967\) 34638.9 1.15193 0.575963 0.817476i \(-0.304627\pi\)
0.575963 + 0.817476i \(0.304627\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −28705.9 −0.948731 −0.474365 0.880328i \(-0.657322\pi\)
−0.474365 + 0.880328i \(0.657322\pi\)
\(972\) 0 0
\(973\) 44904.5 1.47952
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −45305.4 −1.48357 −0.741785 0.670638i \(-0.766021\pi\)
−0.741785 + 0.670638i \(0.766021\pi\)
\(978\) 0 0
\(979\) −55429.8 −1.80954
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −9220.63 −0.299178 −0.149589 0.988748i \(-0.547795\pi\)
−0.149589 + 0.988748i \(0.547795\pi\)
\(984\) 0 0
\(985\) −11843.5 −0.383112
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1608.42 0.0517136
\(990\) 0 0
\(991\) 23444.0 0.751488 0.375744 0.926723i \(-0.377387\pi\)
0.375744 + 0.926723i \(0.377387\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 16869.1 0.537474
\(996\) 0 0
\(997\) 9848.13 0.312832 0.156416 0.987691i \(-0.450006\pi\)
0.156416 + 0.987691i \(0.450006\pi\)
\(998\) 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.bf.1.1 3
3.2 odd 2 2160.4.a.bn.1.1 3
4.3 odd 2 1080.4.a.h.1.3 3
12.11 even 2 1080.4.a.n.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.h.1.3 3 4.3 odd 2
1080.4.a.n.1.3 yes 3 12.11 even 2
2160.4.a.bf.1.1 3 1.1 even 1 trivial
2160.4.a.bn.1.1 3 3.2 odd 2