Properties

Label 8280.2.a.z.1.2
Level $8280$
Weight $2$
Character 8280.1
Self dual yes
Analytic conductor $66.116$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1161328736\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2760)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 8280.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} +3.37228 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +3.37228 q^{7} -4.00000 q^{11} +4.74456 q^{13} -5.37228 q^{17} +4.00000 q^{19} -1.00000 q^{23} +1.00000 q^{25} +5.37228 q^{29} +3.37228 q^{31} -3.37228 q^{35} -5.37228 q^{37} +8.11684 q^{41} -4.00000 q^{43} +6.74456 q^{47} +4.37228 q^{49} -1.37228 q^{53} +4.00000 q^{55} -8.62772 q^{59} -2.00000 q^{61} -4.74456 q^{65} +15.3723 q^{67} -3.37228 q^{71} -12.7446 q^{73} -13.4891 q^{77} -13.4891 q^{79} +12.8614 q^{83} +5.37228 q^{85} +4.74456 q^{89} +16.0000 q^{91} -4.00000 q^{95} +15.4891 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + q^{7} - 8 q^{11} - 2 q^{13} - 5 q^{17} + 8 q^{19} - 2 q^{23} + 2 q^{25} + 5 q^{29} + q^{31} - q^{35} - 5 q^{37} - q^{41} - 8 q^{43} + 2 q^{47} + 3 q^{49} + 3 q^{53} + 8 q^{55} - 23 q^{59} - 4 q^{61} + 2 q^{65} + 25 q^{67} - q^{71} - 14 q^{73} - 4 q^{77} - 4 q^{79} - 3 q^{83} + 5 q^{85} - 2 q^{89} + 32 q^{91} - 8 q^{95} + 8 q^{97}+O(q^{100}) \) 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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.37228 1.27460 0.637301 0.770615i \(-0.280051\pi\)
0.637301 + 0.770615i \(0.280051\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 4.74456 1.31590 0.657952 0.753059i \(-0.271423\pi\)
0.657952 + 0.753059i \(0.271423\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.37228 −1.30297 −0.651485 0.758662i \(-0.725854\pi\)
−0.651485 + 0.758662i \(0.725854\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.37228 0.997608 0.498804 0.866715i \(-0.333773\pi\)
0.498804 + 0.866715i \(0.333773\pi\)
\(30\) 0 0
\(31\) 3.37228 0.605680 0.302840 0.953041i \(-0.402065\pi\)
0.302840 + 0.953041i \(0.402065\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.37228 −0.570020
\(36\) 0 0
\(37\) −5.37228 −0.883198 −0.441599 0.897213i \(-0.645589\pi\)
−0.441599 + 0.897213i \(0.645589\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.11684 1.26764 0.633819 0.773481i \(-0.281486\pi\)
0.633819 + 0.773481i \(0.281486\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.74456 0.983796 0.491898 0.870653i \(-0.336303\pi\)
0.491898 + 0.870653i \(0.336303\pi\)
\(48\) 0 0
\(49\) 4.37228 0.624612
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.37228 −0.188497 −0.0942487 0.995549i \(-0.530045\pi\)
−0.0942487 + 0.995549i \(0.530045\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.62772 −1.12323 −0.561617 0.827398i \(-0.689820\pi\)
−0.561617 + 0.827398i \(0.689820\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.74456 −0.588491
\(66\) 0 0
\(67\) 15.3723 1.87802 0.939012 0.343886i \(-0.111743\pi\)
0.939012 + 0.343886i \(0.111743\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.37228 −0.400216 −0.200108 0.979774i \(-0.564129\pi\)
−0.200108 + 0.979774i \(0.564129\pi\)
\(72\) 0 0
\(73\) −12.7446 −1.49164 −0.745819 0.666149i \(-0.767942\pi\)
−0.745819 + 0.666149i \(0.767942\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −13.4891 −1.53723
\(78\) 0 0
\(79\) −13.4891 −1.51765 −0.758823 0.651297i \(-0.774225\pi\)
−0.758823 + 0.651297i \(0.774225\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.8614 1.41172 0.705861 0.708350i \(-0.250560\pi\)
0.705861 + 0.708350i \(0.250560\pi\)
\(84\) 0 0
\(85\) 5.37228 0.582706
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.74456 0.502923 0.251461 0.967867i \(-0.419089\pi\)
0.251461 + 0.967867i \(0.419089\pi\)
\(90\) 0 0
\(91\) 16.0000 1.67726
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) 15.4891 1.57268 0.786341 0.617792i \(-0.211973\pi\)
0.786341 + 0.617792i \(0.211973\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.3723 1.33059 0.665296 0.746580i \(-0.268305\pi\)
0.665296 + 0.746580i \(0.268305\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.37228 0.712705 0.356353 0.934352i \(-0.384020\pi\)
0.356353 + 0.934352i \(0.384020\pi\)
\(108\) 0 0
\(109\) 4.74456 0.454447 0.227223 0.973843i \(-0.427035\pi\)
0.227223 + 0.973843i \(0.427035\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.11684 0.763568 0.381784 0.924251i \(-0.375310\pi\)
0.381784 + 0.924251i \(0.375310\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −18.1168 −1.66077
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 1.25544 0.111402 0.0557010 0.998447i \(-0.482261\pi\)
0.0557010 + 0.998447i \(0.482261\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 13.4891 1.16966
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.4891 0.981582 0.490791 0.871277i \(-0.336708\pi\)
0.490791 + 0.871277i \(0.336708\pi\)
\(138\) 0 0
\(139\) 8.62772 0.731794 0.365897 0.930655i \(-0.380762\pi\)
0.365897 + 0.930655i \(0.380762\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −18.9783 −1.58704
\(144\) 0 0
\(145\) −5.37228 −0.446144
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.48913 −0.285840 −0.142920 0.989734i \(-0.545649\pi\)
−0.142920 + 0.989734i \(0.545649\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.37228 −0.270868
\(156\) 0 0
\(157\) −17.6060 −1.40511 −0.702555 0.711630i \(-0.747957\pi\)
−0.702555 + 0.711630i \(0.747957\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.37228 −0.265773
\(162\) 0 0
\(163\) 5.25544 0.411638 0.205819 0.978590i \(-0.434014\pi\)
0.205819 + 0.978590i \(0.434014\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.25544 −0.0971487 −0.0485743 0.998820i \(-0.515468\pi\)
−0.0485743 + 0.998820i \(0.515468\pi\)
\(168\) 0 0
\(169\) 9.51087 0.731606
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 14.2337 1.08217 0.541084 0.840969i \(-0.318014\pi\)
0.541084 + 0.840969i \(0.318014\pi\)
\(174\) 0 0
\(175\) 3.37228 0.254921
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −9.48913 −0.709251 −0.354625 0.935009i \(-0.615392\pi\)
−0.354625 + 0.935009i \(0.615392\pi\)
\(180\) 0 0
\(181\) −14.2337 −1.05798 −0.528991 0.848628i \(-0.677429\pi\)
−0.528991 + 0.848628i \(0.677429\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.37228 0.394978
\(186\) 0 0
\(187\) 21.4891 1.57144
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −20.2337 −1.46406 −0.732029 0.681273i \(-0.761427\pi\)
−0.732029 + 0.681273i \(0.761427\pi\)
\(192\) 0 0
\(193\) −20.7446 −1.49323 −0.746613 0.665258i \(-0.768321\pi\)
−0.746613 + 0.665258i \(0.768321\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −11.4891 −0.818566 −0.409283 0.912407i \(-0.634221\pi\)
−0.409283 + 0.912407i \(0.634221\pi\)
\(198\) 0 0
\(199\) 1.25544 0.0889956 0.0444978 0.999009i \(-0.485831\pi\)
0.0444978 + 0.999009i \(0.485831\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 18.1168 1.27155
\(204\) 0 0
\(205\) −8.11684 −0.566905
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 20.8614 1.43616 0.718079 0.695961i \(-0.245022\pi\)
0.718079 + 0.695961i \(0.245022\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 11.3723 0.772001
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −25.4891 −1.71458
\(222\) 0 0
\(223\) 17.2554 1.15551 0.577755 0.816210i \(-0.303929\pi\)
0.577755 + 0.816210i \(0.303929\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 28.7446 1.89949 0.949747 0.313018i \(-0.101340\pi\)
0.949747 + 0.313018i \(0.101340\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 26.2337 1.71863 0.859313 0.511450i \(-0.170891\pi\)
0.859313 + 0.511450i \(0.170891\pi\)
\(234\) 0 0
\(235\) −6.74456 −0.439967
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 11.3723 0.735612 0.367806 0.929903i \(-0.380109\pi\)
0.367806 + 0.929903i \(0.380109\pi\)
\(240\) 0 0
\(241\) −7.25544 −0.467364 −0.233682 0.972313i \(-0.575077\pi\)
−0.233682 + 0.972313i \(0.575077\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.37228 −0.279335
\(246\) 0 0
\(247\) 18.9783 1.20756
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.7446 1.18315 0.591573 0.806251i \(-0.298507\pi\)
0.591573 + 0.806251i \(0.298507\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 11.4891 0.716672 0.358336 0.933593i \(-0.383344\pi\)
0.358336 + 0.933593i \(0.383344\pi\)
\(258\) 0 0
\(259\) −18.1168 −1.12573
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −0.861407 −0.0531166 −0.0265583 0.999647i \(-0.508455\pi\)
−0.0265583 + 0.999647i \(0.508455\pi\)
\(264\) 0 0
\(265\) 1.37228 0.0842986
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.8614 −0.906116 −0.453058 0.891481i \(-0.649667\pi\)
−0.453058 + 0.891481i \(0.649667\pi\)
\(270\) 0 0
\(271\) −5.88316 −0.357376 −0.178688 0.983906i \(-0.557185\pi\)
−0.178688 + 0.983906i \(0.557185\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −22.2337 −1.32635 −0.663175 0.748464i \(-0.730792\pi\)
−0.663175 + 0.748464i \(0.730792\pi\)
\(282\) 0 0
\(283\) 0.627719 0.0373140 0.0186570 0.999826i \(-0.494061\pi\)
0.0186570 + 0.999826i \(0.494061\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 27.3723 1.61573
\(288\) 0 0
\(289\) 11.8614 0.697730
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 18.8614 1.10190 0.550948 0.834540i \(-0.314266\pi\)
0.550948 + 0.834540i \(0.314266\pi\)
\(294\) 0 0
\(295\) 8.62772 0.502325
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.74456 −0.274385
\(300\) 0 0
\(301\) −13.4891 −0.777500
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 2.86141 0.161736 0.0808681 0.996725i \(-0.474231\pi\)
0.0808681 + 0.996725i \(0.474231\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −8.51087 −0.478018 −0.239009 0.971017i \(-0.576823\pi\)
−0.239009 + 0.971017i \(0.576823\pi\)
\(318\) 0 0
\(319\) −21.4891 −1.20316
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −21.4891 −1.19569
\(324\) 0 0
\(325\) 4.74456 0.263181
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 22.7446 1.25395
\(330\) 0 0
\(331\) 1.88316 0.103508 0.0517538 0.998660i \(-0.483519\pi\)
0.0517538 + 0.998660i \(0.483519\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −15.3723 −0.839877
\(336\) 0 0
\(337\) −27.4891 −1.49743 −0.748714 0.662893i \(-0.769328\pi\)
−0.748714 + 0.662893i \(0.769328\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −13.4891 −0.730477
\(342\) 0 0
\(343\) −8.86141 −0.478471
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) −4.11684 −0.220370 −0.110185 0.993911i \(-0.535144\pi\)
−0.110185 + 0.993911i \(0.535144\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) 3.37228 0.178982
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 25.7228 1.35760 0.678799 0.734324i \(-0.262501\pi\)
0.678799 + 0.734324i \(0.262501\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.7446 0.667081
\(366\) 0 0
\(367\) −24.8614 −1.29775 −0.648877 0.760893i \(-0.724761\pi\)
−0.648877 + 0.760893i \(0.724761\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.62772 −0.240259
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 25.4891 1.31276
\(378\) 0 0
\(379\) 9.48913 0.487424 0.243712 0.969848i \(-0.421635\pi\)
0.243712 + 0.969848i \(0.421635\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 18.1168 0.925727 0.462864 0.886429i \(-0.346822\pi\)
0.462864 + 0.886429i \(0.346822\pi\)
\(384\) 0 0
\(385\) 13.4891 0.687469
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.48913 −0.176906 −0.0884528 0.996080i \(-0.528192\pi\)
−0.0884528 + 0.996080i \(0.528192\pi\)
\(390\) 0 0
\(391\) 5.37228 0.271688
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 13.4891 0.678712
\(396\) 0 0
\(397\) 34.2337 1.71814 0.859070 0.511859i \(-0.171043\pi\)
0.859070 + 0.511859i \(0.171043\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −23.4891 −1.17299 −0.586495 0.809953i \(-0.699493\pi\)
−0.586495 + 0.809953i \(0.699493\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 21.4891 1.06518
\(408\) 0 0
\(409\) 14.6277 0.723294 0.361647 0.932315i \(-0.382215\pi\)
0.361647 + 0.932315i \(0.382215\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −29.0951 −1.43168
\(414\) 0 0
\(415\) −12.8614 −0.631342
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −3.25544 −0.158660 −0.0793302 0.996848i \(-0.525278\pi\)
−0.0793302 + 0.996848i \(0.525278\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5.37228 −0.260594
\(426\) 0 0
\(427\) −6.74456 −0.326392
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 0 0
\(433\) 31.0951 1.49433 0.747167 0.664636i \(-0.231413\pi\)
0.747167 + 0.664636i \(0.231413\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) 26.9783 1.28760 0.643801 0.765193i \(-0.277357\pi\)
0.643801 + 0.765193i \(0.277357\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 21.2554 1.00988 0.504938 0.863156i \(-0.331515\pi\)
0.504938 + 0.863156i \(0.331515\pi\)
\(444\) 0 0
\(445\) −4.74456 −0.224914
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.86141 0.323810 0.161905 0.986806i \(-0.448236\pi\)
0.161905 + 0.986806i \(0.448236\pi\)
\(450\) 0 0
\(451\) −32.4674 −1.52883
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −16.0000 −0.750092
\(456\) 0 0
\(457\) −37.6060 −1.75913 −0.879567 0.475776i \(-0.842167\pi\)
−0.879567 + 0.475776i \(0.842167\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 34.4674 1.60531 0.802653 0.596446i \(-0.203421\pi\)
0.802653 + 0.596446i \(0.203421\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 42.3505 1.95975 0.979874 0.199615i \(-0.0639691\pi\)
0.979874 + 0.199615i \(0.0639691\pi\)
\(468\) 0 0
\(469\) 51.8397 2.39373
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 33.7228 1.54083 0.770417 0.637540i \(-0.220048\pi\)
0.770417 + 0.637540i \(0.220048\pi\)
\(480\) 0 0
\(481\) −25.4891 −1.16220
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −15.4891 −0.703325
\(486\) 0 0
\(487\) 33.7228 1.52813 0.764063 0.645141i \(-0.223202\pi\)
0.764063 + 0.645141i \(0.223202\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 16.6277 0.750398 0.375199 0.926944i \(-0.377574\pi\)
0.375199 + 0.926944i \(0.377574\pi\)
\(492\) 0 0
\(493\) −28.8614 −1.29985
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −11.3723 −0.510117
\(498\) 0 0
\(499\) 43.6060 1.95207 0.976036 0.217611i \(-0.0698263\pi\)
0.976036 + 0.217611i \(0.0698263\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 28.6277 1.27645 0.638223 0.769851i \(-0.279670\pi\)
0.638223 + 0.769851i \(0.279670\pi\)
\(504\) 0 0
\(505\) −13.3723 −0.595059
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 20.9783 0.929845 0.464922 0.885351i \(-0.346082\pi\)
0.464922 + 0.885351i \(0.346082\pi\)
\(510\) 0 0
\(511\) −42.9783 −1.90125
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) −26.9783 −1.18650
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −22.2337 −0.974076 −0.487038 0.873381i \(-0.661923\pi\)
−0.487038 + 0.873381i \(0.661923\pi\)
\(522\) 0 0
\(523\) 38.9783 1.70440 0.852200 0.523216i \(-0.175268\pi\)
0.852200 + 0.523216i \(0.175268\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −18.1168 −0.789182
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 38.5109 1.66809
\(534\) 0 0
\(535\) −7.37228 −0.318732
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −17.4891 −0.753310
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4.74456 −0.203235
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 21.4891 0.915468
\(552\) 0 0
\(553\) −45.4891 −1.93439
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −36.3505 −1.54022 −0.770111 0.637910i \(-0.779799\pi\)
−0.770111 + 0.637910i \(0.779799\pi\)
\(558\) 0 0
\(559\) −18.9783 −0.802694
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −7.37228 −0.310705 −0.155352 0.987859i \(-0.549651\pi\)
−0.155352 + 0.987859i \(0.549651\pi\)
\(564\) 0 0
\(565\) −8.11684 −0.341478
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0.510875 0.0214170 0.0107085 0.999943i \(-0.496591\pi\)
0.0107085 + 0.999943i \(0.496591\pi\)
\(570\) 0 0
\(571\) 0.233688 0.00977954 0.00488977 0.999988i \(-0.498444\pi\)
0.00488977 + 0.999988i \(0.498444\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) −7.25544 −0.302048 −0.151024 0.988530i \(-0.548257\pi\)
−0.151024 + 0.988530i \(0.548257\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 43.3723 1.79939
\(582\) 0 0
\(583\) 5.48913 0.227336
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −24.2337 −1.00023 −0.500116 0.865959i \(-0.666709\pi\)
−0.500116 + 0.865959i \(0.666709\pi\)
\(588\) 0 0
\(589\) 13.4891 0.555810
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −12.9783 −0.532953 −0.266476 0.963841i \(-0.585859\pi\)
−0.266476 + 0.963841i \(0.585859\pi\)
\(594\) 0 0
\(595\) 18.1168 0.742718
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 10.5109 0.429463 0.214731 0.976673i \(-0.431112\pi\)
0.214731 + 0.976673i \(0.431112\pi\)
\(600\) 0 0
\(601\) 18.8614 0.769373 0.384686 0.923047i \(-0.374310\pi\)
0.384686 + 0.923047i \(0.374310\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) −33.2554 −1.34980 −0.674898 0.737911i \(-0.735813\pi\)
−0.674898 + 0.737911i \(0.735813\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 32.0000 1.29458
\(612\) 0 0
\(613\) −39.4891 −1.59495 −0.797475 0.603351i \(-0.793832\pi\)
−0.797475 + 0.603351i \(0.793832\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 20.3505 0.819282 0.409641 0.912247i \(-0.365654\pi\)
0.409641 + 0.912247i \(0.365654\pi\)
\(618\) 0 0
\(619\) 38.9783 1.56667 0.783334 0.621601i \(-0.213517\pi\)
0.783334 + 0.621601i \(0.213517\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 16.0000 0.641026
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 28.8614 1.15078
\(630\) 0 0
\(631\) −17.2554 −0.686928 −0.343464 0.939166i \(-0.611600\pi\)
−0.343464 + 0.939166i \(0.611600\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.25544 −0.0498205
\(636\) 0 0
\(637\) 20.7446 0.821929
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −43.7228 −1.72695 −0.863474 0.504394i \(-0.831716\pi\)
−0.863474 + 0.504394i \(0.831716\pi\)
\(642\) 0 0
\(643\) 47.3723 1.86818 0.934090 0.357037i \(-0.116213\pi\)
0.934090 + 0.357037i \(0.116213\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 21.4891 0.844825 0.422412 0.906404i \(-0.361183\pi\)
0.422412 + 0.906404i \(0.361183\pi\)
\(648\) 0 0
\(649\) 34.5109 1.35467
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −10.2337 −0.400475 −0.200238 0.979747i \(-0.564171\pi\)
−0.200238 + 0.979747i \(0.564171\pi\)
\(654\) 0 0
\(655\) −4.00000 −0.156293
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −22.9783 −0.895106 −0.447553 0.894258i \(-0.647704\pi\)
−0.447553 + 0.894258i \(0.647704\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −13.4891 −0.523086
\(666\) 0 0
\(667\) −5.37228 −0.208016
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) −16.5109 −0.636447 −0.318224 0.948016i \(-0.603086\pi\)
−0.318224 + 0.948016i \(0.603086\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −40.1168 −1.54182 −0.770908 0.636947i \(-0.780197\pi\)
−0.770908 + 0.636947i \(0.780197\pi\)
\(678\) 0 0
\(679\) 52.2337 2.00454
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 32.2337 1.23339 0.616694 0.787203i \(-0.288472\pi\)
0.616694 + 0.787203i \(0.288472\pi\)
\(684\) 0 0
\(685\) −11.4891 −0.438977
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6.51087 −0.248045
\(690\) 0 0
\(691\) −1.48913 −0.0566490 −0.0283245 0.999599i \(-0.509017\pi\)
−0.0283245 + 0.999599i \(0.509017\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.62772 −0.327268
\(696\) 0 0
\(697\) −43.6060 −1.65169
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 39.9565 1.50914 0.754568 0.656222i \(-0.227846\pi\)
0.754568 + 0.656222i \(0.227846\pi\)
\(702\) 0 0
\(703\) −21.4891 −0.810478
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 45.0951 1.69598
\(708\) 0 0
\(709\) −20.9783 −0.787855 −0.393927 0.919142i \(-0.628884\pi\)
−0.393927 + 0.919142i \(0.628884\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −3.37228 −0.126293
\(714\) 0 0
\(715\) 18.9783 0.709746
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −29.0951 −1.08506 −0.542532 0.840035i \(-0.682534\pi\)
−0.542532 + 0.840035i \(0.682534\pi\)
\(720\) 0 0
\(721\) −26.9783 −1.00472
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.37228 0.199522
\(726\) 0 0
\(727\) −3.37228 −0.125071 −0.0625355 0.998043i \(-0.519919\pi\)
−0.0625355 + 0.998043i \(0.519919\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 21.4891 0.794804
\(732\) 0 0
\(733\) 0.116844 0.00431573 0.00215787 0.999998i \(-0.499313\pi\)
0.00215787 + 0.999998i \(0.499313\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −61.4891 −2.26498
\(738\) 0 0
\(739\) 39.3723 1.44833 0.724166 0.689625i \(-0.242225\pi\)
0.724166 + 0.689625i \(0.242225\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −34.9783 −1.28323 −0.641614 0.767028i \(-0.721735\pi\)
−0.641614 + 0.767028i \(0.721735\pi\)
\(744\) 0 0
\(745\) 3.48913 0.127832
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 24.8614 0.908416
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) −25.6060 −0.930665 −0.465332 0.885136i \(-0.654065\pi\)
−0.465332 + 0.885136i \(0.654065\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −45.8397 −1.66169 −0.830843 0.556507i \(-0.812141\pi\)
−0.830843 + 0.556507i \(0.812141\pi\)
\(762\) 0 0
\(763\) 16.0000 0.579239
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −40.9348 −1.47807
\(768\) 0 0
\(769\) 20.5109 0.739641 0.369821 0.929103i \(-0.379419\pi\)
0.369821 + 0.929103i \(0.379419\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 4.97825 0.179055 0.0895276 0.995984i \(-0.471464\pi\)
0.0895276 + 0.995984i \(0.471464\pi\)
\(774\) 0 0
\(775\) 3.37228 0.121136
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 32.4674 1.16326
\(780\) 0 0
\(781\) 13.4891 0.482679
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 17.6060 0.628384
\(786\) 0 0
\(787\) −0.627719 −0.0223758 −0.0111879 0.999937i \(-0.503561\pi\)
−0.0111879 + 0.999937i \(0.503561\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 27.3723 0.973246
\(792\) 0 0
\(793\) −9.48913 −0.336969
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16.1168 −0.570888 −0.285444 0.958395i \(-0.592141\pi\)
−0.285444 + 0.958395i \(0.592141\pi\)
\(798\) 0 0
\(799\) −36.2337 −1.28186
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 50.9783 1.79898
\(804\) 0 0
\(805\) 3.37228 0.118857
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −30.6277 −1.07681 −0.538407 0.842685i \(-0.680974\pi\)
−0.538407 + 0.842685i \(0.680974\pi\)
\(810\) 0 0
\(811\) 17.8832 0.627963 0.313981 0.949429i \(-0.398337\pi\)
0.313981 + 0.949429i \(0.398337\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.25544 −0.184090
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −8.97825 −0.313343 −0.156672 0.987651i \(-0.550076\pi\)
−0.156672 + 0.987651i \(0.550076\pi\)
\(822\) 0 0
\(823\) −10.9783 −0.382678 −0.191339 0.981524i \(-0.561283\pi\)
−0.191339 + 0.981524i \(0.561283\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 43.6060 1.51633 0.758164 0.652064i \(-0.226097\pi\)
0.758164 + 0.652064i \(0.226097\pi\)
\(828\) 0 0
\(829\) 43.0951 1.49675 0.748377 0.663273i \(-0.230833\pi\)
0.748377 + 0.663273i \(0.230833\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −23.4891 −0.813850
\(834\) 0 0
\(835\) 1.25544 0.0434462
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −14.7446 −0.509039 −0.254519 0.967068i \(-0.581917\pi\)
−0.254519 + 0.967068i \(0.581917\pi\)
\(840\) 0 0
\(841\) −0.138593 −0.00477908
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −9.51087 −0.327184
\(846\) 0 0
\(847\) 16.8614 0.579365
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 5.37228 0.184159
\(852\) 0 0
\(853\) 32.9783 1.12915 0.564577 0.825380i \(-0.309039\pi\)
0.564577 + 0.825380i \(0.309039\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −50.4674 −1.72393 −0.861966 0.506965i \(-0.830767\pi\)
−0.861966 + 0.506965i \(0.830767\pi\)
\(858\) 0 0
\(859\) −27.6060 −0.941904 −0.470952 0.882159i \(-0.656089\pi\)
−0.470952 + 0.882159i \(0.656089\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 6.74456 0.229588 0.114794 0.993389i \(-0.463379\pi\)
0.114794 + 0.993389i \(0.463379\pi\)
\(864\) 0 0
\(865\) −14.2337 −0.483960
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 53.9565 1.83035
\(870\) 0 0
\(871\) 72.9348 2.47130
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.37228 −0.114004
\(876\) 0 0
\(877\) −20.5109 −0.692603 −0.346302 0.938123i \(-0.612563\pi\)
−0.346302 + 0.938123i \(0.612563\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 22.0000 0.741199 0.370599 0.928793i \(-0.379152\pi\)
0.370599 + 0.928793i \(0.379152\pi\)
\(882\) 0 0
\(883\) −51.2119 −1.72342 −0.861709 0.507402i \(-0.830606\pi\)
−0.861709 + 0.507402i \(0.830606\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 25.7228 0.863688 0.431844 0.901948i \(-0.357863\pi\)
0.431844 + 0.901948i \(0.357863\pi\)
\(888\) 0 0
\(889\) 4.23369 0.141993
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 26.9783 0.902793
\(894\) 0 0
\(895\) 9.48913 0.317186
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 18.1168 0.604231
\(900\) 0 0
\(901\) 7.37228 0.245606
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 14.2337 0.473144
\(906\) 0 0
\(907\) −8.62772 −0.286479 −0.143239 0.989688i \(-0.545752\pi\)
−0.143239 + 0.989688i \(0.545752\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 56.4674 1.87085 0.935424 0.353528i \(-0.115018\pi\)
0.935424 + 0.353528i \(0.115018\pi\)
\(912\) 0 0
\(913\) −51.4456 −1.70260
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 13.4891 0.445450
\(918\) 0 0
\(919\) 37.4891 1.23665 0.618326 0.785922i \(-0.287811\pi\)
0.618326 + 0.785922i \(0.287811\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) −5.37228 −0.176640
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −33.6060 −1.10258 −0.551288 0.834315i \(-0.685863\pi\)
−0.551288 + 0.834315i \(0.685863\pi\)
\(930\) 0 0
\(931\) 17.4891 0.573183
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −21.4891 −0.702770
\(936\) 0 0
\(937\) 34.4674 1.12600 0.563000 0.826457i \(-0.309647\pi\)
0.563000 + 0.826457i \(0.309647\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 12.9783 0.423079 0.211539 0.977369i \(-0.432152\pi\)
0.211539 + 0.977369i \(0.432152\pi\)
\(942\) 0 0
\(943\) −8.11684 −0.264321
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −54.9783 −1.78655 −0.893277 0.449508i \(-0.851599\pi\)
−0.893277 + 0.449508i \(0.851599\pi\)
\(948\) 0 0
\(949\) −60.4674 −1.96285
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −15.4891 −0.501742 −0.250871 0.968021i \(-0.580717\pi\)
−0.250871 + 0.968021i \(0.580717\pi\)
\(954\) 0 0
\(955\) 20.2337 0.654747
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 38.7446 1.25113
\(960\) 0 0
\(961\) −19.6277 −0.633152
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 20.7446 0.667791
\(966\) 0 0
\(967\) 11.7663 0.378379 0.189190 0.981941i \(-0.439414\pi\)
0.189190 + 0.981941i \(0.439414\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1.48913 −0.0477883 −0.0238942 0.999714i \(-0.507606\pi\)
−0.0238942 + 0.999714i \(0.507606\pi\)
\(972\) 0 0
\(973\) 29.0951 0.932746
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −30.6277 −0.979868 −0.489934 0.871760i \(-0.662979\pi\)
−0.489934 + 0.871760i \(0.662979\pi\)
\(978\) 0 0
\(979\) −18.9783 −0.606548
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 48.8614 1.55844 0.779218 0.626752i \(-0.215616\pi\)
0.779218 + 0.626752i \(0.215616\pi\)
\(984\) 0 0
\(985\) 11.4891 0.366074
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −42.1168 −1.33789 −0.668943 0.743314i \(-0.733253\pi\)
−0.668943 + 0.743314i \(0.733253\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1.25544 −0.0398000
\(996\) 0 0
\(997\) −46.2337 −1.46424 −0.732118 0.681178i \(-0.761468\pi\)
−0.732118 + 0.681178i \(0.761468\pi\)
\(998\) 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 8280.2.a.z.1.2 2
3.2 odd 2 2760.2.a.o.1.2 2
12.11 even 2 5520.2.a.bq.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.o.1.2 2 3.2 odd 2
5520.2.a.bq.1.1 2 12.11 even 2
8280.2.a.z.1.2 2 1.1 even 1 trivial