Properties

Label 1183.2.c.b.337.2
Level $1183$
Weight $2$
Character 1183.337
Analytic conductor $9.446$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.44630255912\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1183.337
Dual form 1183.2.c.b.337.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{2} -2.00000 q^{4} +3.00000i q^{5} -1.00000i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} -2.00000 q^{4} +3.00000i q^{5} -1.00000i q^{7} -3.00000 q^{9} -6.00000 q^{10} -6.00000i q^{11} +2.00000 q^{14} -4.00000 q^{16} -4.00000 q^{17} -6.00000i q^{18} -5.00000i q^{19} -6.00000i q^{20} +12.0000 q^{22} -3.00000 q^{23} -4.00000 q^{25} +2.00000i q^{28} -5.00000 q^{29} +3.00000i q^{31} -8.00000i q^{32} -8.00000i q^{34} +3.00000 q^{35} +6.00000 q^{36} -4.00000i q^{37} +10.0000 q^{38} +6.00000i q^{41} +1.00000 q^{43} +12.0000i q^{44} -9.00000i q^{45} -6.00000i q^{46} +7.00000i q^{47} -1.00000 q^{49} -8.00000i q^{50} -9.00000 q^{53} +18.0000 q^{55} -10.0000i q^{58} +8.00000i q^{59} -10.0000 q^{61} -6.00000 q^{62} +3.00000i q^{63} +8.00000 q^{64} +6.00000i q^{67} +8.00000 q^{68} +6.00000i q^{70} +8.00000i q^{71} -13.0000i q^{73} +8.00000 q^{74} +10.0000i q^{76} -6.00000 q^{77} +3.00000 q^{79} -12.0000i q^{80} +9.00000 q^{81} -12.0000 q^{82} -15.0000i q^{83} -12.0000i q^{85} +2.00000i q^{86} +3.00000i q^{89} +18.0000 q^{90} +6.00000 q^{92} -14.0000 q^{94} +15.0000 q^{95} -7.00000i q^{97} -2.00000i q^{98} +18.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 6 q^{9} + O(q^{10}) \) \( 2 q - 4 q^{4} - 6 q^{9} - 12 q^{10} + 4 q^{14} - 8 q^{16} - 8 q^{17} + 24 q^{22} - 6 q^{23} - 8 q^{25} - 10 q^{29} + 6 q^{35} + 12 q^{36} + 20 q^{38} + 2 q^{43} - 2 q^{49} - 18 q^{53} + 36 q^{55} - 20 q^{61} - 12 q^{62} + 16 q^{64} + 16 q^{68} + 16 q^{74} - 12 q^{77} + 6 q^{79} + 18 q^{81} - 24 q^{82} + 36 q^{90} + 12 q^{92} - 28 q^{94} + 30 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −2.00000 −1.00000
\(5\) 3.00000i 1.34164i 0.741620 + 0.670820i \(0.234058\pi\)
−0.741620 + 0.670820i \(0.765942\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) −6.00000 −1.89737
\(11\) − 6.00000i − 1.80907i −0.426401 0.904534i \(-0.640219\pi\)
0.426401 0.904534i \(-0.359781\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) − 6.00000i − 1.41421i
\(19\) − 5.00000i − 1.14708i −0.819178 0.573539i \(-0.805570\pi\)
0.819178 0.573539i \(-0.194430\pi\)
\(20\) − 6.00000i − 1.34164i
\(21\) 0 0
\(22\) 12.0000 2.55841
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 2.00000i 0.377964i
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 3.00000i 0.538816i 0.963026 + 0.269408i \(0.0868280\pi\)
−0.963026 + 0.269408i \(0.913172\pi\)
\(32\) − 8.00000i − 1.41421i
\(33\) 0 0
\(34\) − 8.00000i − 1.37199i
\(35\) 3.00000 0.507093
\(36\) 6.00000 1.00000
\(37\) − 4.00000i − 0.657596i −0.944400 0.328798i \(-0.893356\pi\)
0.944400 0.328798i \(-0.106644\pi\)
\(38\) 10.0000 1.62221
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000i 0.937043i 0.883452 + 0.468521i \(0.155213\pi\)
−0.883452 + 0.468521i \(0.844787\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 12.0000i 1.80907i
\(45\) − 9.00000i − 1.34164i
\(46\) − 6.00000i − 0.884652i
\(47\) 7.00000i 1.02105i 0.859861 + 0.510527i \(0.170550\pi\)
−0.859861 + 0.510527i \(0.829450\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) − 8.00000i − 1.13137i
\(51\) 0 0
\(52\) 0 0
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) 18.0000 2.42712
\(56\) 0 0
\(57\) 0 0
\(58\) − 10.0000i − 1.31306i
\(59\) 8.00000i 1.04151i 0.853706 + 0.520756i \(0.174350\pi\)
−0.853706 + 0.520756i \(0.825650\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −6.00000 −0.762001
\(63\) 3.00000i 0.377964i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 6.00000i 0.733017i 0.930415 + 0.366508i \(0.119447\pi\)
−0.930415 + 0.366508i \(0.880553\pi\)
\(68\) 8.00000 0.970143
\(69\) 0 0
\(70\) 6.00000i 0.717137i
\(71\) 8.00000i 0.949425i 0.880141 + 0.474713i \(0.157448\pi\)
−0.880141 + 0.474713i \(0.842552\pi\)
\(72\) 0 0
\(73\) − 13.0000i − 1.52153i −0.649025 0.760767i \(-0.724823\pi\)
0.649025 0.760767i \(-0.275177\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 10.0000i 1.14708i
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 3.00000 0.337526 0.168763 0.985657i \(-0.446023\pi\)
0.168763 + 0.985657i \(0.446023\pi\)
\(80\) − 12.0000i − 1.34164i
\(81\) 9.00000 1.00000
\(82\) −12.0000 −1.32518
\(83\) − 15.0000i − 1.64646i −0.567705 0.823232i \(-0.692169\pi\)
0.567705 0.823232i \(-0.307831\pi\)
\(84\) 0 0
\(85\) − 12.0000i − 1.30158i
\(86\) 2.00000i 0.215666i
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000i 0.317999i 0.987279 + 0.159000i \(0.0508269\pi\)
−0.987279 + 0.159000i \(0.949173\pi\)
\(90\) 18.0000 1.89737
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) −14.0000 −1.44399
\(95\) 15.0000 1.53897
\(96\) 0 0
\(97\) − 7.00000i − 0.710742i −0.934725 0.355371i \(-0.884354\pi\)
0.934725 0.355371i \(-0.115646\pi\)
\(98\) − 2.00000i − 0.202031i
\(99\) 18.0000i 1.80907i
\(100\) 8.00000 0.800000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) − 18.0000i − 1.74831i
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 2.00000i 0.191565i 0.995402 + 0.0957826i \(0.0305354\pi\)
−0.995402 + 0.0957826i \(0.969465\pi\)
\(110\) 36.0000i 3.43247i
\(111\) 0 0
\(112\) 4.00000i 0.377964i
\(113\) −3.00000 −0.282216 −0.141108 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) 0 0
\(115\) − 9.00000i − 0.839254i
\(116\) 10.0000 0.928477
\(117\) 0 0
\(118\) −16.0000 −1.47292
\(119\) 4.00000i 0.366679i
\(120\) 0 0
\(121\) −25.0000 −2.27273
\(122\) − 20.0000i − 1.81071i
\(123\) 0 0
\(124\) − 6.00000i − 0.538816i
\(125\) 3.00000i 0.268328i
\(126\) −6.00000 −0.534522
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) −5.00000 −0.433555
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 0 0
\(137\) 4.00000i 0.341743i 0.985293 + 0.170872i \(0.0546583\pi\)
−0.985293 + 0.170872i \(0.945342\pi\)
\(138\) 0 0
\(139\) −18.0000 −1.52674 −0.763370 0.645961i \(-0.776457\pi\)
−0.763370 + 0.645961i \(0.776457\pi\)
\(140\) −6.00000 −0.507093
\(141\) 0 0
\(142\) −16.0000 −1.34269
\(143\) 0 0
\(144\) 12.0000 1.00000
\(145\) − 15.0000i − 1.24568i
\(146\) 26.0000 2.15178
\(147\) 0 0
\(148\) 8.00000i 0.657596i
\(149\) 18.0000i 1.47462i 0.675556 + 0.737309i \(0.263904\pi\)
−0.675556 + 0.737309i \(0.736096\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 12.0000 0.970143
\(154\) − 12.0000i − 0.966988i
\(155\) −9.00000 −0.722897
\(156\) 0 0
\(157\) −8.00000 −0.638470 −0.319235 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) 6.00000i 0.477334i
\(159\) 0 0
\(160\) 24.0000 1.89737
\(161\) 3.00000i 0.236433i
\(162\) 18.0000i 1.41421i
\(163\) − 4.00000i − 0.313304i −0.987654 0.156652i \(-0.949930\pi\)
0.987654 0.156652i \(-0.0500701\pi\)
\(164\) − 12.0000i − 0.937043i
\(165\) 0 0
\(166\) 30.0000 2.32845
\(167\) 5.00000i 0.386912i 0.981109 + 0.193456i \(0.0619696\pi\)
−0.981109 + 0.193456i \(0.938030\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 24.0000 1.84072
\(171\) 15.0000i 1.14708i
\(172\) −2.00000 −0.152499
\(173\) 8.00000 0.608229 0.304114 0.952636i \(-0.401639\pi\)
0.304114 + 0.952636i \(0.401639\pi\)
\(174\) 0 0
\(175\) 4.00000i 0.302372i
\(176\) 24.0000i 1.80907i
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) −23.0000 −1.71910 −0.859550 0.511051i \(-0.829256\pi\)
−0.859550 + 0.511051i \(0.829256\pi\)
\(180\) 18.0000i 1.34164i
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) 24.0000i 1.75505i
\(188\) − 14.0000i − 1.02105i
\(189\) 0 0
\(190\) 30.0000i 2.17643i
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) 22.0000i 1.58359i 0.610784 + 0.791797i \(0.290854\pi\)
−0.610784 + 0.791797i \(0.709146\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) − 2.00000i − 0.142494i −0.997459 0.0712470i \(-0.977302\pi\)
0.997459 0.0712470i \(-0.0226979\pi\)
\(198\) −36.0000 −2.55841
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 28.0000i 1.97007i
\(203\) 5.00000i 0.350931i
\(204\) 0 0
\(205\) −18.0000 −1.25717
\(206\) 8.00000i 0.557386i
\(207\) 9.00000 0.625543
\(208\) 0 0
\(209\) −30.0000 −2.07514
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 18.0000 1.23625
\(213\) 0 0
\(214\) − 8.00000i − 0.546869i
\(215\) 3.00000i 0.204598i
\(216\) 0 0
\(217\) 3.00000 0.203653
\(218\) −4.00000 −0.270914
\(219\) 0 0
\(220\) −36.0000 −2.42712
\(221\) 0 0
\(222\) 0 0
\(223\) − 15.0000i − 1.00447i −0.864730 0.502237i \(-0.832510\pi\)
0.864730 0.502237i \(-0.167490\pi\)
\(224\) −8.00000 −0.534522
\(225\) 12.0000 0.800000
\(226\) − 6.00000i − 0.399114i
\(227\) − 20.0000i − 1.32745i −0.747978 0.663723i \(-0.768975\pi\)
0.747978 0.663723i \(-0.231025\pi\)
\(228\) 0 0
\(229\) 14.0000i 0.925146i 0.886581 + 0.462573i \(0.153074\pi\)
−0.886581 + 0.462573i \(0.846926\pi\)
\(230\) 18.0000 1.18688
\(231\) 0 0
\(232\) 0 0
\(233\) −15.0000 −0.982683 −0.491341 0.870967i \(-0.663493\pi\)
−0.491341 + 0.870967i \(0.663493\pi\)
\(234\) 0 0
\(235\) −21.0000 −1.36989
\(236\) − 16.0000i − 1.04151i
\(237\) 0 0
\(238\) −8.00000 −0.518563
\(239\) 4.00000i 0.258738i 0.991596 + 0.129369i \(0.0412952\pi\)
−0.991596 + 0.129369i \(0.958705\pi\)
\(240\) 0 0
\(241\) − 17.0000i − 1.09507i −0.836784 0.547533i \(-0.815567\pi\)
0.836784 0.547533i \(-0.184433\pi\)
\(242\) − 50.0000i − 3.21412i
\(243\) 0 0
\(244\) 20.0000 1.28037
\(245\) − 3.00000i − 0.191663i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −6.00000 −0.379473
\(251\) 26.0000 1.64111 0.820553 0.571571i \(-0.193666\pi\)
0.820553 + 0.571571i \(0.193666\pi\)
\(252\) − 6.00000i − 0.377964i
\(253\) 18.0000i 1.13165i
\(254\) 8.00000i 0.501965i
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 15.0000 0.928477
\(262\) 16.0000i 0.988483i
\(263\) −15.0000 −0.924940 −0.462470 0.886635i \(-0.653037\pi\)
−0.462470 + 0.886635i \(0.653037\pi\)
\(264\) 0 0
\(265\) − 27.0000i − 1.65860i
\(266\) − 10.0000i − 0.613139i
\(267\) 0 0
\(268\) − 12.0000i − 0.733017i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 8.00000i 0.485965i 0.970031 + 0.242983i \(0.0781258\pi\)
−0.970031 + 0.242983i \(0.921874\pi\)
\(272\) 16.0000 0.970143
\(273\) 0 0
\(274\) −8.00000 −0.483298
\(275\) 24.0000i 1.44725i
\(276\) 0 0
\(277\) −1.00000 −0.0600842 −0.0300421 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) − 36.0000i − 2.15914i
\(279\) − 9.00000i − 0.538816i
\(280\) 0 0
\(281\) − 30.0000i − 1.78965i −0.446417 0.894825i \(-0.647300\pi\)
0.446417 0.894825i \(-0.352700\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) − 16.0000i − 0.949425i
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 24.0000i 1.41421i
\(289\) −1.00000 −0.0588235
\(290\) 30.0000 1.76166
\(291\) 0 0
\(292\) 26.0000i 1.52153i
\(293\) − 19.0000i − 1.10999i −0.831853 0.554996i \(-0.812720\pi\)
0.831853 0.554996i \(-0.187280\pi\)
\(294\) 0 0
\(295\) −24.0000 −1.39733
\(296\) 0 0
\(297\) 0 0
\(298\) −36.0000 −2.08542
\(299\) 0 0
\(300\) 0 0
\(301\) − 1.00000i − 0.0576390i
\(302\) 0 0
\(303\) 0 0
\(304\) 20.0000i 1.14708i
\(305\) − 30.0000i − 1.71780i
\(306\) 24.0000i 1.37199i
\(307\) − 33.0000i − 1.88341i −0.336440 0.941705i \(-0.609223\pi\)
0.336440 0.941705i \(-0.390777\pi\)
\(308\) 12.0000 0.683763
\(309\) 0 0
\(310\) − 18.0000i − 1.02233i
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) − 16.0000i − 0.902932i
\(315\) −9.00000 −0.507093
\(316\) −6.00000 −0.337526
\(317\) 24.0000i 1.34797i 0.738743 + 0.673987i \(0.235420\pi\)
−0.738743 + 0.673987i \(0.764580\pi\)
\(318\) 0 0
\(319\) 30.0000i 1.67968i
\(320\) 24.0000i 1.34164i
\(321\) 0 0
\(322\) −6.00000 −0.334367
\(323\) 20.0000i 1.11283i
\(324\) −18.0000 −1.00000
\(325\) 0 0
\(326\) 8.00000 0.443079
\(327\) 0 0
\(328\) 0 0
\(329\) 7.00000 0.385922
\(330\) 0 0
\(331\) − 22.0000i − 1.20923i −0.796518 0.604615i \(-0.793327\pi\)
0.796518 0.604615i \(-0.206673\pi\)
\(332\) 30.0000i 1.64646i
\(333\) 12.0000i 0.657596i
\(334\) −10.0000 −0.547176
\(335\) −18.0000 −0.983445
\(336\) 0 0
\(337\) −17.0000 −0.926049 −0.463025 0.886345i \(-0.653236\pi\)
−0.463025 + 0.886345i \(0.653236\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 24.0000i 1.30158i
\(341\) 18.0000 0.974755
\(342\) −30.0000 −1.62221
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 16.0000i 0.860165i
\(347\) −32.0000 −1.71785 −0.858925 0.512101i \(-0.828867\pi\)
−0.858925 + 0.512101i \(0.828867\pi\)
\(348\) 0 0
\(349\) 11.0000i 0.588817i 0.955680 + 0.294408i \(0.0951225\pi\)
−0.955680 + 0.294408i \(0.904877\pi\)
\(350\) −8.00000 −0.427618
\(351\) 0 0
\(352\) −48.0000 −2.55841
\(353\) 10.0000i 0.532246i 0.963939 + 0.266123i \(0.0857428\pi\)
−0.963939 + 0.266123i \(0.914257\pi\)
\(354\) 0 0
\(355\) −24.0000 −1.27379
\(356\) − 6.00000i − 0.317999i
\(357\) 0 0
\(358\) − 46.0000i − 2.43118i
\(359\) 20.0000i 1.05556i 0.849381 + 0.527780i \(0.176975\pi\)
−0.849381 + 0.527780i \(0.823025\pi\)
\(360\) 0 0
\(361\) −6.00000 −0.315789
\(362\) − 28.0000i − 1.47165i
\(363\) 0 0
\(364\) 0 0
\(365\) 39.0000 2.04135
\(366\) 0 0
\(367\) 14.0000 0.730794 0.365397 0.930852i \(-0.380933\pi\)
0.365397 + 0.930852i \(0.380933\pi\)
\(368\) 12.0000 0.625543
\(369\) − 18.0000i − 0.937043i
\(370\) 24.0000i 1.24770i
\(371\) 9.00000i 0.467257i
\(372\) 0 0
\(373\) 30.0000 1.55334 0.776671 0.629907i \(-0.216907\pi\)
0.776671 + 0.629907i \(0.216907\pi\)
\(374\) −48.0000 −2.48202
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 6.00000i 0.308199i 0.988055 + 0.154100i \(0.0492477\pi\)
−0.988055 + 0.154100i \(0.950752\pi\)
\(380\) −30.0000 −1.53897
\(381\) 0 0
\(382\) − 16.0000i − 0.818631i
\(383\) 36.0000i 1.83951i 0.392488 + 0.919757i \(0.371614\pi\)
−0.392488 + 0.919757i \(0.628386\pi\)
\(384\) 0 0
\(385\) − 18.0000i − 0.917365i
\(386\) −44.0000 −2.23954
\(387\) −3.00000 −0.152499
\(388\) 14.0000i 0.710742i
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) 0 0
\(394\) 4.00000 0.201517
\(395\) 9.00000i 0.452839i
\(396\) − 36.0000i − 1.80907i
\(397\) − 13.0000i − 0.652451i −0.945292 0.326226i \(-0.894223\pi\)
0.945292 0.326226i \(-0.105777\pi\)
\(398\) − 8.00000i − 0.401004i
\(399\) 0 0
\(400\) 16.0000 0.800000
\(401\) − 32.0000i − 1.59800i −0.601329 0.799002i \(-0.705362\pi\)
0.601329 0.799002i \(-0.294638\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −28.0000 −1.39305
\(405\) 27.0000i 1.34164i
\(406\) −10.0000 −0.496292
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) 13.0000i 0.642809i 0.946942 + 0.321404i \(0.104155\pi\)
−0.946942 + 0.321404i \(0.895845\pi\)
\(410\) − 36.0000i − 1.77791i
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 8.00000 0.393654
\(414\) 18.0000i 0.884652i
\(415\) 45.0000 2.20896
\(416\) 0 0
\(417\) 0 0
\(418\) − 60.0000i − 2.93470i
\(419\) −10.0000 −0.488532 −0.244266 0.969708i \(-0.578547\pi\)
−0.244266 + 0.969708i \(0.578547\pi\)
\(420\) 0 0
\(421\) 12.0000i 0.584844i 0.956289 + 0.292422i \(0.0944612\pi\)
−0.956289 + 0.292422i \(0.905539\pi\)
\(422\) − 10.0000i − 0.486792i
\(423\) − 21.0000i − 1.02105i
\(424\) 0 0
\(425\) 16.0000 0.776114
\(426\) 0 0
\(427\) 10.0000i 0.483934i
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) − 6.00000i − 0.289010i −0.989504 0.144505i \(-0.953841\pi\)
0.989504 0.144505i \(-0.0461589\pi\)
\(432\) 0 0
\(433\) −12.0000 −0.576683 −0.288342 0.957528i \(-0.593104\pi\)
−0.288342 + 0.957528i \(0.593104\pi\)
\(434\) 6.00000i 0.288009i
\(435\) 0 0
\(436\) − 4.00000i − 0.191565i
\(437\) 15.0000i 0.717547i
\(438\) 0 0
\(439\) 22.0000 1.05000 0.525001 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 19.0000 0.902717 0.451359 0.892343i \(-0.350940\pi\)
0.451359 + 0.892343i \(0.350940\pi\)
\(444\) 0 0
\(445\) −9.00000 −0.426641
\(446\) 30.0000 1.42054
\(447\) 0 0
\(448\) − 8.00000i − 0.377964i
\(449\) 36.0000i 1.69895i 0.527633 + 0.849473i \(0.323080\pi\)
−0.527633 + 0.849473i \(0.676920\pi\)
\(450\) 24.0000i 1.13137i
\(451\) 36.0000 1.69517
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) 40.0000 1.87729
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) −28.0000 −1.30835
\(459\) 0 0
\(460\) 18.0000i 0.839254i
\(461\) 22.0000i 1.02464i 0.858794 + 0.512321i \(0.171214\pi\)
−0.858794 + 0.512321i \(0.828786\pi\)
\(462\) 0 0
\(463\) − 14.0000i − 0.650635i −0.945605 0.325318i \(-0.894529\pi\)
0.945605 0.325318i \(-0.105471\pi\)
\(464\) 20.0000 0.928477
\(465\) 0 0
\(466\) − 30.0000i − 1.38972i
\(467\) 22.0000 1.01804 0.509019 0.860755i \(-0.330008\pi\)
0.509019 + 0.860755i \(0.330008\pi\)
\(468\) 0 0
\(469\) 6.00000 0.277054
\(470\) − 42.0000i − 1.93732i
\(471\) 0 0
\(472\) 0 0
\(473\) − 6.00000i − 0.275880i
\(474\) 0 0
\(475\) 20.0000i 0.917663i
\(476\) − 8.00000i − 0.366679i
\(477\) 27.0000 1.23625
\(478\) −8.00000 −0.365911
\(479\) − 11.0000i − 0.502603i −0.967909 0.251301i \(-0.919141\pi\)
0.967909 0.251301i \(-0.0808585\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 34.0000 1.54866
\(483\) 0 0
\(484\) 50.0000 2.27273
\(485\) 21.0000 0.953561
\(486\) 0 0
\(487\) 26.0000i 1.17817i 0.808070 + 0.589086i \(0.200512\pi\)
−0.808070 + 0.589086i \(0.799488\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 6.00000 0.271052
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) 0 0
\(495\) −54.0000 −2.42712
\(496\) − 12.0000i − 0.538816i
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 16.0000i 0.716258i 0.933672 + 0.358129i \(0.116585\pi\)
−0.933672 + 0.358129i \(0.883415\pi\)
\(500\) − 6.00000i − 0.268328i
\(501\) 0 0
\(502\) 52.0000i 2.32087i
\(503\) 2.00000 0.0891756 0.0445878 0.999005i \(-0.485803\pi\)
0.0445878 + 0.999005i \(0.485803\pi\)
\(504\) 0 0
\(505\) 42.0000i 1.86898i
\(506\) −36.0000 −1.60040
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) 19.0000i 0.842160i 0.907023 + 0.421080i \(0.138349\pi\)
−0.907023 + 0.421080i \(0.861651\pi\)
\(510\) 0 0
\(511\) −13.0000 −0.575086
\(512\) 32.0000i 1.41421i
\(513\) 0 0
\(514\) 4.00000i 0.176432i
\(515\) 12.0000i 0.528783i
\(516\) 0 0
\(517\) 42.0000 1.84716
\(518\) − 8.00000i − 0.351500i
\(519\) 0 0
\(520\) 0 0
\(521\) 40.0000 1.75243 0.876216 0.481919i \(-0.160060\pi\)
0.876216 + 0.481919i \(0.160060\pi\)
\(522\) 30.0000i 1.31306i
\(523\) 10.0000 0.437269 0.218635 0.975807i \(-0.429840\pi\)
0.218635 + 0.975807i \(0.429840\pi\)
\(524\) −16.0000 −0.698963
\(525\) 0 0
\(526\) − 30.0000i − 1.30806i
\(527\) − 12.0000i − 0.522728i
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 54.0000 2.34561
\(531\) − 24.0000i − 1.04151i
\(532\) 10.0000 0.433555
\(533\) 0 0
\(534\) 0 0
\(535\) − 12.0000i − 0.518805i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6.00000i 0.258438i
\(540\) 0 0
\(541\) − 40.0000i − 1.71973i −0.510518 0.859867i \(-0.670546\pi\)
0.510518 0.859867i \(-0.329454\pi\)
\(542\) −16.0000 −0.687259
\(543\) 0 0
\(544\) 32.0000i 1.37199i
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) −7.00000 −0.299298 −0.149649 0.988739i \(-0.547814\pi\)
−0.149649 + 0.988739i \(0.547814\pi\)
\(548\) − 8.00000i − 0.341743i
\(549\) 30.0000 1.28037
\(550\) −48.0000 −2.04673
\(551\) 25.0000i 1.06504i
\(552\) 0 0
\(553\) − 3.00000i − 0.127573i
\(554\) − 2.00000i − 0.0849719i
\(555\) 0 0
\(556\) 36.0000 1.52674
\(557\) − 12.0000i − 0.508456i −0.967144 0.254228i \(-0.918179\pi\)
0.967144 0.254228i \(-0.0818214\pi\)
\(558\) 18.0000 0.762001
\(559\) 0 0
\(560\) −12.0000 −0.507093
\(561\) 0 0
\(562\) 60.0000 2.53095
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) − 9.00000i − 0.378633i
\(566\) − 32.0000i − 1.34506i
\(567\) − 9.00000i − 0.377964i
\(568\) 0 0
\(569\) −7.00000 −0.293455 −0.146728 0.989177i \(-0.546874\pi\)
−0.146728 + 0.989177i \(0.546874\pi\)
\(570\) 0 0
\(571\) 17.0000 0.711428 0.355714 0.934595i \(-0.384238\pi\)
0.355714 + 0.934595i \(0.384238\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 12.0000i 0.500870i
\(575\) 12.0000 0.500435
\(576\) −24.0000 −1.00000
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) − 2.00000i − 0.0831890i
\(579\) 0 0
\(580\) 30.0000i 1.24568i
\(581\) −15.0000 −0.622305
\(582\) 0 0
\(583\) 54.0000i 2.23645i
\(584\) 0 0
\(585\) 0 0
\(586\) 38.0000 1.56977
\(587\) − 39.0000i − 1.60970i −0.593477 0.804851i \(-0.702245\pi\)
0.593477 0.804851i \(-0.297755\pi\)
\(588\) 0 0
\(589\) 15.0000 0.618064
\(590\) − 48.0000i − 1.97613i
\(591\) 0 0
\(592\) 16.0000i 0.657596i
\(593\) − 27.0000i − 1.10876i −0.832265 0.554379i \(-0.812956\pi\)
0.832265 0.554379i \(-0.187044\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) − 36.0000i − 1.47462i
\(597\) 0 0
\(598\) 0 0
\(599\) 11.0000 0.449448 0.224724 0.974422i \(-0.427852\pi\)
0.224724 + 0.974422i \(0.427852\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 2.00000 0.0815139
\(603\) − 18.0000i − 0.733017i
\(604\) 0 0
\(605\) − 75.0000i − 3.04918i
\(606\) 0 0
\(607\) 2.00000 0.0811775 0.0405887 0.999176i \(-0.487077\pi\)
0.0405887 + 0.999176i \(0.487077\pi\)
\(608\) −40.0000 −1.62221
\(609\) 0 0
\(610\) 60.0000 2.42933
\(611\) 0 0
\(612\) −24.0000 −0.970143
\(613\) − 8.00000i − 0.323117i −0.986863 0.161558i \(-0.948348\pi\)
0.986863 0.161558i \(-0.0516520\pi\)
\(614\) 66.0000 2.66354
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000i 1.20775i 0.797077 + 0.603877i \(0.206378\pi\)
−0.797077 + 0.603877i \(0.793622\pi\)
\(618\) 0 0
\(619\) 20.0000i 0.803868i 0.915669 + 0.401934i \(0.131662\pi\)
−0.915669 + 0.401934i \(0.868338\pi\)
\(620\) 18.0000 0.722897
\(621\) 0 0
\(622\) 12.0000i 0.481156i
\(623\) 3.00000 0.120192
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 44.0000i 1.75859i
\(627\) 0 0
\(628\) 16.0000 0.638470
\(629\) 16.0000i 0.637962i
\(630\) − 18.0000i − 0.717137i
\(631\) 22.0000i 0.875806i 0.899022 + 0.437903i \(0.144279\pi\)
−0.899022 + 0.437903i \(0.855721\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −48.0000 −1.90632
\(635\) 12.0000i 0.476205i
\(636\) 0 0
\(637\) 0 0
\(638\) −60.0000 −2.37542
\(639\) − 24.0000i − 0.949425i
\(640\) 0 0
\(641\) −9.00000 −0.355479 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(642\) 0 0
\(643\) − 8.00000i − 0.315489i −0.987480 0.157745i \(-0.949578\pi\)
0.987480 0.157745i \(-0.0504223\pi\)
\(644\) − 6.00000i − 0.236433i
\(645\) 0 0
\(646\) −40.0000 −1.57378
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 0 0
\(649\) 48.0000 1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) 8.00000i 0.313304i
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) 0 0
\(655\) 24.0000i 0.937758i
\(656\) − 24.0000i − 0.937043i
\(657\) 39.0000i 1.52153i
\(658\) 14.0000i 0.545777i
\(659\) 17.0000 0.662226 0.331113 0.943591i \(-0.392576\pi\)
0.331113 + 0.943591i \(0.392576\pi\)
\(660\) 0 0
\(661\) − 33.0000i − 1.28355i −0.766892 0.641776i \(-0.778198\pi\)
0.766892 0.641776i \(-0.221802\pi\)
\(662\) 44.0000 1.71011
\(663\) 0 0
\(664\) 0 0
\(665\) − 15.0000i − 0.581675i
\(666\) −24.0000 −0.929981
\(667\) 15.0000 0.580802
\(668\) − 10.0000i − 0.386912i
\(669\) 0 0
\(670\) − 36.0000i − 1.39080i
\(671\) 60.0000i 2.31627i
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) − 34.0000i − 1.30963i
\(675\) 0 0
\(676\) 0 0
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 0 0
\(679\) −7.00000 −0.268635
\(680\) 0 0
\(681\) 0 0
\(682\) 36.0000i 1.37851i
\(683\) − 12.0000i − 0.459167i −0.973289 0.229584i \(-0.926264\pi\)
0.973289 0.229584i \(-0.0737364\pi\)
\(684\) − 30.0000i − 1.14708i
\(685\) −12.0000 −0.458496
\(686\) −2.00000 −0.0763604
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) − 11.0000i − 0.418460i −0.977866 0.209230i \(-0.932904\pi\)
0.977866 0.209230i \(-0.0670957\pi\)
\(692\) −16.0000 −0.608229
\(693\) 18.0000 0.683763
\(694\) − 64.0000i − 2.42941i
\(695\) − 54.0000i − 2.04834i
\(696\) 0 0
\(697\) − 24.0000i − 0.909065i
\(698\) −22.0000 −0.832712
\(699\) 0 0
\(700\) − 8.00000i − 0.302372i
\(701\) 27.0000 1.01978 0.509888 0.860241i \(-0.329687\pi\)
0.509888 + 0.860241i \(0.329687\pi\)
\(702\) 0 0
\(703\) −20.0000 −0.754314
\(704\) − 48.0000i − 1.80907i
\(705\) 0 0
\(706\) −20.0000 −0.752710
\(707\) − 14.0000i − 0.526524i
\(708\) 0 0
\(709\) − 10.0000i − 0.375558i −0.982211 0.187779i \(-0.939871\pi\)
0.982211 0.187779i \(-0.0601289\pi\)
\(710\) − 48.0000i − 1.80141i
\(711\) −9.00000 −0.337526
\(712\) 0 0
\(713\) − 9.00000i − 0.337053i
\(714\) 0 0
\(715\) 0 0
\(716\) 46.0000 1.71910
\(717\) 0 0
\(718\) −40.0000 −1.49279
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 36.0000i 1.34164i
\(721\) − 4.00000i − 0.148968i
\(722\) − 12.0000i − 0.446594i
\(723\) 0 0
\(724\) 28.0000 1.04061
\(725\) 20.0000 0.742781
\(726\) 0 0
\(727\) −46.0000 −1.70605 −0.853023 0.521874i \(-0.825233\pi\)
−0.853023 + 0.521874i \(0.825233\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 78.0000i 2.88691i
\(731\) −4.00000 −0.147945
\(732\) 0 0
\(733\) − 51.0000i − 1.88373i −0.335994 0.941864i \(-0.609072\pi\)
0.335994 0.941864i \(-0.390928\pi\)
\(734\) 28.0000i 1.03350i
\(735\) 0 0
\(736\) 24.0000i 0.884652i
\(737\) 36.0000 1.32608
\(738\) 36.0000 1.32518
\(739\) − 26.0000i − 0.956425i −0.878244 0.478213i \(-0.841285\pi\)
0.878244 0.478213i \(-0.158715\pi\)
\(740\) −24.0000 −0.882258
\(741\) 0 0
\(742\) −18.0000 −0.660801
\(743\) − 36.0000i − 1.32071i −0.750953 0.660356i \(-0.770405\pi\)
0.750953 0.660356i \(-0.229595\pi\)
\(744\) 0 0
\(745\) −54.0000 −1.97841
\(746\) 60.0000i 2.19676i
\(747\) 45.0000i 1.64646i
\(748\) − 48.0000i − 1.75505i
\(749\) 4.00000i 0.146157i
\(750\) 0 0
\(751\) 17.0000 0.620339 0.310169 0.950681i \(-0.399614\pi\)
0.310169 + 0.950681i \(0.399614\pi\)
\(752\) − 28.0000i − 1.02105i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −15.0000 −0.545184 −0.272592 0.962130i \(-0.587881\pi\)
−0.272592 + 0.962130i \(0.587881\pi\)
\(758\) −12.0000 −0.435860
\(759\) 0 0
\(760\) 0 0
\(761\) 9.00000i 0.326250i 0.986605 + 0.163125i \(0.0521573\pi\)
−0.986605 + 0.163125i \(0.947843\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) 16.0000 0.578860
\(765\) 36.0000i 1.30158i
\(766\) −72.0000 −2.60147
\(767\) 0 0
\(768\) 0 0
\(769\) 35.0000i 1.26213i 0.775729 + 0.631066i \(0.217382\pi\)
−0.775729 + 0.631066i \(0.782618\pi\)
\(770\) 36.0000 1.29735
\(771\) 0 0
\(772\) − 44.0000i − 1.58359i
\(773\) − 54.0000i − 1.94225i −0.238581 0.971123i \(-0.576682\pi\)
0.238581 0.971123i \(-0.423318\pi\)
\(774\) − 6.00000i − 0.215666i
\(775\) − 12.0000i − 0.431053i
\(776\) 0 0
\(777\) 0 0
\(778\) − 60.0000i − 2.15110i
\(779\) 30.0000 1.07486
\(780\) 0 0
\(781\) 48.0000 1.71758
\(782\) 24.0000i 0.858238i
\(783\) 0 0
\(784\) 4.00000 0.142857
\(785\) − 24.0000i − 0.856597i
\(786\) 0 0
\(787\) 37.0000i 1.31891i 0.751745 + 0.659454i \(0.229212\pi\)
−0.751745 + 0.659454i \(0.770788\pi\)
\(788\) 4.00000i 0.142494i
\(789\) 0 0
\(790\) −18.0000 −0.640411
\(791\) 3.00000i 0.106668i
\(792\) 0 0
\(793\) 0 0
\(794\) 26.0000 0.922705
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) − 28.0000i − 0.990569i
\(800\) 32.0000i 1.13137i
\(801\) − 9.00000i − 0.317999i
\(802\) 64.0000 2.25992
\(803\) −78.0000 −2.75256
\(804\) 0 0
\(805\) −9.00000 −0.317208
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −31.0000 −1.08990 −0.544951 0.838468i \(-0.683452\pi\)
−0.544951 + 0.838468i \(0.683452\pi\)
\(810\) −54.0000 −1.89737
\(811\) 52.0000i 1.82597i 0.407997 + 0.912983i \(0.366228\pi\)
−0.407997 + 0.912983i \(0.633772\pi\)
\(812\) − 10.0000i − 0.350931i
\(813\) 0 0
\(814\) − 48.0000i − 1.68240i
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) − 5.00000i − 0.174928i
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) 36.0000 1.25717
\(821\) 6.00000i 0.209401i 0.994504 + 0.104701i \(0.0333885\pi\)
−0.994504 + 0.104701i \(0.966612\pi\)
\(822\) 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
\(826\) 16.0000i 0.556711i
\(827\) 4.00000i 0.139094i 0.997579 + 0.0695468i \(0.0221553\pi\)
−0.997579 + 0.0695468i \(0.977845\pi\)
\(828\) −18.0000 −0.625543
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 90.0000i 3.12395i
\(831\) 0 0
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) −15.0000 −0.519096
\(836\) 60.0000 2.07514
\(837\) 0 0
\(838\) − 20.0000i − 0.690889i
\(839\) − 8.00000i − 0.276191i −0.990419 0.138095i \(-0.955902\pi\)
0.990419 0.138095i \(-0.0440980\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −24.0000 −0.827095
\(843\) 0 0
\(844\) 10.0000 0.344214
\(845\) 0 0
\(846\) 42.0000 1.44399
\(847\) 25.0000i 0.859010i
\(848\) 36.0000 1.23625
\(849\) 0 0
\(850\) 32.0000i 1.09759i
\(851\) 12.0000i 0.411355i
\(852\) 0 0
\(853\) 45.0000i 1.54077i 0.637579 + 0.770385i \(0.279936\pi\)
−0.637579 + 0.770385i \(0.720064\pi\)
\(854\) −20.0000 −0.684386
\(855\) −45.0000 −1.53897
\(856\) 0 0
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) −2.00000 −0.0682391 −0.0341196 0.999418i \(-0.510863\pi\)
−0.0341196 + 0.999418i \(0.510863\pi\)
\(860\) − 6.00000i − 0.204598i
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 32.0000i 1.08929i 0.838666 + 0.544646i \(0.183336\pi\)
−0.838666 + 0.544646i \(0.816664\pi\)
\(864\) 0 0
\(865\) 24.0000i 0.816024i
\(866\) − 24.0000i − 0.815553i
\(867\) 0 0
\(868\) −6.00000 −0.203653
\(869\) − 18.0000i − 0.610608i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 21.0000i 0.710742i
\(874\) −30.0000 −1.01477
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 44.0000i 1.48493i
\(879\) 0 0
\(880\) −72.0000 −2.42712
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 6.00000i 0.202031i
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 38.0000i 1.27663i
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) − 4.00000i − 0.134156i
\(890\) − 18.0000i − 0.603361i
\(891\) − 54.0000i − 1.80907i
\(892\) 30.0000i 1.00447i
\(893\) 35.0000 1.17123
\(894\) 0 0
\(895\) − 69.0000i − 2.30642i
\(896\) 0 0
\(897\) 0 0
\(898\) −72.0000 −2.40267
\(899\) − 15.0000i − 0.500278i
\(900\) −24.0000 −0.800000
\(901\) 36.0000 1.19933
\(902\) 72.0000i 2.39734i
\(903\) 0 0
\(904\) 0 0
\(905\) − 42.0000i − 1.39613i
\(906\) 0 0
\(907\) −7.00000 −0.232431 −0.116216 0.993224i \(-0.537076\pi\)
−0.116216 + 0.993224i \(0.537076\pi\)
\(908\) 40.0000i 1.32745i
\(909\) −42.0000 −1.39305
\(910\) 0 0
\(911\) −15.0000 −0.496972 −0.248486 0.968635i \(-0.579933\pi\)
−0.248486 + 0.968635i \(0.579933\pi\)
\(912\) 0 0
\(913\) −90.0000 −2.97857
\(914\) 0 0
\(915\) 0 0
\(916\) − 28.0000i − 0.925146i
\(917\) − 8.00000i − 0.264183i
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −44.0000 −1.44906
\(923\) 0 0
\(924\) 0 0
\(925\) 16.0000i 0.526077i
\(926\) 28.0000 0.920137
\(927\) −12.0000 −0.394132
\(928\) 40.0000i 1.31306i
\(929\) − 5.00000i − 0.164045i −0.996630 0.0820223i \(-0.973862\pi\)
0.996630 0.0820223i \(-0.0261379\pi\)
\(930\) 0 0
\(931\) 5.00000i 0.163868i
\(932\) 30.0000 0.982683
\(933\) 0 0
\(934\) 44.0000i 1.43972i
\(935\) −72.0000 −2.35465
\(936\) 0 0
\(937\) −8.00000 −0.261349 −0.130674 0.991425i \(-0.541714\pi\)
−0.130674 + 0.991425i \(0.541714\pi\)
\(938\) 12.0000i 0.391814i
\(939\) 0 0
\(940\) 42.0000 1.36989
\(941\) − 55.0000i − 1.79295i −0.443096 0.896474i \(-0.646120\pi\)
0.443096 0.896474i \(-0.353880\pi\)
\(942\) 0 0
\(943\) − 18.0000i − 0.586161i
\(944\) − 32.0000i − 1.04151i
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) − 18.0000i − 0.584921i −0.956278 0.292461i \(-0.905526\pi\)
0.956278 0.292461i \(-0.0944741\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −40.0000 −1.29777
\(951\) 0 0
\(952\) 0 0
\(953\) 39.0000 1.26333 0.631667 0.775240i \(-0.282371\pi\)
0.631667 + 0.775240i \(0.282371\pi\)
\(954\) 54.0000i 1.74831i
\(955\) − 24.0000i − 0.776622i
\(956\) − 8.00000i − 0.258738i
\(957\) 0 0
\(958\) 22.0000 0.710788
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) 22.0000 0.709677
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 34.0000i 1.09507i
\(965\) −66.0000 −2.12462
\(966\) 0 0
\(967\) − 22.0000i − 0.707472i −0.935345 0.353736i \(-0.884911\pi\)
0.935345 0.353736i \(-0.115089\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 42.0000i 1.34854i
\(971\) 38.0000 1.21948 0.609739 0.792602i \(-0.291274\pi\)
0.609739 + 0.792602i \(0.291274\pi\)
\(972\) 0 0
\(973\) 18.0000i 0.577054i
\(974\) −52.0000 −1.66619
\(975\) 0 0
\(976\) 40.0000 1.28037
\(977\) − 10.0000i − 0.319928i −0.987123 0.159964i \(-0.948862\pi\)
0.987123 0.159964i \(-0.0511379\pi\)
\(978\) 0 0
\(979\) 18.0000 0.575282
\(980\) 6.00000i 0.191663i
\(981\) − 6.00000i − 0.191565i
\(982\) 24.0000i 0.765871i
\(983\) 17.0000i 0.542216i 0.962549 + 0.271108i \(0.0873900\pi\)
−0.962549 + 0.271108i \(0.912610\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 40.0000i 1.27386i
\(987\) 0 0
\(988\) 0 0
\(989\) −3.00000 −0.0953945
\(990\) − 108.000i − 3.43247i
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 24.0000 0.762001
\(993\) 0 0
\(994\) 16.0000i 0.507489i
\(995\) − 12.0000i − 0.380426i
\(996\) 0 0
\(997\) −28.0000 −0.886769 −0.443384 0.896332i \(-0.646222\pi\)
−0.443384 + 0.896332i \(0.646222\pi\)
\(998\) −32.0000 −1.01294
\(999\) 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 1183.2.c.b.337.2 2
13.5 odd 4 1183.2.a.b.1.1 1
13.8 odd 4 91.2.a.a.1.1 1
13.12 even 2 inner 1183.2.c.b.337.1 2
39.8 even 4 819.2.a.f.1.1 1
52.47 even 4 1456.2.a.g.1.1 1
65.34 odd 4 2275.2.a.h.1.1 1
91.34 even 4 637.2.a.a.1.1 1
91.47 even 12 637.2.e.d.508.1 2
91.60 odd 12 637.2.e.e.79.1 2
91.73 even 12 637.2.e.d.79.1 2
91.83 even 4 8281.2.a.l.1.1 1
91.86 odd 12 637.2.e.e.508.1 2
104.21 odd 4 5824.2.a.s.1.1 1
104.99 even 4 5824.2.a.t.1.1 1
273.125 odd 4 5733.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.a.a.1.1 1 13.8 odd 4
637.2.a.a.1.1 1 91.34 even 4
637.2.e.d.79.1 2 91.73 even 12
637.2.e.d.508.1 2 91.47 even 12
637.2.e.e.79.1 2 91.60 odd 12
637.2.e.e.508.1 2 91.86 odd 12
819.2.a.f.1.1 1 39.8 even 4
1183.2.a.b.1.1 1 13.5 odd 4
1183.2.c.b.337.1 2 13.12 even 2 inner
1183.2.c.b.337.2 2 1.1 even 1 trivial
1456.2.a.g.1.1 1 52.47 even 4
2275.2.a.h.1.1 1 65.34 odd 4
5733.2.a.l.1.1 1 273.125 odd 4
5824.2.a.s.1.1 1 104.21 odd 4
5824.2.a.t.1.1 1 104.99 even 4
8281.2.a.l.1.1 1 91.83 even 4