Properties

Label 3528.2.a.bk.1.1
Level $3528$
Weight $2$
Character 3528.1
Self dual yes
Analytic conductor $28.171$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1712218331\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 3528.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.27492 q^{5} +O(q^{10})\) \(q-3.27492 q^{5} -3.27492 q^{11} +6.27492 q^{13} +4.00000 q^{17} -6.27492 q^{19} -4.00000 q^{23} +5.72508 q^{25} -5.27492 q^{29} -1.00000 q^{31} -2.27492 q^{37} +4.54983 q^{41} +0.274917 q^{43} +6.00000 q^{47} -9.27492 q^{53} +10.7251 q^{55} +1.27492 q^{59} +10.0000 q^{61} -20.5498 q^{65} -0.274917 q^{67} -2.00000 q^{71} -4.27492 q^{73} +11.5498 q^{79} -7.27492 q^{83} -13.0997 q^{85} +10.5498 q^{89} +20.5498 q^{95} +8.72508 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{5} + O(q^{10}) \) \( 2q + q^{5} + q^{11} + 5q^{13} + 8q^{17} - 5q^{19} - 8q^{23} + 19q^{25} - 3q^{29} - 2q^{31} + 3q^{37} - 6q^{41} - 7q^{43} + 12q^{47} - 11q^{53} + 29q^{55} - 5q^{59} + 20q^{61} - 26q^{65} + 7q^{67} - 4q^{71} - q^{73} + 8q^{79} - 7q^{83} + 4q^{85} + 6q^{89} + 26q^{95} + 25q^{97} + O(q^{100}) \)

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
\(5\) −3.27492 −1.46459 −0.732294 0.680989i \(-0.761550\pi\)
−0.732294 + 0.680989i \(0.761550\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.27492 −0.987425 −0.493712 0.869625i \(-0.664360\pi\)
−0.493712 + 0.869625i \(0.664360\pi\)
\(12\) 0 0
\(13\) 6.27492 1.74035 0.870174 0.492744i \(-0.164006\pi\)
0.870174 + 0.492744i \(0.164006\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −6.27492 −1.43956 −0.719782 0.694200i \(-0.755758\pi\)
−0.719782 + 0.694200i \(0.755758\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 5.72508 1.14502
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.27492 −0.979528 −0.489764 0.871855i \(-0.662917\pi\)
−0.489764 + 0.871855i \(0.662917\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.27492 −0.373994 −0.186997 0.982360i \(-0.559875\pi\)
−0.186997 + 0.982360i \(0.559875\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.54983 0.710565 0.355282 0.934759i \(-0.384385\pi\)
0.355282 + 0.934759i \(0.384385\pi\)
\(42\) 0 0
\(43\) 0.274917 0.0419245 0.0209622 0.999780i \(-0.493327\pi\)
0.0209622 + 0.999780i \(0.493327\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.27492 −1.27401 −0.637004 0.770861i \(-0.719827\pi\)
−0.637004 + 0.770861i \(0.719827\pi\)
\(54\) 0 0
\(55\) 10.7251 1.44617
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.27492 0.165980 0.0829900 0.996550i \(-0.473553\pi\)
0.0829900 + 0.996550i \(0.473553\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −20.5498 −2.54889
\(66\) 0 0
\(67\) −0.274917 −0.0335865 −0.0167932 0.999859i \(-0.505346\pi\)
−0.0167932 + 0.999859i \(0.505346\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) −4.27492 −0.500341 −0.250171 0.968202i \(-0.580487\pi\)
−0.250171 + 0.968202i \(0.580487\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 11.5498 1.29946 0.649729 0.760166i \(-0.274882\pi\)
0.649729 + 0.760166i \(0.274882\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.27492 −0.798526 −0.399263 0.916836i \(-0.630734\pi\)
−0.399263 + 0.916836i \(0.630734\pi\)
\(84\) 0 0
\(85\) −13.0997 −1.42086
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.5498 1.11828 0.559140 0.829073i \(-0.311131\pi\)
0.559140 + 0.829073i \(0.311131\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 20.5498 2.10837
\(96\) 0 0
\(97\) 8.72508 0.885898 0.442949 0.896547i \(-0.353932\pi\)
0.442949 + 0.896547i \(0.353932\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 10.8248 1.06659 0.533297 0.845928i \(-0.320953\pi\)
0.533297 + 0.845928i \(0.320953\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.8248 1.52984 0.764918 0.644127i \(-0.222779\pi\)
0.764918 + 0.644127i \(0.222779\pi\)
\(108\) 0 0
\(109\) 16.8248 1.61152 0.805759 0.592243i \(-0.201757\pi\)
0.805759 + 0.592243i \(0.201757\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.54983 0.428012 0.214006 0.976832i \(-0.431349\pi\)
0.214006 + 0.976832i \(0.431349\pi\)
\(114\) 0 0
\(115\) 13.0997 1.22155
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.274917 −0.0249925
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2.37459 −0.212389
\(126\) 0 0
\(127\) −6.45017 −0.572360 −0.286180 0.958176i \(-0.592385\pi\)
−0.286180 + 0.958176i \(0.592385\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −7.27492 −0.635612 −0.317806 0.948156i \(-0.602946\pi\)
−0.317806 + 0.948156i \(0.602946\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.45017 −0.123896 −0.0619480 0.998079i \(-0.519731\pi\)
−0.0619480 + 0.998079i \(0.519731\pi\)
\(138\) 0 0
\(139\) 8.27492 0.701869 0.350935 0.936400i \(-0.385864\pi\)
0.350935 + 0.936400i \(0.385864\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −20.5498 −1.71846
\(144\) 0 0
\(145\) 17.2749 1.43460
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 14.5498 1.19197 0.595984 0.802996i \(-0.296762\pi\)
0.595984 + 0.802996i \(0.296762\pi\)
\(150\) 0 0
\(151\) 22.3746 1.82082 0.910409 0.413709i \(-0.135767\pi\)
0.910409 + 0.413709i \(0.135767\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.27492 0.263048
\(156\) 0 0
\(157\) −0.549834 −0.0438816 −0.0219408 0.999759i \(-0.506985\pi\)
−0.0219408 + 0.999759i \(0.506985\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) 0 0
\(169\) 26.3746 2.02881
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 22.5498 1.71443 0.857216 0.514957i \(-0.172192\pi\)
0.857216 + 0.514957i \(0.172192\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 16.5498 1.23699 0.618496 0.785788i \(-0.287742\pi\)
0.618496 + 0.785788i \(0.287742\pi\)
\(180\) 0 0
\(181\) −18.8248 −1.39923 −0.699616 0.714519i \(-0.746646\pi\)
−0.699616 + 0.714519i \(0.746646\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 7.45017 0.547747
\(186\) 0 0
\(187\) −13.0997 −0.957943
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.54983 −0.329214 −0.164607 0.986359i \(-0.552636\pi\)
−0.164607 + 0.986359i \(0.552636\pi\)
\(192\) 0 0
\(193\) 0.450166 0.0324036 0.0162018 0.999869i \(-0.494843\pi\)
0.0162018 + 0.999869i \(0.494843\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.45017 0.103320 0.0516600 0.998665i \(-0.483549\pi\)
0.0516600 + 0.998665i \(0.483549\pi\)
\(198\) 0 0
\(199\) −5.09967 −0.361506 −0.180753 0.983529i \(-0.557853\pi\)
−0.180753 + 0.983529i \(0.557853\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −14.9003 −1.04068
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 20.5498 1.42146
\(210\) 0 0
\(211\) −27.6495 −1.90347 −0.951735 0.306921i \(-0.900701\pi\)
−0.951735 + 0.306921i \(0.900701\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.900331 −0.0614021
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 25.0997 1.68839
\(222\) 0 0
\(223\) −1.27492 −0.0853748 −0.0426874 0.999088i \(-0.513592\pi\)
−0.0426874 + 0.999088i \(0.513592\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.2749 0.748343 0.374171 0.927360i \(-0.377927\pi\)
0.374171 + 0.927360i \(0.377927\pi\)
\(228\) 0 0
\(229\) 18.2749 1.20764 0.603820 0.797121i \(-0.293644\pi\)
0.603820 + 0.797121i \(0.293644\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.549834 0.0360209 0.0180104 0.999838i \(-0.494267\pi\)
0.0180104 + 0.999838i \(0.494267\pi\)
\(234\) 0 0
\(235\) −19.6495 −1.28179
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −15.4502 −0.999388 −0.499694 0.866202i \(-0.666554\pi\)
−0.499694 + 0.866202i \(0.666554\pi\)
\(240\) 0 0
\(241\) 9.82475 0.632868 0.316434 0.948615i \(-0.397514\pi\)
0.316434 + 0.948615i \(0.397514\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −39.3746 −2.50534
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.3746 1.15979 0.579897 0.814690i \(-0.303093\pi\)
0.579897 + 0.814690i \(0.303093\pi\)
\(252\) 0 0
\(253\) 13.0997 0.823569
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −11.0997 −0.692378 −0.346189 0.938165i \(-0.612524\pi\)
−0.346189 + 0.938165i \(0.612524\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.45017 0.582722 0.291361 0.956613i \(-0.405892\pi\)
0.291361 + 0.956613i \(0.405892\pi\)
\(264\) 0 0
\(265\) 30.3746 1.86590
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −20.7251 −1.26363 −0.631815 0.775119i \(-0.717690\pi\)
−0.631815 + 0.775119i \(0.717690\pi\)
\(270\) 0 0
\(271\) 1.27492 0.0774457 0.0387229 0.999250i \(-0.487671\pi\)
0.0387229 + 0.999250i \(0.487671\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −18.7492 −1.13062
\(276\) 0 0
\(277\) −26.8248 −1.61174 −0.805872 0.592090i \(-0.798303\pi\)
−0.805872 + 0.592090i \(0.798303\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 26.5498 1.58383 0.791915 0.610631i \(-0.209084\pi\)
0.791915 + 0.610631i \(0.209084\pi\)
\(282\) 0 0
\(283\) −25.9244 −1.54105 −0.770523 0.637412i \(-0.780005\pi\)
−0.770523 + 0.637412i \(0.780005\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 27.8248 1.62554 0.812770 0.582585i \(-0.197959\pi\)
0.812770 + 0.582585i \(0.197959\pi\)
\(294\) 0 0
\(295\) −4.17525 −0.243092
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −25.0997 −1.45155
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −32.7492 −1.87521
\(306\) 0 0
\(307\) 11.3746 0.649182 0.324591 0.945854i \(-0.394773\pi\)
0.324591 + 0.945854i \(0.394773\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.54983 0.257997 0.128999 0.991645i \(-0.458824\pi\)
0.128999 + 0.991645i \(0.458824\pi\)
\(312\) 0 0
\(313\) −19.5498 −1.10502 −0.552511 0.833506i \(-0.686330\pi\)
−0.552511 + 0.833506i \(0.686330\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 27.8248 1.56279 0.781397 0.624034i \(-0.214507\pi\)
0.781397 + 0.624034i \(0.214507\pi\)
\(318\) 0 0
\(319\) 17.2749 0.967210
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −25.0997 −1.39658
\(324\) 0 0
\(325\) 35.9244 1.99273
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.17525 0.0645975 0.0322987 0.999478i \(-0.489717\pi\)
0.0322987 + 0.999478i \(0.489717\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0.900331 0.0491903
\(336\) 0 0
\(337\) −24.0997 −1.31279 −0.656396 0.754416i \(-0.727920\pi\)
−0.656396 + 0.754416i \(0.727920\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.27492 0.177347
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −30.1993 −1.62119 −0.810593 0.585610i \(-0.800855\pi\)
−0.810593 + 0.585610i \(0.800855\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 20.5498 1.09376 0.546879 0.837212i \(-0.315816\pi\)
0.546879 + 0.837212i \(0.315816\pi\)
\(354\) 0 0
\(355\) 6.54983 0.347629
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 19.6495 1.03706 0.518531 0.855059i \(-0.326479\pi\)
0.518531 + 0.855059i \(0.326479\pi\)
\(360\) 0 0
\(361\) 20.3746 1.07235
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 14.0000 0.732793
\(366\) 0 0
\(367\) 22.0997 1.15359 0.576797 0.816888i \(-0.304302\pi\)
0.576797 + 0.816888i \(0.304302\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6.27492 −0.324903 −0.162451 0.986717i \(-0.551940\pi\)
−0.162451 + 0.986717i \(0.551940\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −33.0997 −1.70472
\(378\) 0 0
\(379\) 13.1752 0.676767 0.338384 0.941008i \(-0.390120\pi\)
0.338384 + 0.941008i \(0.390120\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −10.5498 −0.539071 −0.269536 0.962990i \(-0.586870\pi\)
−0.269536 + 0.962990i \(0.586870\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −37.8248 −1.90317
\(396\) 0 0
\(397\) −3.37459 −0.169366 −0.0846828 0.996408i \(-0.526988\pi\)
−0.0846828 + 0.996408i \(0.526988\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 0 0
\(403\) −6.27492 −0.312576
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.45017 0.369291
\(408\) 0 0
\(409\) 10.4502 0.516727 0.258364 0.966048i \(-0.416817\pi\)
0.258364 + 0.966048i \(0.416817\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 23.8248 1.16951
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −28.5498 −1.39475 −0.697375 0.716706i \(-0.745649\pi\)
−0.697375 + 0.716706i \(0.745649\pi\)
\(420\) 0 0
\(421\) 8.82475 0.430092 0.215046 0.976604i \(-0.431010\pi\)
0.215046 + 0.976604i \(0.431010\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 22.9003 1.11083
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 17.6495 0.850147 0.425073 0.905159i \(-0.360248\pi\)
0.425073 + 0.905159i \(0.360248\pi\)
\(432\) 0 0
\(433\) 3.17525 0.152593 0.0762963 0.997085i \(-0.475690\pi\)
0.0762963 + 0.997085i \(0.475690\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 25.0997 1.20068
\(438\) 0 0
\(439\) 17.2749 0.824487 0.412243 0.911074i \(-0.364745\pi\)
0.412243 + 0.911074i \(0.364745\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.37459 0.302866 0.151433 0.988468i \(-0.451611\pi\)
0.151433 + 0.988468i \(0.451611\pi\)
\(444\) 0 0
\(445\) −34.5498 −1.63782
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −20.5498 −0.969807 −0.484903 0.874568i \(-0.661145\pi\)
−0.484903 + 0.874568i \(0.661145\pi\)
\(450\) 0 0
\(451\) −14.9003 −0.701629
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 36.6495 1.71439 0.857196 0.514991i \(-0.172205\pi\)
0.857196 + 0.514991i \(0.172205\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.64950 0.169974 0.0849872 0.996382i \(-0.472915\pi\)
0.0849872 + 0.996382i \(0.472915\pi\)
\(462\) 0 0
\(463\) 13.1752 0.612306 0.306153 0.951982i \(-0.400958\pi\)
0.306153 + 0.951982i \(0.400958\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −40.5498 −1.87642 −0.938211 0.346063i \(-0.887518\pi\)
−0.938211 + 0.346063i \(0.887518\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −0.900331 −0.0413973
\(474\) 0 0
\(475\) −35.9244 −1.64833
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5.45017 0.249024 0.124512 0.992218i \(-0.460263\pi\)
0.124512 + 0.992218i \(0.460263\pi\)
\(480\) 0 0
\(481\) −14.2749 −0.650880
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −28.5739 −1.29748
\(486\) 0 0
\(487\) 1.00000 0.0453143 0.0226572 0.999743i \(-0.492787\pi\)
0.0226572 + 0.999743i \(0.492787\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −36.9244 −1.66638 −0.833188 0.552990i \(-0.813487\pi\)
−0.833188 + 0.552990i \(0.813487\pi\)
\(492\) 0 0
\(493\) −21.0997 −0.950281
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 32.2749 1.44482 0.722412 0.691463i \(-0.243034\pi\)
0.722412 + 0.691463i \(0.243034\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 37.6495 1.67871 0.839354 0.543585i \(-0.182933\pi\)
0.839354 + 0.543585i \(0.182933\pi\)
\(504\) 0 0
\(505\) −19.6495 −0.874391
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 11.2749 0.499752 0.249876 0.968278i \(-0.419610\pi\)
0.249876 + 0.968278i \(0.419610\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −35.4502 −1.56212
\(516\) 0 0
\(517\) −19.6495 −0.864184
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 14.5498 0.637440 0.318720 0.947849i \(-0.396747\pi\)
0.318720 + 0.947849i \(0.396747\pi\)
\(522\) 0 0
\(523\) −17.7251 −0.775064 −0.387532 0.921856i \(-0.626672\pi\)
−0.387532 + 0.921856i \(0.626672\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 28.5498 1.23663
\(534\) 0 0
\(535\) −51.8248 −2.24058
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −8.27492 −0.355766 −0.177883 0.984052i \(-0.556925\pi\)
−0.177883 + 0.984052i \(0.556925\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −55.0997 −2.36021
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 33.0997 1.41009
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 29.8248 1.26372 0.631858 0.775084i \(-0.282293\pi\)
0.631858 + 0.775084i \(0.282293\pi\)
\(558\) 0 0
\(559\) 1.72508 0.0729632
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −15.2749 −0.643761 −0.321881 0.946780i \(-0.604315\pi\)
−0.321881 + 0.946780i \(0.604315\pi\)
\(564\) 0 0
\(565\) −14.9003 −0.626862
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −11.4502 −0.480016 −0.240008 0.970771i \(-0.577150\pi\)
−0.240008 + 0.970771i \(0.577150\pi\)
\(570\) 0 0
\(571\) 8.27492 0.346295 0.173147 0.984896i \(-0.444606\pi\)
0.173147 + 0.984896i \(0.444606\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −22.9003 −0.955010
\(576\) 0 0
\(577\) −25.0000 −1.04076 −0.520382 0.853934i \(-0.674210\pi\)
−0.520382 + 0.853934i \(0.674210\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 30.3746 1.25799
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −9.27492 −0.382817 −0.191408 0.981510i \(-0.561305\pi\)
−0.191408 + 0.981510i \(0.561305\pi\)
\(588\) 0 0
\(589\) 6.27492 0.258553
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0.549834 0.0225790 0.0112895 0.999936i \(-0.496406\pi\)
0.0112895 + 0.999936i \(0.496406\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −22.5498 −0.921361 −0.460681 0.887566i \(-0.652395\pi\)
−0.460681 + 0.887566i \(0.652395\pi\)
\(600\) 0 0
\(601\) 4.09967 0.167229 0.0836145 0.996498i \(-0.473354\pi\)
0.0836145 + 0.996498i \(0.473354\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.900331 0.0366037
\(606\) 0 0
\(607\) −7.00000 −0.284121 −0.142061 0.989858i \(-0.545373\pi\)
−0.142061 + 0.989858i \(0.545373\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 37.6495 1.52314
\(612\) 0 0
\(613\) −8.54983 −0.345325 −0.172662 0.984981i \(-0.555237\pi\)
−0.172662 + 0.984981i \(0.555237\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 28.8248 1.15856 0.579282 0.815127i \(-0.303333\pi\)
0.579282 + 0.815127i \(0.303333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −20.8488 −0.833954
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −9.09967 −0.362828
\(630\) 0 0
\(631\) 19.8248 0.789211 0.394605 0.918851i \(-0.370881\pi\)
0.394605 + 0.918851i \(0.370881\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 21.1238 0.838271
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.64950 0.144147 0.0720734 0.997399i \(-0.477038\pi\)
0.0720734 + 0.997399i \(0.477038\pi\)
\(642\) 0 0
\(643\) 5.37459 0.211953 0.105976 0.994369i \(-0.466203\pi\)
0.105976 + 0.994369i \(0.466203\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 34.0000 1.33668 0.668339 0.743857i \(-0.267006\pi\)
0.668339 + 0.743857i \(0.267006\pi\)
\(648\) 0 0
\(649\) −4.17525 −0.163893
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.92442 0.270974 0.135487 0.990779i \(-0.456740\pi\)
0.135487 + 0.990779i \(0.456740\pi\)
\(654\) 0 0
\(655\) 23.8248 0.930910
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 42.1993 1.64385 0.821926 0.569594i \(-0.192899\pi\)
0.821926 + 0.569594i \(0.192899\pi\)
\(660\) 0 0
\(661\) 7.17525 0.279085 0.139542 0.990216i \(-0.455437\pi\)
0.139542 + 0.990216i \(0.455437\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 21.0997 0.816982
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −32.7492 −1.26427
\(672\) 0 0
\(673\) 41.5498 1.60163 0.800814 0.598913i \(-0.204400\pi\)
0.800814 + 0.598913i \(0.204400\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13.2749 0.510197 0.255098 0.966915i \(-0.417892\pi\)
0.255098 + 0.966915i \(0.417892\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12.1752 −0.465873 −0.232936 0.972492i \(-0.574833\pi\)
−0.232936 + 0.972492i \(0.574833\pi\)
\(684\) 0 0
\(685\) 4.74917 0.181457
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −58.1993 −2.21722
\(690\) 0 0
\(691\) −10.8248 −0.411793 −0.205896 0.978574i \(-0.566011\pi\)
−0.205896 + 0.978574i \(0.566011\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −27.0997 −1.02795
\(696\) 0 0
\(697\) 18.1993 0.689349
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 12.9244 0.488149 0.244074 0.969757i \(-0.421516\pi\)
0.244074 + 0.969757i \(0.421516\pi\)
\(702\) 0 0
\(703\) 14.2749 0.538389
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −28.1993 −1.05905 −0.529524 0.848295i \(-0.677630\pi\)
−0.529524 + 0.848295i \(0.677630\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) 67.2990 2.51684
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −28.1993 −1.05166 −0.525829 0.850590i \(-0.676245\pi\)
−0.525829 + 0.850590i \(0.676245\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −30.1993 −1.12158
\(726\) 0 0
\(727\) 31.5498 1.17012 0.585059 0.810991i \(-0.301071\pi\)
0.585059 + 0.810991i \(0.301071\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.09967 0.0406727
\(732\) 0 0
\(733\) 5.92442 0.218823 0.109412 0.993997i \(-0.465103\pi\)
0.109412 + 0.993997i \(0.465103\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.900331 0.0331641
\(738\) 0 0
\(739\) 1.37459 0.0505650 0.0252825 0.999680i \(-0.491951\pi\)
0.0252825 + 0.999680i \(0.491951\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 44.1993 1.62152 0.810758 0.585381i \(-0.199055\pi\)
0.810758 + 0.585381i \(0.199055\pi\)
\(744\) 0 0
\(745\) −47.6495 −1.74574
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −10.4502 −0.381332 −0.190666 0.981655i \(-0.561065\pi\)
−0.190666 + 0.981655i \(0.561065\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −73.2749 −2.66675
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 25.0997 0.909862 0.454931 0.890527i \(-0.349664\pi\)
0.454931 + 0.890527i \(0.349664\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) −32.6495 −1.17737 −0.588686 0.808362i \(-0.700354\pi\)
−0.588686 + 0.808362i \(0.700354\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −3.09967 −0.111487 −0.0557437 0.998445i \(-0.517753\pi\)
−0.0557437 + 0.998445i \(0.517753\pi\)
\(774\) 0 0
\(775\) −5.72508 −0.205651
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −28.5498 −1.02290
\(780\) 0 0
\(781\) 6.54983 0.234372
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.80066 0.0642684
\(786\) 0 0
\(787\) 2.54983 0.0908918 0.0454459 0.998967i \(-0.485529\pi\)
0.0454459 + 0.998967i \(0.485529\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 62.7492 2.22829
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 31.4743 1.11488 0.557438 0.830219i \(-0.311785\pi\)
0.557438 + 0.830219i \(0.311785\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 14.0000 0.494049
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 29.6495 1.04242 0.521211 0.853428i \(-0.325481\pi\)
0.521211 + 0.853428i \(0.325481\pi\)
\(810\) 0 0
\(811\) −42.5498 −1.49413 −0.747063 0.664753i \(-0.768537\pi\)
−0.747063 + 0.664753i \(0.768537\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 39.2990 1.37658
\(816\) 0 0
\(817\) −1.72508 −0.0603530
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −12.7251 −0.444108 −0.222054 0.975034i \(-0.571276\pi\)
−0.222054 + 0.975034i \(0.571276\pi\)
\(822\) 0 0
\(823\) −34.1993 −1.19211 −0.596057 0.802942i \(-0.703267\pi\)
−0.596057 + 0.802942i \(0.703267\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 44.0241 1.53087 0.765434 0.643515i \(-0.222524\pi\)
0.765434 + 0.643515i \(0.222524\pi\)
\(828\) 0 0
\(829\) 30.2749 1.05149 0.525746 0.850642i \(-0.323786\pi\)
0.525746 + 0.850642i \(0.323786\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −19.6495 −0.679999
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 30.1993 1.04260 0.521298 0.853374i \(-0.325448\pi\)
0.521298 + 0.853374i \(0.325448\pi\)
\(840\) 0 0
\(841\) −1.17525 −0.0405258
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −86.3746 −2.97138
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 9.09967 0.311933
\(852\) 0 0
\(853\) 13.3746 0.457937 0.228969 0.973434i \(-0.426465\pi\)
0.228969 + 0.973434i \(0.426465\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 13.4502 0.459449 0.229724 0.973256i \(-0.426218\pi\)
0.229724 + 0.973256i \(0.426218\pi\)
\(858\) 0 0
\(859\) −55.6495 −1.89874 −0.949368 0.314165i \(-0.898275\pi\)
−0.949368 + 0.314165i \(0.898275\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −19.0997 −0.650160 −0.325080 0.945686i \(-0.605391\pi\)
−0.325080 + 0.945686i \(0.605391\pi\)
\(864\) 0 0
\(865\) −73.8488 −2.51094
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −37.8248 −1.28312
\(870\) 0 0
\(871\) −1.72508 −0.0584522
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 46.7492 1.57861 0.789304 0.614003i \(-0.210442\pi\)
0.789304 + 0.614003i \(0.210442\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −9.45017 −0.318384 −0.159192 0.987248i \(-0.550889\pi\)
−0.159192 + 0.987248i \(0.550889\pi\)
\(882\) 0 0
\(883\) 7.37459 0.248175 0.124087 0.992271i \(-0.460400\pi\)
0.124087 + 0.992271i \(0.460400\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −33.6495 −1.12984 −0.564920 0.825146i \(-0.691093\pi\)
−0.564920 + 0.825146i \(0.691093\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −37.6495 −1.25989
\(894\) 0 0
\(895\) −54.1993 −1.81168
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5.27492 0.175928
\(900\) 0 0
\(901\) −37.0997 −1.23597
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 61.6495 2.04930
\(906\) 0 0
\(907\) −29.7251 −0.987005 −0.493503 0.869744i \(-0.664284\pi\)
−0.493503 + 0.869744i \(0.664284\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −53.8488 −1.78409 −0.892046 0.451945i \(-0.850730\pi\)
−0.892046 + 0.451945i \(0.850730\pi\)
\(912\) 0 0
\(913\) 23.8248 0.788484
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −2.82475 −0.0931800 −0.0465900 0.998914i \(-0.514835\pi\)
−0.0465900 + 0.998914i \(0.514835\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −12.5498 −0.413083
\(924\) 0 0
\(925\) −13.0241 −0.428229
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 40.1993 1.31890 0.659449 0.751750i \(-0.270790\pi\)
0.659449 + 0.751750i \(0.270790\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 42.9003 1.40299
\(936\) 0 0
\(937\) −6.09967 −0.199267 −0.0996337 0.995024i \(-0.531767\pi\)
−0.0996337 + 0.995024i \(0.531767\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −51.8248 −1.68944 −0.844719 0.535210i \(-0.820233\pi\)
−0.844719 + 0.535210i \(0.820233\pi\)
\(942\) 0 0
\(943\) −18.1993 −0.592652
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 19.4502 0.632045 0.316023 0.948752i \(-0.397652\pi\)
0.316023 + 0.948752i \(0.397652\pi\)
\(948\) 0 0
\(949\) −26.8248 −0.870768
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −55.6495 −1.80266 −0.901332 0.433129i \(-0.857410\pi\)
−0.901332 + 0.433129i \(0.857410\pi\)
\(954\) 0 0
\(955\) 14.9003 0.482163
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.47425 −0.0474579
\(966\) 0 0
\(967\) −53.5498 −1.72205 −0.861023 0.508566i \(-0.830176\pi\)
−0.861023 + 0.508566i \(0.830176\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −56.5739 −1.81554 −0.907772 0.419464i \(-0.862218\pi\)
−0.907772 + 0.419464i \(0.862218\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −8.90033 −0.284747 −0.142373 0.989813i \(-0.545473\pi\)
−0.142373 + 0.989813i \(0.545473\pi\)
\(978\) 0 0
\(979\) −34.5498 −1.10422
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −39.2990 −1.25344 −0.626722 0.779243i \(-0.715604\pi\)
−0.626722 + 0.779243i \(0.715604\pi\)
\(984\) 0 0
\(985\) −4.74917 −0.151321
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.09967 −0.0349674
\(990\) 0 0
\(991\) −14.0997 −0.447891 −0.223945 0.974602i \(-0.571894\pi\)
−0.223945 + 0.974602i \(0.571894\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 16.7010 0.529457
\(996\) 0 0
\(997\) −44.8248 −1.41961 −0.709807 0.704396i \(-0.751218\pi\)
−0.709807 + 0.704396i \(0.751218\pi\)
\(998\) 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 3528.2.a.bk.1.1 2
3.2 odd 2 1176.2.a.k.1.2 2
4.3 odd 2 7056.2.a.cu.1.1 2
7.2 even 3 504.2.s.i.361.2 4
7.3 odd 6 3528.2.s.bk.3313.1 4
7.4 even 3 504.2.s.i.289.2 4
7.5 odd 6 3528.2.s.bk.361.1 4
7.6 odd 2 3528.2.a.bd.1.2 2
12.11 even 2 2352.2.a.bf.1.2 2
21.2 odd 6 168.2.q.c.25.1 4
21.5 even 6 1176.2.q.l.361.2 4
21.11 odd 6 168.2.q.c.121.1 yes 4
21.17 even 6 1176.2.q.l.961.2 4
21.20 even 2 1176.2.a.n.1.1 2
24.5 odd 2 9408.2.a.ec.1.1 2
24.11 even 2 9408.2.a.dp.1.1 2
28.11 odd 6 1008.2.s.r.289.2 4
28.23 odd 6 1008.2.s.r.865.2 4
28.27 even 2 7056.2.a.ch.1.2 2
84.11 even 6 336.2.q.g.289.1 4
84.23 even 6 336.2.q.g.193.1 4
84.47 odd 6 2352.2.q.bf.1537.2 4
84.59 odd 6 2352.2.q.bf.961.2 4
84.83 odd 2 2352.2.a.ba.1.1 2
168.11 even 6 1344.2.q.x.961.2 4
168.53 odd 6 1344.2.q.w.961.2 4
168.83 odd 2 9408.2.a.dw.1.2 2
168.107 even 6 1344.2.q.x.193.2 4
168.125 even 2 9408.2.a.dj.1.2 2
168.149 odd 6 1344.2.q.w.193.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.q.c.25.1 4 21.2 odd 6
168.2.q.c.121.1 yes 4 21.11 odd 6
336.2.q.g.193.1 4 84.23 even 6
336.2.q.g.289.1 4 84.11 even 6
504.2.s.i.289.2 4 7.4 even 3
504.2.s.i.361.2 4 7.2 even 3
1008.2.s.r.289.2 4 28.11 odd 6
1008.2.s.r.865.2 4 28.23 odd 6
1176.2.a.k.1.2 2 3.2 odd 2
1176.2.a.n.1.1 2 21.20 even 2
1176.2.q.l.361.2 4 21.5 even 6
1176.2.q.l.961.2 4 21.17 even 6
1344.2.q.w.193.2 4 168.149 odd 6
1344.2.q.w.961.2 4 168.53 odd 6
1344.2.q.x.193.2 4 168.107 even 6
1344.2.q.x.961.2 4 168.11 even 6
2352.2.a.ba.1.1 2 84.83 odd 2
2352.2.a.bf.1.2 2 12.11 even 2
2352.2.q.bf.961.2 4 84.59 odd 6
2352.2.q.bf.1537.2 4 84.47 odd 6
3528.2.a.bd.1.2 2 7.6 odd 2
3528.2.a.bk.1.1 2 1.1 even 1 trivial
3528.2.s.bk.361.1 4 7.5 odd 6
3528.2.s.bk.3313.1 4 7.3 odd 6
7056.2.a.ch.1.2 2 28.27 even 2
7056.2.a.cu.1.1 2 4.3 odd 2
9408.2.a.dj.1.2 2 168.125 even 2
9408.2.a.dp.1.1 2 24.11 even 2
9408.2.a.dw.1.2 2 168.83 odd 2
9408.2.a.ec.1.1 2 24.5 odd 2