Properties

Label 1521.2.b.f.1351.1
Level $1521$
Weight $2$
Character 1521.1351
Analytic conductor $12.145$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1521.1351
Dual form 1521.2.b.f.1351.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{4} -3.46410i q^{5} -1.73205i q^{7} +O(q^{10})\) \(q+2.00000 q^{4} -3.46410i q^{5} -1.73205i q^{7} -3.46410i q^{11} +4.00000 q^{16} +3.46410i q^{19} -6.92820i q^{20} +6.00000 q^{23} -7.00000 q^{25} -3.46410i q^{28} -6.00000 q^{29} +1.73205i q^{31} -6.00000 q^{35} -6.92820i q^{41} -1.00000 q^{43} -6.92820i q^{44} -3.46410i q^{47} +4.00000 q^{49} -12.0000 q^{53} -12.0000 q^{55} +3.46410i q^{59} +1.00000 q^{61} +8.00000 q^{64} -8.66025i q^{67} +10.3923i q^{71} +1.73205i q^{73} +6.92820i q^{76} -6.00000 q^{77} -11.0000 q^{79} -13.8564i q^{80} +13.8564i q^{83} +6.92820i q^{89} +12.0000 q^{92} +12.0000 q^{95} -5.19615i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} + 8 q^{16} + 12 q^{23} - 14 q^{25} - 12 q^{29} - 12 q^{35} - 2 q^{43} + 8 q^{49} - 24 q^{53} - 24 q^{55} + 2 q^{61} + 16 q^{64} - 12 q^{77} - 22 q^{79} + 24 q^{92} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) − 3.46410i − 1.54919i −0.632456 0.774597i \(-0.717953\pi\)
0.632456 0.774597i \(-0.282047\pi\)
\(6\) 0 0
\(7\) − 1.73205i − 0.654654i −0.944911 0.327327i \(-0.893852\pi\)
0.944911 0.327327i \(-0.106148\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 3.46410i − 1.04447i −0.852803 0.522233i \(-0.825099\pi\)
0.852803 0.522233i \(-0.174901\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 3.46410i 0.794719i 0.917663 + 0.397360i \(0.130073\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) − 6.92820i − 1.54919i
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) 0 0
\(28\) − 3.46410i − 0.654654i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 1.73205i 0.311086i 0.987829 + 0.155543i \(0.0497126\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6.00000 −1.01419
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 6.92820i − 1.08200i −0.841021 0.541002i \(-0.818045\pi\)
0.841021 0.541002i \(-0.181955\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) − 6.92820i − 1.04447i
\(45\) 0 0
\(46\) 0 0
\(47\) − 3.46410i − 0.505291i −0.967559 0.252646i \(-0.918699\pi\)
0.967559 0.252646i \(-0.0813007\pi\)
\(48\) 0 0
\(49\) 4.00000 0.571429
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.46410i 0.450988i 0.974245 + 0.225494i \(0.0723995\pi\)
−0.974245 + 0.225494i \(0.927600\pi\)
\(60\) 0 0
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) − 8.66025i − 1.05802i −0.848616 0.529009i \(-0.822564\pi\)
0.848616 0.529009i \(-0.177436\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.3923i 1.23334i 0.787222 + 0.616670i \(0.211519\pi\)
−0.787222 + 0.616670i \(0.788481\pi\)
\(72\) 0 0
\(73\) 1.73205i 0.202721i 0.994850 + 0.101361i \(0.0323196\pi\)
−0.994850 + 0.101361i \(0.967680\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 6.92820i 0.794719i
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) −11.0000 −1.23760 −0.618798 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) − 13.8564i − 1.54919i
\(81\) 0 0
\(82\) 0 0
\(83\) 13.8564i 1.52094i 0.649374 + 0.760469i \(0.275031\pi\)
−0.649374 + 0.760469i \(0.724969\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.92820i 0.734388i 0.930144 + 0.367194i \(0.119682\pi\)
−0.930144 + 0.367194i \(0.880318\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 12.0000 1.25109
\(93\) 0 0
\(94\) 0 0
\(95\) 12.0000 1.23117
\(96\) 0 0
\(97\) − 5.19615i − 0.527589i −0.964579 0.263795i \(-0.915026\pi\)
0.964579 0.263795i \(-0.0849741\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −14.0000 −1.40000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) −1.00000 −0.0985329 −0.0492665 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) − 15.5885i − 1.49310i −0.665327 0.746552i \(-0.731708\pi\)
0.665327 0.746552i \(-0.268292\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 6.92820i − 0.654654i
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) − 20.7846i − 1.93817i
\(116\) −12.0000 −1.11417
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 3.46410i 0.311086i
\(125\) 6.92820i 0.619677i
\(126\) 0 0
\(127\) 13.0000 1.15356 0.576782 0.816898i \(-0.304308\pi\)
0.576782 + 0.816898i \(0.304308\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 6.00000 0.520266
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) −12.0000 −1.01419
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 20.7846i 1.72607i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.92820i 0.567581i 0.958886 + 0.283790i \(0.0915919\pi\)
−0.958886 + 0.283790i \(0.908408\pi\)
\(150\) 0 0
\(151\) − 3.46410i − 0.281905i −0.990016 0.140952i \(-0.954984\pi\)
0.990016 0.140952i \(-0.0450164\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) 11.0000 0.877896 0.438948 0.898513i \(-0.355351\pi\)
0.438948 + 0.898513i \(0.355351\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 10.3923i − 0.819028i
\(162\) 0 0
\(163\) − 19.0526i − 1.49231i −0.665771 0.746156i \(-0.731897\pi\)
0.665771 0.746156i \(-0.268103\pi\)
\(164\) − 13.8564i − 1.08200i
\(165\) 0 0
\(166\) 0 0
\(167\) − 6.92820i − 0.536120i −0.963402 0.268060i \(-0.913617\pi\)
0.963402 0.268060i \(-0.0863826\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0 0
\(172\) −2.00000 −0.152499
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 12.1244i 0.916515i
\(176\) − 13.8564i − 1.04447i
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) − 6.92820i − 0.505291i
\(189\) 0 0
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) 15.5885i 1.12208i 0.827788 + 0.561041i \(0.189599\pi\)
−0.827788 + 0.561041i \(0.810401\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 8.00000 0.571429
\(197\) 13.8564i 0.987228i 0.869681 + 0.493614i \(0.164324\pi\)
−0.869681 + 0.493614i \(0.835676\pi\)
\(198\) 0 0
\(199\) 7.00000 0.496217 0.248108 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 10.3923i 0.729397i
\(204\) 0 0
\(205\) −24.0000 −1.67623
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 13.0000 0.894957 0.447478 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) −24.0000 −1.64833
\(213\) 0 0
\(214\) 0 0
\(215\) 3.46410i 0.236250i
\(216\) 0 0
\(217\) 3.00000 0.203653
\(218\) 0 0
\(219\) 0 0
\(220\) −24.0000 −1.61808
\(221\) 0 0
\(222\) 0 0
\(223\) − 17.3205i − 1.15987i −0.814664 0.579934i \(-0.803079\pi\)
0.814664 0.579934i \(-0.196921\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 20.7846i − 1.37952i −0.724037 0.689761i \(-0.757715\pi\)
0.724037 0.689761i \(-0.242285\pi\)
\(228\) 0 0
\(229\) 27.7128i 1.83131i 0.401960 + 0.915657i \(0.368329\pi\)
−0.401960 + 0.915657i \(0.631671\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 6.92820i 0.450988i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 20.7846i 1.33885i 0.742878 + 0.669427i \(0.233460\pi\)
−0.742878 + 0.669427i \(0.766540\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) − 13.8564i − 0.885253i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) − 20.7846i − 1.30672i
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 41.5692i 2.55358i
\(266\) 0 0
\(267\) 0 0
\(268\) − 17.3205i − 1.05802i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 5.19615i 0.315644i 0.987468 + 0.157822i \(0.0504472\pi\)
−0.987468 + 0.157822i \(0.949553\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 24.2487i 1.46225i
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 24.2487i 1.44656i 0.690557 + 0.723278i \(0.257366\pi\)
−0.690557 + 0.723278i \(0.742634\pi\)
\(282\) 0 0
\(283\) −11.0000 −0.653882 −0.326941 0.945045i \(-0.606018\pi\)
−0.326941 + 0.945045i \(0.606018\pi\)
\(284\) 20.7846i 1.23334i
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 3.46410i 0.202721i
\(293\) 17.3205i 1.01187i 0.862570 + 0.505937i \(0.168853\pi\)
−0.862570 + 0.505937i \(0.831147\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.73205i 0.0998337i
\(302\) 0 0
\(303\) 0 0
\(304\) 13.8564i 0.794719i
\(305\) − 3.46410i − 0.198354i
\(306\) 0 0
\(307\) − 1.73205i − 0.0988534i −0.998778 0.0494267i \(-0.984261\pi\)
0.998778 0.0494267i \(-0.0157394\pi\)
\(308\) −12.0000 −0.683763
\(309\) 0 0
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 13.0000 0.734803 0.367402 0.930062i \(-0.380247\pi\)
0.367402 + 0.930062i \(0.380247\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −22.0000 −1.23760
\(317\) − 6.92820i − 0.389127i −0.980890 0.194563i \(-0.937671\pi\)
0.980890 0.194563i \(-0.0623290\pi\)
\(318\) 0 0
\(319\) 20.7846i 1.16371i
\(320\) − 27.7128i − 1.54919i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 5.19615i 0.285606i 0.989751 + 0.142803i \(0.0456116\pi\)
−0.989751 + 0.142803i \(0.954388\pi\)
\(332\) 27.7128i 1.52094i
\(333\) 0 0
\(334\) 0 0
\(335\) −30.0000 −1.63908
\(336\) 0 0
\(337\) −5.00000 −0.272367 −0.136184 0.990684i \(-0.543484\pi\)
−0.136184 + 0.990684i \(0.543484\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) − 19.0526i − 1.02874i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 0 0
\(349\) 19.0526i 1.01986i 0.860216 + 0.509930i \(0.170329\pi\)
−0.860216 + 0.509930i \(0.829671\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.3923i 0.553127i 0.960996 + 0.276563i \(0.0891955\pi\)
−0.960996 + 0.276563i \(0.910804\pi\)
\(354\) 0 0
\(355\) 36.0000 1.91068
\(356\) 13.8564i 0.734388i
\(357\) 0 0
\(358\) 0 0
\(359\) − 6.92820i − 0.365657i −0.983145 0.182828i \(-0.941475\pi\)
0.983145 0.182828i \(-0.0585252\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 17.0000 0.887393 0.443696 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) 24.0000 1.25109
\(369\) 0 0
\(370\) 0 0
\(371\) 20.7846i 1.07908i
\(372\) 0 0
\(373\) −11.0000 −0.569558 −0.284779 0.958593i \(-0.591920\pi\)
−0.284779 + 0.958593i \(0.591920\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 22.5167i 1.15660i 0.815823 + 0.578302i \(0.196284\pi\)
−0.815823 + 0.578302i \(0.803716\pi\)
\(380\) 24.0000 1.23117
\(381\) 0 0
\(382\) 0 0
\(383\) − 27.7128i − 1.41606i −0.706183 0.708029i \(-0.749584\pi\)
0.706183 0.708029i \(-0.250416\pi\)
\(384\) 0 0
\(385\) 20.7846i 1.05928i
\(386\) 0 0
\(387\) 0 0
\(388\) − 10.3923i − 0.527589i
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 38.1051i 1.91728i
\(396\) 0 0
\(397\) 15.5885i 0.782362i 0.920314 + 0.391181i \(0.127933\pi\)
−0.920314 + 0.391181i \(0.872067\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −28.0000 −1.40000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 36.0000 1.79107
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 8.66025i 0.428222i 0.976809 + 0.214111i \(0.0686854\pi\)
−0.976809 + 0.214111i \(0.931315\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2.00000 −0.0985329
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) 48.0000 2.35623
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) − 12.1244i − 0.590905i −0.955357 0.295452i \(-0.904530\pi\)
0.955357 0.295452i \(-0.0954704\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 1.73205i − 0.0838198i
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 20.7846i 1.00116i 0.865690 + 0.500580i \(0.166880\pi\)
−0.865690 + 0.500580i \(0.833120\pi\)
\(432\) 0 0
\(433\) −23.0000 −1.10531 −0.552655 0.833410i \(-0.686385\pi\)
−0.552655 + 0.833410i \(0.686385\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) − 31.1769i − 1.49310i
\(437\) 20.7846i 0.994263i
\(438\) 0 0
\(439\) 35.0000 1.67046 0.835229 0.549902i \(-0.185335\pi\)
0.835229 + 0.549902i \(0.185335\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 24.0000 1.13771
\(446\) 0 0
\(447\) 0 0
\(448\) − 13.8564i − 0.654654i
\(449\) − 38.1051i − 1.79829i −0.437649 0.899146i \(-0.644189\pi\)
0.437649 0.899146i \(-0.355811\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) −12.0000 −0.564433
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 36.3731i 1.70146i 0.525603 + 0.850730i \(0.323840\pi\)
−0.525603 + 0.850730i \(0.676160\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) − 41.5692i − 1.93817i
\(461\) 41.5692i 1.93607i 0.250812 + 0.968036i \(0.419302\pi\)
−0.250812 + 0.968036i \(0.580698\pi\)
\(462\) 0 0
\(463\) − 36.3731i − 1.69040i −0.534450 0.845200i \(-0.679481\pi\)
0.534450 0.845200i \(-0.320519\pi\)
\(464\) −24.0000 −1.11417
\(465\) 0 0
\(466\) 0 0
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 0 0
\(469\) −15.0000 −0.692636
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.46410i 0.159280i
\(474\) 0 0
\(475\) − 24.2487i − 1.11261i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 34.6410i 1.58279i 0.611306 + 0.791394i \(0.290644\pi\)
−0.611306 + 0.791394i \(0.709356\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) −18.0000 −0.817338
\(486\) 0 0
\(487\) − 24.2487i − 1.09881i −0.835555 0.549407i \(-0.814854\pi\)
0.835555 0.549407i \(-0.185146\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 6.92820i 0.311086i
\(497\) 18.0000 0.807410
\(498\) 0 0
\(499\) 31.1769i 1.39567i 0.716258 + 0.697835i \(0.245853\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 13.8564i 0.619677i
\(501\) 0 0
\(502\) 0 0
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) − 62.3538i − 2.77471i
\(506\) 0 0
\(507\) 0 0
\(508\) 26.0000 1.15356
\(509\) − 17.3205i − 0.767718i −0.923392 0.383859i \(-0.874595\pi\)
0.923392 0.383859i \(-0.125405\pi\)
\(510\) 0 0
\(511\) 3.00000 0.132712
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.46410i 0.152647i
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 24.0000 1.05146 0.525730 0.850652i \(-0.323792\pi\)
0.525730 + 0.850652i \(0.323792\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 12.0000 0.520266
\(533\) 0 0
\(534\) 0 0
\(535\) − 20.7846i − 0.898597i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 13.8564i − 0.596838i
\(540\) 0 0
\(541\) − 29.4449i − 1.26593i −0.774179 0.632967i \(-0.781837\pi\)
0.774179 0.632967i \(-0.218163\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −54.0000 −2.31311
\(546\) 0 0
\(547\) −19.0000 −0.812381 −0.406191 0.913788i \(-0.633143\pi\)
−0.406191 + 0.913788i \(0.633143\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 20.7846i − 0.885454i
\(552\) 0 0
\(553\) 19.0526i 0.810197i
\(554\) 0 0
\(555\) 0 0
\(556\) 10.0000 0.424094
\(557\) − 27.7128i − 1.17423i −0.809504 0.587115i \(-0.800264\pi\)
0.809504 0.587115i \(-0.199736\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −24.0000 −1.01419
\(561\) 0 0
\(562\) 0 0
\(563\) 42.0000 1.77009 0.885044 0.465506i \(-0.154128\pi\)
0.885044 + 0.465506i \(0.154128\pi\)
\(564\) 0 0
\(565\) 20.7846i 0.874415i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −44.0000 −1.84134 −0.920671 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −42.0000 −1.75152
\(576\) 0 0
\(577\) − 34.6410i − 1.44212i −0.692870 0.721062i \(-0.743654\pi\)
0.692870 0.721062i \(-0.256346\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 41.5692i 1.72607i
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) 41.5692i 1.72162i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 31.1769i − 1.28681i −0.765526 0.643404i \(-0.777521\pi\)
0.765526 0.643404i \(-0.222479\pi\)
\(588\) 0 0
\(589\) −6.00000 −0.247226
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 3.46410i − 0.142254i −0.997467 0.0711268i \(-0.977341\pi\)
0.997467 0.0711268i \(-0.0226595\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 13.8564i 0.567581i
\(597\) 0 0
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) − 6.92820i − 0.281905i
\(605\) 3.46410i 0.140836i
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 8.66025i − 0.349784i −0.984588 0.174892i \(-0.944042\pi\)
0.984588 0.174892i \(-0.0559577\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 10.3923i 0.418378i 0.977875 + 0.209189i \(0.0670825\pi\)
−0.977875 + 0.209189i \(0.932918\pi\)
\(618\) 0 0
\(619\) 25.9808i 1.04425i 0.852867 + 0.522127i \(0.174861\pi\)
−0.852867 + 0.522127i \(0.825139\pi\)
\(620\) 12.0000 0.481932
\(621\) 0 0
\(622\) 0 0
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 22.0000 0.877896
\(629\) 0 0
\(630\) 0 0
\(631\) 1.73205i 0.0689519i 0.999406 + 0.0344759i \(0.0109762\pi\)
−0.999406 + 0.0344759i \(0.989024\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 45.0333i − 1.78709i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 19.0526i 0.751360i 0.926750 + 0.375680i \(0.122591\pi\)
−0.926750 + 0.375680i \(0.877409\pi\)
\(644\) − 20.7846i − 0.819028i
\(645\) 0 0
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) − 38.1051i − 1.49231i
\(653\) −36.0000 −1.40879 −0.704394 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) 0 0
\(655\) 20.7846i 0.812122i
\(656\) − 27.7128i − 1.08200i
\(657\) 0 0
\(658\) 0 0
\(659\) −48.0000 −1.86981 −0.934907 0.354892i \(-0.884518\pi\)
−0.934907 + 0.354892i \(0.884518\pi\)
\(660\) 0 0
\(661\) 25.9808i 1.01053i 0.862963 + 0.505267i \(0.168606\pi\)
−0.862963 + 0.505267i \(0.831394\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 20.7846i − 0.805993i
\(666\) 0 0
\(667\) −36.0000 −1.39393
\(668\) − 13.8564i − 0.536120i
\(669\) 0 0
\(670\) 0 0
\(671\) − 3.46410i − 0.133730i
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −48.0000 −1.84479 −0.922395 0.386248i \(-0.873771\pi\)
−0.922395 + 0.386248i \(0.873771\pi\)
\(678\) 0 0
\(679\) −9.00000 −0.345388
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 24.2487i − 0.927851i −0.885874 0.463926i \(-0.846441\pi\)
0.885874 0.463926i \(-0.153559\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) − 43.3013i − 1.64726i −0.567129 0.823629i \(-0.691946\pi\)
0.567129 0.823629i \(-0.308054\pi\)
\(692\) 12.0000 0.456172
\(693\) 0 0
\(694\) 0 0
\(695\) − 17.3205i − 0.657004i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 24.2487i 0.916515i
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) − 27.7128i − 1.04447i
\(705\) 0 0
\(706\) 0 0
\(707\) − 31.1769i − 1.17253i
\(708\) 0 0
\(709\) − 19.0526i − 0.715534i −0.933811 0.357767i \(-0.883538\pi\)
0.933811 0.357767i \(-0.116462\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 10.3923i 0.389195i
\(714\) 0 0
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) 0 0
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) 1.73205i 0.0645049i
\(722\) 0 0
\(723\) 0 0
\(724\) 28.0000 1.04061
\(725\) 42.0000 1.55984
\(726\) 0 0
\(727\) −35.0000 −1.29808 −0.649039 0.760755i \(-0.724829\pi\)
−0.649039 + 0.760755i \(0.724829\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 39.8372i 1.47142i 0.677297 + 0.735710i \(0.263151\pi\)
−0.677297 + 0.735710i \(0.736849\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −30.0000 −1.10506
\(738\) 0 0
\(739\) − 45.0333i − 1.65658i −0.560301 0.828289i \(-0.689315\pi\)
0.560301 0.828289i \(-0.310685\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 10.3923i − 0.381257i −0.981662 0.190628i \(-0.938947\pi\)
0.981662 0.190628i \(-0.0610525\pi\)
\(744\) 0 0
\(745\) 24.0000 0.879292
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 10.3923i − 0.379727i
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) − 13.8564i − 0.505291i
\(753\) 0 0
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) 34.0000 1.23575 0.617876 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 20.7846i 0.753442i 0.926327 + 0.376721i \(0.122948\pi\)
−0.926327 + 0.376721i \(0.877052\pi\)
\(762\) 0 0
\(763\) −27.0000 −0.977466
\(764\) 36.0000 1.30243
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 6.92820i 0.249837i 0.992167 + 0.124919i \(0.0398670\pi\)
−0.992167 + 0.124919i \(0.960133\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 31.1769i 1.12208i
\(773\) 51.9615i 1.86893i 0.356060 + 0.934463i \(0.384120\pi\)
−0.356060 + 0.934463i \(0.615880\pi\)
\(774\) 0 0
\(775\) − 12.1244i − 0.435520i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) 0 0
\(783\) 0 0
\(784\) 16.0000 0.571429
\(785\) − 38.1051i − 1.36003i
\(786\) 0 0
\(787\) 32.9090i 1.17308i 0.809921 + 0.586539i \(0.199510\pi\)
−0.809921 + 0.586539i \(0.800490\pi\)
\(788\) 27.7128i 0.987228i
\(789\) 0 0
\(790\) 0 0
\(791\) 10.3923i 0.369508i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 14.0000 0.496217
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.00000 0.211735
\(804\) 0 0
\(805\) −36.0000 −1.26883
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −12.0000 −0.421898 −0.210949 0.977497i \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) 0 0
\(811\) − 25.9808i − 0.912308i −0.889901 0.456154i \(-0.849227\pi\)
0.889901 0.456154i \(-0.150773\pi\)
\(812\) 20.7846i 0.729397i
\(813\) 0 0
\(814\) 0 0
\(815\) −66.0000 −2.31188
\(816\) 0 0
\(817\) − 3.46410i − 0.121194i
\(818\) 0 0
\(819\) 0 0
\(820\) −48.0000 −1.67623
\(821\) 24.2487i 0.846286i 0.906063 + 0.423143i \(0.139073\pi\)
−0.906063 + 0.423143i \(0.860927\pi\)
\(822\) 0 0
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 48.4974i − 1.68642i −0.537584 0.843210i \(-0.680663\pi\)
0.537584 0.843210i \(-0.319337\pi\)
\(828\) 0 0
\(829\) 31.0000 1.07667 0.538337 0.842729i \(-0.319053\pi\)
0.538337 + 0.842729i \(0.319053\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) 24.0000 0.830057
\(837\) 0 0
\(838\) 0 0
\(839\) − 31.1769i − 1.07635i −0.842834 0.538173i \(-0.819115\pi\)
0.842834 0.538173i \(-0.180885\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 26.0000 0.894957
\(845\) 0 0
\(846\) 0 0
\(847\) 1.73205i 0.0595140i
\(848\) −48.0000 −1.64833
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 25.9808i − 0.889564i −0.895639 0.444782i \(-0.853281\pi\)
0.895639 0.444782i \(-0.146719\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 13.0000 0.443554 0.221777 0.975097i \(-0.428814\pi\)
0.221777 + 0.975097i \(0.428814\pi\)
\(860\) 6.92820i 0.236250i
\(861\) 0 0
\(862\) 0 0
\(863\) 17.3205i 0.589597i 0.955559 + 0.294798i \(0.0952525\pi\)
−0.955559 + 0.294798i \(0.904747\pi\)
\(864\) 0 0
\(865\) − 20.7846i − 0.706698i
\(866\) 0 0
\(867\) 0 0
\(868\) 6.00000 0.203653
\(869\) 38.1051i 1.29263i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) 41.5692i 1.40369i 0.712328 + 0.701846i \(0.247641\pi\)
−0.712328 + 0.701846i \(0.752359\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −48.0000 −1.61808
\(881\) 12.0000 0.404290 0.202145 0.979356i \(-0.435209\pi\)
0.202145 + 0.979356i \(0.435209\pi\)
\(882\) 0 0
\(883\) −5.00000 −0.168263 −0.0841317 0.996455i \(-0.526812\pi\)
−0.0841317 + 0.996455i \(0.526812\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 42.0000 1.41022 0.705111 0.709097i \(-0.250897\pi\)
0.705111 + 0.709097i \(0.250897\pi\)
\(888\) 0 0
\(889\) − 22.5167i − 0.755185i
\(890\) 0 0
\(891\) 0 0
\(892\) − 34.6410i − 1.15987i
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) 41.5692i 1.38951i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 10.3923i − 0.346603i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 48.4974i − 1.61211i
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) − 41.5692i − 1.37952i
\(909\) 0 0
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) 48.0000 1.58857
\(914\) 0 0
\(915\) 0 0
\(916\) 55.4256i 1.83131i
\(917\) 10.3923i 0.343184i
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 20.7846i 0.681921i 0.940078 + 0.340960i \(0.110752\pi\)
−0.940078 + 0.340960i \(0.889248\pi\)
\(930\) 0 0
\(931\) 13.8564i 0.454125i
\(932\) 36.0000 1.17922
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −24.0000 −0.782794
\(941\) 10.3923i 0.338779i 0.985549 + 0.169390i \(0.0541797\pi\)
−0.985549 + 0.169390i \(0.945820\pi\)
\(942\) 0 0
\(943\) − 41.5692i − 1.35368i
\(944\) 13.8564i 0.450988i
\(945\) 0 0
\(946\) 0 0
\(947\) − 48.4974i − 1.57595i −0.615704 0.787977i \(-0.711128\pi\)
0.615704 0.787977i \(-0.288872\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) − 62.3538i − 2.01772i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 28.0000 0.903226
\(962\) 0 0
\(963\) 0 0
\(964\) 41.5692i 1.33885i
\(965\) 54.0000 1.73832
\(966\) 0 0
\(967\) − 24.2487i − 0.779786i −0.920860 0.389893i \(-0.872512\pi\)
0.920860 0.389893i \(-0.127488\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) − 8.66025i − 0.277635i
\(974\) 0 0
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) − 27.7128i − 0.886611i −0.896370 0.443306i \(-0.853806\pi\)
0.896370 0.443306i \(-0.146194\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) − 27.7128i − 0.885253i
\(981\) 0 0
\(982\) 0 0
\(983\) 10.3923i 0.331463i 0.986171 + 0.165732i \(0.0529985\pi\)
−0.986171 + 0.165732i \(0.947001\pi\)
\(984\) 0 0
\(985\) 48.0000 1.52941
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 24.2487i − 0.768736i
\(996\) 0 0
\(997\) 35.0000 1.10846 0.554231 0.832363i \(-0.313013\pi\)
0.554231 + 0.832363i \(0.313013\pi\)
\(998\) 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 1521.2.b.f.1351.1 2
3.2 odd 2 507.2.b.c.337.2 2
13.3 even 3 117.2.q.a.82.1 2
13.4 even 6 117.2.q.a.10.1 2
13.5 odd 4 1521.2.a.h.1.1 2
13.8 odd 4 1521.2.a.h.1.2 2
13.12 even 2 inner 1521.2.b.f.1351.2 2
39.2 even 12 507.2.e.f.22.2 4
39.5 even 4 507.2.a.e.1.2 2
39.8 even 4 507.2.a.e.1.1 2
39.11 even 12 507.2.e.f.22.1 4
39.17 odd 6 39.2.j.a.10.1 yes 2
39.20 even 12 507.2.e.f.484.1 4
39.23 odd 6 507.2.j.b.316.1 2
39.29 odd 6 39.2.j.a.4.1 2
39.32 even 12 507.2.e.f.484.2 4
39.35 odd 6 507.2.j.b.361.1 2
39.38 odd 2 507.2.b.c.337.1 2
52.3 odd 6 1872.2.by.f.433.1 2
52.43 odd 6 1872.2.by.f.1297.1 2
156.47 odd 4 8112.2.a.bu.1.1 2
156.83 odd 4 8112.2.a.bu.1.2 2
156.95 even 6 624.2.bv.b.49.1 2
156.107 even 6 624.2.bv.b.433.1 2
195.17 even 12 975.2.w.d.49.2 4
195.29 odd 6 975.2.bc.c.901.1 2
195.68 even 12 975.2.w.d.199.2 4
195.107 even 12 975.2.w.d.199.1 4
195.134 odd 6 975.2.bc.c.751.1 2
195.173 even 12 975.2.w.d.49.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.j.a.4.1 2 39.29 odd 6
39.2.j.a.10.1 yes 2 39.17 odd 6
117.2.q.a.10.1 2 13.4 even 6
117.2.q.a.82.1 2 13.3 even 3
507.2.a.e.1.1 2 39.8 even 4
507.2.a.e.1.2 2 39.5 even 4
507.2.b.c.337.1 2 39.38 odd 2
507.2.b.c.337.2 2 3.2 odd 2
507.2.e.f.22.1 4 39.11 even 12
507.2.e.f.22.2 4 39.2 even 12
507.2.e.f.484.1 4 39.20 even 12
507.2.e.f.484.2 4 39.32 even 12
507.2.j.b.316.1 2 39.23 odd 6
507.2.j.b.361.1 2 39.35 odd 6
624.2.bv.b.49.1 2 156.95 even 6
624.2.bv.b.433.1 2 156.107 even 6
975.2.w.d.49.1 4 195.173 even 12
975.2.w.d.49.2 4 195.17 even 12
975.2.w.d.199.1 4 195.107 even 12
975.2.w.d.199.2 4 195.68 even 12
975.2.bc.c.751.1 2 195.134 odd 6
975.2.bc.c.901.1 2 195.29 odd 6
1521.2.a.h.1.1 2 13.5 odd 4
1521.2.a.h.1.2 2 13.8 odd 4
1521.2.b.f.1351.1 2 1.1 even 1 trivial
1521.2.b.f.1351.2 2 13.12 even 2 inner
1872.2.by.f.433.1 2 52.3 odd 6
1872.2.by.f.1297.1 2 52.43 odd 6
8112.2.a.bu.1.1 2 156.47 odd 4
8112.2.a.bu.1.2 2 156.83 odd 4