Properties

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

Related objects

Downloads

Learn more about

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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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
Character \(\chi\) \(=\) 3528.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{5} +O(q^{10})\) \(q+2.00000 q^{5} +6.00000 q^{11} +3.00000 q^{13} +4.00000 q^{17} +5.00000 q^{19} +4.00000 q^{23} -1.00000 q^{25} +4.00000 q^{29} -7.00000 q^{31} -9.00000 q^{37} -2.00000 q^{41} -1.00000 q^{43} +2.00000 q^{47} -8.00000 q^{53} +12.0000 q^{55} -10.0000 q^{61} +6.00000 q^{65} -15.0000 q^{67} +6.00000 q^{71} +11.0000 q^{73} +1.00000 q^{79} +6.00000 q^{83} +8.00000 q^{85} -8.00000 q^{89} +10.0000 q^{95} +14.0000 q^{97} +O(q^{100})\)

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
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) 3.00000 0.832050 0.416025 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 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
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −9.00000 −1.47959 −0.739795 0.672832i \(-0.765078\pi\)
−0.739795 + 0.672832i \(0.765078\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.00000 −1.09888 −0.549442 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) 0 0
\(55\) 12.0000 1.61808
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) −15.0000 −1.83254 −0.916271 0.400559i \(-0.868816\pi\)
−0.916271 + 0.400559i \(0.868816\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.00000 0.112509 0.0562544 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 10.0000 1.02598
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 9.00000 0.886796 0.443398 0.896325i \(-0.353773\pi\)
0.443398 + 0.896325i \(0.353773\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −1.00000 −0.0887357 −0.0443678 0.999015i \(-0.514127\pi\)
−0.0443678 + 0.999015i \(0.514127\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 14.0000 1.22319 0.611593 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −20.0000 −1.70872 −0.854358 0.519685i \(-0.826049\pi\)
−0.854358 + 0.519685i \(0.826049\pi\)
\(138\) 0 0
\(139\) 9.00000 0.763370 0.381685 0.924292i \(-0.375344\pi\)
0.381685 + 0.924292i \(0.375344\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 18.0000 1.50524
\(144\) 0 0
\(145\) 8.00000 0.664364
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −14.0000 −1.12451
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 20.0000 1.52057 0.760286 0.649589i \(-0.225059\pi\)
0.760286 + 0.649589i \(0.225059\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −26.0000 −1.94333 −0.971666 0.236360i \(-0.924046\pi\)
−0.971666 + 0.236360i \(0.924046\pi\)
\(180\) 0 0
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −18.0000 −1.32339
\(186\) 0 0
\(187\) 24.0000 1.75505
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 0 0
\(193\) 3.00000 0.215945 0.107972 0.994154i \(-0.465564\pi\)
0.107972 + 0.994154i \(0.465564\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 30.0000 2.07514
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.00000 −0.136399
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.0000 0.929213 0.464606 0.885517i \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) 0 0
\(229\) 7.00000 0.462573 0.231287 0.972886i \(-0.425707\pi\)
0.231287 + 0.972886i \(0.425707\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2.00000 −0.129369 −0.0646846 0.997906i \(-0.520604\pi\)
−0.0646846 + 0.997906i \(0.520604\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 15.0000 0.954427
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) −16.0000 −0.982872
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.00000 −0.361814
\(276\) 0 0
\(277\) 1.00000 0.0600842 0.0300421 0.999549i \(-0.490436\pi\)
0.0300421 + 0.999549i \(0.490436\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 1.00000 0.0594438 0.0297219 0.999558i \(-0.490538\pi\)
0.0297219 + 0.999558i \(0.490538\pi\)
\(284\) 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
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −20.0000 −1.14520
\(306\) 0 0
\(307\) 11.0000 0.627803 0.313902 0.949456i \(-0.398364\pi\)
0.313902 + 0.949456i \(0.398364\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 0 0
\(313\) 1.00000 0.0565233 0.0282617 0.999601i \(-0.491003\pi\)
0.0282617 + 0.999601i \(0.491003\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 20.0000 1.11283
\(324\) 0 0
\(325\) −3.00000 −0.166410
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 5.00000 0.274825 0.137412 0.990514i \(-0.456121\pi\)
0.137412 + 0.990514i \(0.456121\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −30.0000 −1.63908
\(336\) 0 0
\(337\) 29.0000 1.57973 0.789865 0.613280i \(-0.210150\pi\)
0.789865 + 0.613280i \(0.210150\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −42.0000 −2.27443
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 22.0000 1.15153
\(366\) 0 0
\(367\) 7.00000 0.365397 0.182699 0.983169i \(-0.441517\pi\)
0.182699 + 0.983169i \(0.441517\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −13.0000 −0.673114 −0.336557 0.941663i \(-0.609263\pi\)
−0.336557 + 0.941663i \(0.609263\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −15.0000 −0.770498 −0.385249 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.00000 0.100631
\(396\) 0 0
\(397\) 5.00000 0.250943 0.125471 0.992097i \(-0.459956\pi\)
0.125471 + 0.992097i \(0.459956\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −21.0000 −1.04608
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −54.0000 −2.67668
\(408\) 0 0
\(409\) 3.00000 0.148340 0.0741702 0.997246i \(-0.476369\pi\)
0.0741702 + 0.997246i \(0.476369\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 26.0000 1.27018 0.635092 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) 0 0
\(421\) −35.0000 −1.70580 −0.852898 0.522078i \(-0.825157\pi\)
−0.852898 + 0.522078i \(0.825157\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) −31.0000 −1.48976 −0.744882 0.667196i \(-0.767494\pi\)
−0.744882 + 0.667196i \(0.767494\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 20.0000 0.956730
\(438\) 0 0
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 16.0000 0.760183 0.380091 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 0 0
\(445\) −16.0000 −0.758473
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 38.0000 1.79333 0.896665 0.442709i \(-0.145982\pi\)
0.896665 + 0.442709i \(0.145982\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 13.0000 0.608114 0.304057 0.952654i \(-0.401659\pi\)
0.304057 + 0.952654i \(0.401659\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 17.0000 0.790057 0.395029 0.918669i \(-0.370735\pi\)
0.395029 + 0.918669i \(0.370735\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 30.0000 1.38823 0.694117 0.719862i \(-0.255795\pi\)
0.694117 + 0.719862i \(0.255795\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) −5.00000 −0.229416
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) −27.0000 −1.23109
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 28.0000 1.27141
\(486\) 0 0
\(487\) 25.0000 1.13286 0.566429 0.824110i \(-0.308325\pi\)
0.566429 + 0.824110i \(0.308325\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) 16.0000 0.720604
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −17.0000 −0.761025 −0.380512 0.924776i \(-0.624252\pi\)
−0.380512 + 0.924776i \(0.624252\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −14.0000 −0.624229 −0.312115 0.950044i \(-0.601037\pi\)
−0.312115 + 0.950044i \(0.601037\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 18.0000 0.793175
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) −29.0000 −1.26808 −0.634041 0.773300i \(-0.718605\pi\)
−0.634041 + 0.773300i \(0.718605\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −28.0000 −1.21970
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) 24.0000 1.03761
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.00000 0.0429934 0.0214967 0.999769i \(-0.493157\pi\)
0.0214967 + 0.999769i \(0.493157\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −22.0000 −0.942376
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 20.0000 0.852029
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) −3.00000 −0.126886
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2.00000 −0.0842900 −0.0421450 0.999112i \(-0.513419\pi\)
−0.0421450 + 0.999112i \(0.513419\pi\)
\(564\) 0 0
\(565\) −12.0000 −0.504844
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 23.0000 0.962520 0.481260 0.876578i \(-0.340179\pi\)
0.481260 + 0.876578i \(0.340179\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) −39.0000 −1.62359 −0.811796 0.583942i \(-0.801510\pi\)
−0.811796 + 0.583942i \(0.801510\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −48.0000 −1.98796
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.0000 0.660391 0.330195 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) 0 0
\(589\) −35.0000 −1.44215
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 0 0
\(601\) −31.0000 −1.26452 −0.632258 0.774758i \(-0.717872\pi\)
−0.632258 + 0.774758i \(0.717872\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 50.0000 2.03279
\(606\) 0 0
\(607\) −1.00000 −0.0405887 −0.0202944 0.999794i \(-0.506460\pi\)
−0.0202944 + 0.999794i \(0.506460\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.00000 0.242734
\(612\) 0 0
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 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
\(619\) −9.00000 −0.361741 −0.180870 0.983507i \(-0.557891\pi\)
−0.180870 + 0.983507i \(0.557891\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2.00000 −0.0793676
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20.0000 0.789953 0.394976 0.918691i \(-0.370753\pi\)
0.394976 + 0.918691i \(0.370753\pi\)
\(642\) 0 0
\(643\) 17.0000 0.670415 0.335207 0.942144i \(-0.391194\pi\)
0.335207 + 0.942144i \(0.391194\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 22.0000 0.860927 0.430463 0.902608i \(-0.358350\pi\)
0.430463 + 0.902608i \(0.358350\pi\)
\(654\) 0 0
\(655\) 28.0000 1.09405
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 40.0000 1.55818 0.779089 0.626913i \(-0.215682\pi\)
0.779089 + 0.626913i \(0.215682\pi\)
\(660\) 0 0
\(661\) −35.0000 −1.36134 −0.680671 0.732589i \(-0.738312\pi\)
−0.680671 + 0.732589i \(0.738312\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16.0000 0.619522
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −60.0000 −2.31627
\(672\) 0 0
\(673\) 7.00000 0.269830 0.134915 0.990857i \(-0.456924\pi\)
0.134915 + 0.990857i \(0.456924\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −40.0000 −1.52832
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 7.00000 0.266293 0.133146 0.991096i \(-0.457492\pi\)
0.133146 + 0.991096i \(0.457492\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 18.0000 0.682779
\(696\) 0 0
\(697\) −8.00000 −0.303022
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −28.0000 −1.05755 −0.528773 0.848763i \(-0.677348\pi\)
−0.528773 + 0.848763i \(0.677348\pi\)
\(702\) 0 0
\(703\) −45.0000 −1.69721
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −50.0000 −1.87779 −0.938895 0.344204i \(-0.888149\pi\)
−0.938895 + 0.344204i \(0.888149\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −28.0000 −1.04861
\(714\) 0 0
\(715\) 36.0000 1.34632
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) −5.00000 −0.185440 −0.0927199 0.995692i \(-0.529556\pi\)
−0.0927199 + 0.995692i \(0.529556\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −4.00000 −0.147945
\(732\) 0 0
\(733\) 11.0000 0.406294 0.203147 0.979148i \(-0.434883\pi\)
0.203147 + 0.979148i \(0.434883\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −90.0000 −3.31519
\(738\) 0 0
\(739\) −5.00000 −0.183928 −0.0919640 0.995762i \(-0.529314\pi\)
−0.0919640 + 0.995762i \(0.529314\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 34.0000 1.24734 0.623670 0.781688i \(-0.285641\pi\)
0.623670 + 0.781688i \(0.285641\pi\)
\(744\) 0 0
\(745\) −8.00000 −0.293097
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −37.0000 −1.35015 −0.675075 0.737749i \(-0.735889\pi\)
−0.675075 + 0.737749i \(0.735889\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −7.00000 −0.252426 −0.126213 0.992003i \(-0.540282\pi\)
−0.126213 + 0.992003i \(0.540282\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −50.0000 −1.79838 −0.899188 0.437564i \(-0.855842\pi\)
−0.899188 + 0.437564i \(0.855842\pi\)
\(774\) 0 0
\(775\) 7.00000 0.251447
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −10.0000 −0.358287
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −36.0000 −1.28490
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −30.0000 −1.06533
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 66.0000 2.32909
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −24.0000 −0.842754 −0.421377 0.906886i \(-0.638453\pi\)
−0.421377 + 0.906886i \(0.638453\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 8.00000 0.280228
\(816\) 0 0
\(817\) −5.00000 −0.174928
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −18.0000 −0.625921 −0.312961 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) 0 0
\(829\) 43.0000 1.49345 0.746726 0.665132i \(-0.231625\pi\)
0.746726 + 0.665132i \(0.231625\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 36.0000 1.24583
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −8.00000 −0.275208
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −36.0000 −1.23406
\(852\) 0 0
\(853\) 17.0000 0.582069 0.291034 0.956713i \(-0.406001\pi\)
0.291034 + 0.956713i \(0.406001\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −16.0000 −0.545913 −0.272956 0.962026i \(-0.588002\pi\)
−0.272956 + 0.962026i \(0.588002\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −14.0000 −0.476566 −0.238283 0.971196i \(-0.576585\pi\)
−0.238283 + 0.971196i \(0.576585\pi\)
\(864\) 0 0
\(865\) 40.0000 1.36004
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 6.00000 0.203536
\(870\) 0 0
\(871\) −45.0000 −1.52477
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 52.0000 1.75192 0.875962 0.482380i \(-0.160227\pi\)
0.875962 + 0.482380i \(0.160227\pi\)
\(882\) 0 0
\(883\) 1.00000 0.0336527 0.0168263 0.999858i \(-0.494644\pi\)
0.0168263 + 0.999858i \(0.494644\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −54.0000 −1.81314 −0.906571 0.422053i \(-0.861310\pi\)
−0.906571 + 0.422053i \(0.861310\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 10.0000 0.334637
\(894\) 0 0
\(895\) −52.0000 −1.73817
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −28.0000 −0.933852
\(900\) 0 0
\(901\) −32.0000 −1.06607
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) 17.0000 0.564476 0.282238 0.959344i \(-0.408923\pi\)
0.282238 + 0.959344i \(0.408923\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −28.0000 −0.927681 −0.463841 0.885919i \(-0.653529\pi\)
−0.463841 + 0.885919i \(0.653529\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.00000 0.0329870 0.0164935 0.999864i \(-0.494750\pi\)
0.0164935 + 0.999864i \(0.494750\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 18.0000 0.592477
\(924\) 0 0
\(925\) 9.00000 0.295918
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 58.0000 1.90292 0.951459 0.307775i \(-0.0995844\pi\)
0.951459 + 0.307775i \(0.0995844\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 48.0000 1.56977
\(936\) 0 0
\(937\) 33.0000 1.07806 0.539032 0.842286i \(-0.318790\pi\)
0.539032 + 0.842286i \(0.318790\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 48.0000 1.56476 0.782378 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 38.0000 1.23483 0.617417 0.786636i \(-0.288179\pi\)
0.617417 + 0.786636i \(0.288179\pi\)
\(948\) 0 0
\(949\) 33.0000 1.07123
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 8.00000 0.259145 0.129573 0.991570i \(-0.458639\pi\)
0.129573 + 0.991570i \(0.458639\pi\)
\(954\) 0 0
\(955\) −20.0000 −0.647185
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 6.00000 0.193147
\(966\) 0 0
\(967\) −27.0000 −0.868261 −0.434131 0.900850i \(-0.642944\pi\)
−0.434131 + 0.900850i \(0.642944\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −56.0000 −1.79713 −0.898563 0.438845i \(-0.855388\pi\)
−0.898563 + 0.438845i \(0.855388\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) −48.0000 −1.53409
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 24.0000 0.764704
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) −33.0000 −1.04828 −0.524140 0.851632i \(-0.675613\pi\)
−0.524140 + 0.851632i \(0.675613\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 32.0000 1.01447
\(996\) 0 0
\(997\) 17.0000 0.538395 0.269198 0.963085i \(-0.413241\pi\)
0.269198 + 0.963085i \(0.413241\pi\)
\(998\) 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.y.1.1 1
3.2 odd 2 1176.2.a.e.1.1 1
4.3 odd 2 7056.2.a.bn.1.1 1
7.2 even 3 3528.2.s.d.361.1 2
7.3 odd 6 504.2.s.g.289.1 2
7.4 even 3 3528.2.s.d.3313.1 2
7.5 odd 6 504.2.s.g.361.1 2
7.6 odd 2 3528.2.a.f.1.1 1
12.11 even 2 2352.2.a.e.1.1 1
21.2 odd 6 1176.2.q.e.361.1 2
21.5 even 6 168.2.q.b.25.1 2
21.11 odd 6 1176.2.q.e.961.1 2
21.17 even 6 168.2.q.b.121.1 yes 2
21.20 even 2 1176.2.a.d.1.1 1
24.5 odd 2 9408.2.a.bk.1.1 1
24.11 even 2 9408.2.a.cs.1.1 1
28.3 even 6 1008.2.s.m.289.1 2
28.19 even 6 1008.2.s.m.865.1 2
28.27 even 2 7056.2.a.i.1.1 1
84.11 even 6 2352.2.q.v.961.1 2
84.23 even 6 2352.2.q.v.1537.1 2
84.47 odd 6 336.2.q.a.193.1 2
84.59 odd 6 336.2.q.a.289.1 2
84.83 odd 2 2352.2.a.x.1.1 1
168.5 even 6 1344.2.q.i.193.1 2
168.59 odd 6 1344.2.q.t.961.1 2
168.83 odd 2 9408.2.a.f.1.1 1
168.101 even 6 1344.2.q.i.961.1 2
168.125 even 2 9408.2.a.cd.1.1 1
168.131 odd 6 1344.2.q.t.193.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.q.b.25.1 2 21.5 even 6
168.2.q.b.121.1 yes 2 21.17 even 6
336.2.q.a.193.1 2 84.47 odd 6
336.2.q.a.289.1 2 84.59 odd 6
504.2.s.g.289.1 2 7.3 odd 6
504.2.s.g.361.1 2 7.5 odd 6
1008.2.s.m.289.1 2 28.3 even 6
1008.2.s.m.865.1 2 28.19 even 6
1176.2.a.d.1.1 1 21.20 even 2
1176.2.a.e.1.1 1 3.2 odd 2
1176.2.q.e.361.1 2 21.2 odd 6
1176.2.q.e.961.1 2 21.11 odd 6
1344.2.q.i.193.1 2 168.5 even 6
1344.2.q.i.961.1 2 168.101 even 6
1344.2.q.t.193.1 2 168.131 odd 6
1344.2.q.t.961.1 2 168.59 odd 6
2352.2.a.e.1.1 1 12.11 even 2
2352.2.a.x.1.1 1 84.83 odd 2
2352.2.q.v.961.1 2 84.11 even 6
2352.2.q.v.1537.1 2 84.23 even 6
3528.2.a.f.1.1 1 7.6 odd 2
3528.2.a.y.1.1 1 1.1 even 1 trivial
3528.2.s.d.361.1 2 7.2 even 3
3528.2.s.d.3313.1 2 7.4 even 3
7056.2.a.i.1.1 1 28.27 even 2
7056.2.a.bn.1.1 1 4.3 odd 2
9408.2.a.f.1.1 1 168.83 odd 2
9408.2.a.bk.1.1 1 24.5 odd 2
9408.2.a.cd.1.1 1 168.125 even 2
9408.2.a.cs.1.1 1 24.11 even 2