Properties

Label 4600.2.e.a.4049.1
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 184)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.a.4049.2

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{3} -2.00000i q^{7} -6.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} -2.00000i q^{7} -6.00000 q^{9} +5.00000i q^{13} -6.00000i q^{17} -6.00000 q^{19} -6.00000 q^{21} -1.00000i q^{23} +9.00000i q^{27} -9.00000 q^{29} +3.00000 q^{31} -8.00000i q^{37} +15.0000 q^{39} +3.00000 q^{41} +8.00000i q^{43} +7.00000i q^{47} +3.00000 q^{49} -18.0000 q^{51} +2.00000i q^{53} +18.0000i q^{57} -4.00000 q^{59} -10.0000 q^{61} +12.0000i q^{63} +8.00000i q^{67} -3.00000 q^{69} +7.00000 q^{71} -9.00000i q^{73} +6.00000 q^{79} +9.00000 q^{81} +14.0000i q^{83} +27.0000i q^{87} -16.0000 q^{89} +10.0000 q^{91} -9.00000i q^{93} +6.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{9} - 12 q^{19} - 12 q^{21} - 18 q^{29} + 6 q^{31} + 30 q^{39} + 6 q^{41} + 6 q^{49} - 36 q^{51} - 8 q^{59} - 20 q^{61} - 6 q^{69} + 14 q^{71} + 12 q^{79} + 18 q^{81} - 32 q^{89} + 20 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 3.00000i − 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 0 0
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 5.00000i 1.38675i 0.720577 + 0.693375i \(0.243877\pi\)
−0.720577 + 0.693375i \(0.756123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) −6.00000 −1.30931
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 9.00000i 1.73205i
\(28\) 0 0
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) 15.0000 2.40192
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.00000i 1.02105i 0.859861 + 0.510527i \(0.170550\pi\)
−0.859861 + 0.510527i \(0.829450\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −18.0000 −2.52050
\(52\) 0 0
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 18.0000i 2.38416i
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 12.0000i 1.51186i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 7.00000 0.830747 0.415374 0.909651i \(-0.363651\pi\)
0.415374 + 0.909651i \(0.363651\pi\)
\(72\) 0 0
\(73\) − 9.00000i − 1.05337i −0.850060 0.526685i \(-0.823435\pi\)
0.850060 0.526685i \(-0.176565\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 14.0000i 1.53670i 0.640030 + 0.768350i \(0.278922\pi\)
−0.640030 + 0.768350i \(0.721078\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 27.0000i 2.89470i
\(88\) 0 0
\(89\) −16.0000 −1.69600 −0.847998 0.529999i \(-0.822192\pi\)
−0.847998 + 0.529999i \(0.822192\pi\)
\(90\) 0 0
\(91\) 10.0000 1.04828
\(92\) 0 0
\(93\) − 9.00000i − 0.933257i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.00000i 0.609208i 0.952479 + 0.304604i \(0.0985241\pi\)
−0.952479 + 0.304604i \(0.901476\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) − 14.0000i − 1.37946i −0.724066 0.689730i \(-0.757729\pi\)
0.724066 0.689730i \(-0.242271\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.0000i 1.35343i 0.736245 + 0.676716i \(0.236597\pi\)
−0.736245 + 0.676716i \(0.763403\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −24.0000 −2.27798
\(112\) 0 0
\(113\) − 2.00000i − 0.188144i −0.995565 0.0940721i \(-0.970012\pi\)
0.995565 0.0940721i \(-0.0299884\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 30.0000i − 2.77350i
\(118\) 0 0
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) − 9.00000i − 0.811503i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 5.00000i 0.443678i 0.975083 + 0.221839i \(0.0712060\pi\)
−0.975083 + 0.221839i \(0.928794\pi\)
\(128\) 0 0
\(129\) 24.0000 2.11308
\(130\) 0 0
\(131\) −9.00000 −0.786334 −0.393167 0.919467i \(-0.628621\pi\)
−0.393167 + 0.919467i \(0.628621\pi\)
\(132\) 0 0
\(133\) 12.0000i 1.04053i
\(134\) 0 0
\(135\) 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
\(139\) 23.0000 1.95083 0.975417 0.220366i \(-0.0707252\pi\)
0.975417 + 0.220366i \(0.0707252\pi\)
\(140\) 0 0
\(141\) 21.0000 1.76852
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 9.00000i − 0.742307i
\(148\) 0 0
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) 7.00000 0.569652 0.284826 0.958579i \(-0.408064\pi\)
0.284826 + 0.958579i \(0.408064\pi\)
\(152\) 0 0
\(153\) 36.0000i 2.91043i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 12.0000i 0.957704i 0.877896 + 0.478852i \(0.158947\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) 11.0000i 0.861586i 0.902451 + 0.430793i \(0.141766\pi\)
−0.902451 + 0.430793i \(0.858234\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.00000i 0.309529i 0.987951 + 0.154765i \(0.0494619\pi\)
−0.987951 + 0.154765i \(0.950538\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 36.0000 2.75299
\(172\) 0 0
\(173\) 10.0000i 0.760286i 0.924928 + 0.380143i \(0.124125\pi\)
−0.924928 + 0.380143i \(0.875875\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000i 0.901975i
\(178\) 0 0
\(179\) 3.00000 0.224231 0.112115 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 30.0000i 2.21766i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 18.0000 1.30931
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) 7.00000i 0.503871i 0.967744 + 0.251936i \(0.0810671\pi\)
−0.967744 + 0.251936i \(0.918933\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 27.0000i − 1.92367i −0.273629 0.961835i \(-0.588224\pi\)
0.273629 0.961835i \(-0.411776\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 24.0000 1.69283
\(202\) 0 0
\(203\) 18.0000i 1.26335i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.00000i 0.417029i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) − 21.0000i − 1.43890i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 6.00000i − 0.407307i
\(218\) 0 0
\(219\) −27.0000 −1.82449
\(220\) 0 0
\(221\) 30.0000 2.01802
\(222\) 0 0
\(223\) − 8.00000i − 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 10.0000i − 0.663723i −0.943328 0.331862i \(-0.892323\pi\)
0.943328 0.331862i \(-0.107677\pi\)
\(228\) 0 0
\(229\) 20.0000 1.32164 0.660819 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 15.0000i − 0.982683i −0.870967 0.491341i \(-0.836507\pi\)
0.870967 0.491341i \(-0.163493\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 18.0000i − 1.16923i
\(238\) 0 0
\(239\) 13.0000 0.840900 0.420450 0.907316i \(-0.361872\pi\)
0.420450 + 0.907316i \(0.361872\pi\)
\(240\) 0 0
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 30.0000i − 1.90885i
\(248\) 0 0
\(249\) 42.0000 2.66164
\(250\) 0 0
\(251\) 10.0000 0.631194 0.315597 0.948893i \(-0.397795\pi\)
0.315597 + 0.948893i \(0.397795\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 13.0000i − 0.810918i −0.914113 0.405459i \(-0.867112\pi\)
0.914113 0.405459i \(-0.132888\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) 54.0000 3.34252
\(262\) 0 0
\(263\) − 6.00000i − 0.369976i −0.982741 0.184988i \(-0.940775\pi\)
0.982741 0.184988i \(-0.0592246\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 48.0000i 2.93755i
\(268\) 0 0
\(269\) −5.00000 −0.304855 −0.152428 0.988315i \(-0.548709\pi\)
−0.152428 + 0.988315i \(0.548709\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 0 0
\(273\) − 30.0000i − 1.81568i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 23.0000i − 1.38194i −0.722885 0.690968i \(-0.757185\pi\)
0.722885 0.690968i \(-0.242815\pi\)
\(278\) 0 0
\(279\) −18.0000 −1.07763
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 6.00000i − 0.354169i
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 18.0000 1.05518
\(292\) 0 0
\(293\) − 12.0000i − 0.701047i −0.936554 0.350524i \(-0.886004\pi\)
0.936554 0.350524i \(-0.113996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.00000 0.289157
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) 0 0
\(303\) − 18.0000i − 1.03407i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 0 0
\(309\) −42.0000 −2.38930
\(310\) 0 0
\(311\) −17.0000 −0.963982 −0.481991 0.876176i \(-0.660086\pi\)
−0.481991 + 0.876176i \(0.660086\pi\)
\(312\) 0 0
\(313\) 20.0000i 1.13047i 0.824931 + 0.565233i \(0.191214\pi\)
−0.824931 + 0.565233i \(0.808786\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 2.00000i − 0.112331i −0.998421 0.0561656i \(-0.982113\pi\)
0.998421 0.0561656i \(-0.0178875\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 42.0000 2.34421
\(322\) 0 0
\(323\) 36.0000i 2.00309i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 14.0000 0.771845
\(330\) 0 0
\(331\) −19.0000 −1.04433 −0.522167 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) 0 0
\(333\) 48.0000i 2.63038i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.00000i 0.435788i 0.975972 + 0.217894i \(0.0699187\pi\)
−0.975972 + 0.217894i \(0.930081\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 20.0000i − 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) 7.00000 0.374701 0.187351 0.982293i \(-0.440010\pi\)
0.187351 + 0.982293i \(0.440010\pi\)
\(350\) 0 0
\(351\) −45.0000 −2.40192
\(352\) 0 0
\(353\) 11.0000i 0.585471i 0.956193 + 0.292735i \(0.0945655\pi\)
−0.956193 + 0.292735i \(0.905434\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 36.0000i 1.90532i
\(358\) 0 0
\(359\) 18.0000 0.950004 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 33.0000i 1.73205i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 10.0000i − 0.521996i −0.965339 0.260998i \(-0.915948\pi\)
0.965339 0.260998i \(-0.0840516\pi\)
\(368\) 0 0
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) 4.00000 0.207670
\(372\) 0 0
\(373\) − 4.00000i − 0.207112i −0.994624 0.103556i \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 45.0000i − 2.31762i
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 15.0000 0.768473
\(382\) 0 0
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 48.0000i − 2.43998i
\(388\) 0 0
\(389\) 4.00000 0.202808 0.101404 0.994845i \(-0.467667\pi\)
0.101404 + 0.994845i \(0.467667\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) 27.0000i 1.36197i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 31.0000i 1.55585i 0.628360 + 0.777923i \(0.283727\pi\)
−0.628360 + 0.777923i \(0.716273\pi\)
\(398\) 0 0
\(399\) 36.0000 1.80225
\(400\) 0 0
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 0 0
\(403\) 15.0000i 0.747203i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −25.0000 −1.23617 −0.618085 0.786111i \(-0.712091\pi\)
−0.618085 + 0.786111i \(0.712091\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) 8.00000i 0.393654i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 69.0000i − 3.37894i
\(418\) 0 0
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) − 42.0000i − 2.04211i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 20.0000i 0.967868i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) 16.0000i 0.768911i 0.923144 + 0.384455i \(0.125611\pi\)
−0.923144 + 0.384455i \(0.874389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.00000i 0.287019i
\(438\) 0 0
\(439\) −5.00000 −0.238637 −0.119318 0.992856i \(-0.538071\pi\)
−0.119318 + 0.992856i \(0.538071\pi\)
\(440\) 0 0
\(441\) −18.0000 −0.857143
\(442\) 0 0
\(443\) 9.00000i 0.427603i 0.976877 + 0.213801i \(0.0685846\pi\)
−0.976877 + 0.213801i \(0.931415\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 42.0000i 1.98653i
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) − 21.0000i − 0.986666i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 38.0000i 1.77757i 0.458329 + 0.888783i \(0.348448\pi\)
−0.458329 + 0.888783i \(0.651552\pi\)
\(458\) 0 0
\(459\) 54.0000 2.52050
\(460\) 0 0
\(461\) 1.00000 0.0465746 0.0232873 0.999729i \(-0.492587\pi\)
0.0232873 + 0.999729i \(0.492587\pi\)
\(462\) 0 0
\(463\) − 16.0000i − 0.743583i −0.928316 0.371792i \(-0.878744\pi\)
0.928316 0.371792i \(-0.121256\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.00000i 0.277647i 0.990317 + 0.138823i \(0.0443321\pi\)
−0.990317 + 0.138823i \(0.955668\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 36.0000 1.65879
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 12.0000i − 0.549442i
\(478\) 0 0
\(479\) −38.0000 −1.73626 −0.868132 0.496333i \(-0.834679\pi\)
−0.868132 + 0.496333i \(0.834679\pi\)
\(480\) 0 0
\(481\) 40.0000 1.82384
\(482\) 0 0
\(483\) 6.00000i 0.273009i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 17.0000i 0.770344i 0.922845 + 0.385172i \(0.125858\pi\)
−0.922845 + 0.385172i \(0.874142\pi\)
\(488\) 0 0
\(489\) 33.0000 1.49231
\(490\) 0 0
\(491\) −7.00000 −0.315906 −0.157953 0.987447i \(-0.550489\pi\)
−0.157953 + 0.987447i \(0.550489\pi\)
\(492\) 0 0
\(493\) 54.0000i 2.43204i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 14.0000i − 0.627986i
\(498\) 0 0
\(499\) 1.00000 0.0447661 0.0223831 0.999749i \(-0.492875\pi\)
0.0223831 + 0.999749i \(0.492875\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) − 10.0000i − 0.445878i −0.974832 0.222939i \(-0.928435\pi\)
0.974832 0.222939i \(-0.0715651\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 36.0000i 1.59882i
\(508\) 0 0
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) −18.0000 −0.796273
\(512\) 0 0
\(513\) − 54.0000i − 2.38416i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 30.0000 1.31685
\(520\) 0 0
\(521\) 8.00000 0.350486 0.175243 0.984525i \(-0.443929\pi\)
0.175243 + 0.984525i \(0.443929\pi\)
\(522\) 0 0
\(523\) − 18.0000i − 0.787085i −0.919306 0.393543i \(-0.871249\pi\)
0.919306 0.393543i \(-0.128751\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 18.0000i − 0.784092i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) 0 0
\(533\) 15.0000i 0.649722i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 9.00000i − 0.388379i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −43.0000 −1.84871 −0.924357 0.381528i \(-0.875398\pi\)
−0.924357 + 0.381528i \(0.875398\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 19.0000i − 0.812381i −0.913788 0.406191i \(-0.866857\pi\)
0.913788 0.406191i \(-0.133143\pi\)
\(548\) 0 0
\(549\) 60.0000 2.56074
\(550\) 0 0
\(551\) 54.0000 2.30048
\(552\) 0 0
\(553\) − 12.0000i − 0.510292i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.00000i 0.338971i 0.985533 + 0.169485i \(0.0542106\pi\)
−0.985533 + 0.169485i \(0.945789\pi\)
\(558\) 0 0
\(559\) −40.0000 −1.69182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 44.0000i − 1.85438i −0.374593 0.927189i \(-0.622217\pi\)
0.374593 0.927189i \(-0.377783\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 18.0000i − 0.755929i
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 26.0000 1.08807 0.544033 0.839064i \(-0.316897\pi\)
0.544033 + 0.839064i \(0.316897\pi\)
\(572\) 0 0
\(573\) 48.0000i 2.00523i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 33.0000i 1.37381i 0.726748 + 0.686904i \(0.241031\pi\)
−0.726748 + 0.686904i \(0.758969\pi\)
\(578\) 0 0
\(579\) 21.0000 0.872730
\(580\) 0 0
\(581\) 28.0000 1.16164
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15.0000i 0.619116i 0.950881 + 0.309558i \(0.100181\pi\)
−0.950881 + 0.309558i \(0.899819\pi\)
\(588\) 0 0
\(589\) −18.0000 −0.741677
\(590\) 0 0
\(591\) −81.0000 −3.33189
\(592\) 0 0
\(593\) − 22.0000i − 0.903432i −0.892162 0.451716i \(-0.850812\pi\)
0.892162 0.451716i \(-0.149188\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 60.0000i 2.45564i
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 13.0000 0.530281 0.265141 0.964210i \(-0.414582\pi\)
0.265141 + 0.964210i \(0.414582\pi\)
\(602\) 0 0
\(603\) − 48.0000i − 1.95471i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 40.0000i 1.62355i 0.583970 + 0.811775i \(0.301498\pi\)
−0.583970 + 0.811775i \(0.698502\pi\)
\(608\) 0 0
\(609\) 54.0000 2.18819
\(610\) 0 0
\(611\) −35.0000 −1.41595
\(612\) 0 0
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 9.00000 0.361158
\(622\) 0 0
\(623\) 32.0000i 1.28205i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −48.0000 −1.91389
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) − 24.0000i − 0.953914i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 15.0000i 0.594322i
\(638\) 0 0
\(639\) −42.0000 −1.66149
\(640\) 0 0
\(641\) −40.0000 −1.57991 −0.789953 0.613168i \(-0.789895\pi\)
−0.789953 + 0.613168i \(0.789895\pi\)
\(642\) 0 0
\(643\) 44.0000i 1.73519i 0.497271 + 0.867595i \(0.334335\pi\)
−0.497271 + 0.867595i \(0.665665\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 33.0000i − 1.29736i −0.761060 0.648682i \(-0.775321\pi\)
0.761060 0.648682i \(-0.224679\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −18.0000 −0.705476
\(652\) 0 0
\(653\) − 39.0000i − 1.52619i −0.646288 0.763094i \(-0.723679\pi\)
0.646288 0.763094i \(-0.276321\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 54.0000i 2.10674i
\(658\) 0 0
\(659\) −50.0000 −1.94772 −0.973862 0.227142i \(-0.927062\pi\)
−0.973862 + 0.227142i \(0.927062\pi\)
\(660\) 0 0
\(661\) −42.0000 −1.63361 −0.816805 0.576913i \(-0.804257\pi\)
−0.816805 + 0.576913i \(0.804257\pi\)
\(662\) 0 0
\(663\) − 90.0000i − 3.49531i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9.00000i 0.348481i
\(668\) 0 0
\(669\) −24.0000 −0.927894
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 5.00000i 0.192736i 0.995346 + 0.0963679i \(0.0307225\pi\)
−0.995346 + 0.0963679i \(0.969277\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 22.0000i − 0.845529i −0.906240 0.422764i \(-0.861060\pi\)
0.906240 0.422764i \(-0.138940\pi\)
\(678\) 0 0
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) −30.0000 −1.14960
\(682\) 0 0
\(683\) − 29.0000i − 1.10965i −0.831966 0.554827i \(-0.812784\pi\)
0.831966 0.554827i \(-0.187216\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 60.0000i − 2.28914i
\(688\) 0 0
\(689\) −10.0000 −0.380970
\(690\) 0 0
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 18.0000i − 0.681799i
\(698\) 0 0
\(699\) −45.0000 −1.70206
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) 48.0000i 1.81035i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 12.0000i − 0.451306i
\(708\) 0 0
\(709\) −14.0000 −0.525781 −0.262891 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) −36.0000 −1.35011
\(712\) 0 0
\(713\) − 3.00000i − 0.112351i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 39.0000i − 1.45648i
\(718\) 0 0
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 0 0
\(721\) −28.0000 −1.04277
\(722\) 0 0
\(723\) 12.0000i 0.446285i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 36.0000i − 1.33517i −0.744535 0.667583i \(-0.767329\pi\)
0.744535 0.667583i \(-0.232671\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) 0 0
\(733\) 14.0000i 0.517102i 0.965998 + 0.258551i \(0.0832450\pi\)
−0.965998 + 0.258551i \(0.916755\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −29.0000 −1.06678 −0.533391 0.845869i \(-0.679083\pi\)
−0.533391 + 0.845869i \(0.679083\pi\)
\(740\) 0 0
\(741\) −90.0000 −3.30623
\(742\) 0 0
\(743\) 6.00000i 0.220119i 0.993925 + 0.110059i \(0.0351041\pi\)
−0.993925 + 0.110059i \(0.964896\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 84.0000i − 3.07340i
\(748\) 0 0
\(749\) 28.0000 1.02310
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 0 0
\(753\) − 30.0000i − 1.09326i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 12.0000i − 0.436147i −0.975932 0.218074i \(-0.930023\pi\)
0.975932 0.218074i \(-0.0699773\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 41.0000 1.48625 0.743124 0.669153i \(-0.233343\pi\)
0.743124 + 0.669153i \(0.233343\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 20.0000i − 0.722158i
\(768\) 0 0
\(769\) 28.0000 1.00971 0.504853 0.863205i \(-0.331547\pi\)
0.504853 + 0.863205i \(0.331547\pi\)
\(770\) 0 0
\(771\) −39.0000 −1.40455
\(772\) 0 0
\(773\) − 8.00000i − 0.287740i −0.989597 0.143870i \(-0.954045\pi\)
0.989597 0.143870i \(-0.0459547\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 48.0000i 1.72199i
\(778\) 0 0
\(779\) −18.0000 −0.644917
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 81.0000i − 2.89470i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 40.0000i 1.42585i 0.701242 + 0.712923i \(0.252629\pi\)
−0.701242 + 0.712923i \(0.747371\pi\)
\(788\) 0 0
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) −4.00000 −0.142224
\(792\) 0 0
\(793\) − 50.0000i − 1.77555i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 42.0000i 1.48772i 0.668338 + 0.743858i \(0.267006\pi\)
−0.668338 + 0.743858i \(0.732994\pi\)
\(798\) 0 0
\(799\) 42.0000 1.48585
\(800\) 0 0
\(801\) 96.0000 3.39199
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 15.0000i 0.528025i
\(808\) 0 0
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) −5.00000 −0.175574 −0.0877869 0.996139i \(-0.527979\pi\)
−0.0877869 + 0.996139i \(0.527979\pi\)
\(812\) 0 0
\(813\) 24.0000i 0.841717i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 48.0000i − 1.67931i
\(818\) 0 0
\(819\) −60.0000 −2.09657
\(820\) 0 0
\(821\) −54.0000 −1.88461 −0.942306 0.334751i \(-0.891348\pi\)
−0.942306 + 0.334751i \(0.891348\pi\)
\(822\) 0 0
\(823\) 57.0000i 1.98690i 0.114289 + 0.993448i \(0.463541\pi\)
−0.114289 + 0.993448i \(0.536459\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 12.0000i − 0.417281i −0.977992 0.208640i \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) 0 0
\(829\) −54.0000 −1.87550 −0.937749 0.347314i \(-0.887094\pi\)
−0.937749 + 0.347314i \(0.887094\pi\)
\(830\) 0 0
\(831\) −69.0000 −2.39358
\(832\) 0 0
\(833\) − 18.0000i − 0.623663i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 27.0000i 0.933257i
\(838\) 0 0
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 54.0000i 1.85986i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 22.0000i 0.755929i
\(848\) 0 0
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 0 0
\(853\) 14.0000i 0.479351i 0.970853 + 0.239675i \(0.0770410\pi\)
−0.970853 + 0.239675i \(0.922959\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 23.0000i 0.785665i 0.919610 + 0.392833i \(0.128505\pi\)
−0.919610 + 0.392833i \(0.871495\pi\)
\(858\) 0 0
\(859\) 17.0000 0.580033 0.290016 0.957022i \(-0.406339\pi\)
0.290016 + 0.957022i \(0.406339\pi\)
\(860\) 0 0
\(861\) −18.0000 −0.613438
\(862\) 0 0
\(863\) − 49.0000i − 1.66798i −0.551780 0.833990i \(-0.686051\pi\)
0.551780 0.833990i \(-0.313949\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 57.0000i 1.93582i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −40.0000 −1.35535
\(872\) 0 0
\(873\) − 36.0000i − 1.21842i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 18.0000i 0.607817i 0.952701 + 0.303908i \(0.0982917\pi\)
−0.952701 + 0.303908i \(0.901708\pi\)
\(878\) 0 0
\(879\) −36.0000 −1.21425
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) 36.0000i 1.21150i 0.795656 + 0.605748i \(0.207126\pi\)
−0.795656 + 0.605748i \(0.792874\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 27.0000i 0.906571i 0.891365 + 0.453286i \(0.149748\pi\)
−0.891365 + 0.453286i \(0.850252\pi\)
\(888\) 0 0
\(889\) 10.0000 0.335389
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 42.0000i − 1.40548i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 15.0000i − 0.500835i
\(898\) 0 0
\(899\) −27.0000 −0.900500
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) − 48.0000i − 1.59734i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 10.0000i − 0.332045i −0.986122 0.166022i \(-0.946908\pi\)
0.986122 0.166022i \(-0.0530924\pi\)
\(908\) 0 0
\(909\) −36.0000 −1.19404
\(910\) 0 0
\(911\) −50.0000 −1.65657 −0.828287 0.560304i \(-0.810684\pi\)
−0.828287 + 0.560304i \(0.810684\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 18.0000i 0.594412i
\(918\) 0 0
\(919\) −10.0000 −0.329870 −0.164935 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) 35.0000i 1.15204i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 84.0000i 2.75892i
\(928\) 0 0
\(929\) −7.00000 −0.229663 −0.114831 0.993385i \(-0.536633\pi\)
−0.114831 + 0.993385i \(0.536633\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) 0 0
\(933\) 51.0000i 1.66967i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 4.00000i 0.130674i 0.997863 + 0.0653372i \(0.0208123\pi\)
−0.997863 + 0.0653372i \(0.979188\pi\)
\(938\) 0 0
\(939\) 60.0000 1.95803
\(940\) 0 0
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 0 0
\(943\) − 3.00000i − 0.0976934i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 7.00000i 0.227469i 0.993511 + 0.113735i \(0.0362814\pi\)
−0.993511 + 0.113735i \(0.963719\pi\)
\(948\) 0 0
\(949\) 45.0000 1.46076
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) − 42.0000i − 1.36051i −0.732974 0.680257i \(-0.761868\pi\)
0.732974 0.680257i \(-0.238132\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 8.00000 0.258333
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) − 84.0000i − 2.70686i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 47.0000i − 1.51142i −0.654907 0.755709i \(-0.727292\pi\)
0.654907 0.755709i \(-0.272708\pi\)
\(968\) 0 0
\(969\) 108.000 3.46946
\(970\) 0 0
\(971\) −26.0000 −0.834380 −0.417190 0.908819i \(-0.636985\pi\)
−0.417190 + 0.908819i \(0.636985\pi\)
\(972\) 0 0
\(973\) − 46.0000i − 1.47469i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 14.0000i 0.446531i 0.974758 + 0.223265i \(0.0716716\pi\)
−0.974758 + 0.223265i \(0.928328\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 42.0000i − 1.33687i
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 57.0000i 1.80884i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 50.0000i − 1.58352i −0.610835 0.791758i \(-0.709166\pi\)
0.610835 0.791758i \(-0.290834\pi\)
\(998\) 0 0
\(999\) 72.0000 2.27798
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 4600.2.e.a.4049.1 2
5.2 odd 4 4600.2.a.a.1.1 1
5.3 odd 4 184.2.a.d.1.1 1
5.4 even 2 inner 4600.2.e.a.4049.2 2
15.8 even 4 1656.2.a.c.1.1 1
20.3 even 4 368.2.a.a.1.1 1
20.7 even 4 9200.2.a.bj.1.1 1
35.13 even 4 9016.2.a.b.1.1 1
40.3 even 4 1472.2.a.m.1.1 1
40.13 odd 4 1472.2.a.a.1.1 1
60.23 odd 4 3312.2.a.i.1.1 1
115.68 even 4 4232.2.a.j.1.1 1
460.183 odd 4 8464.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
184.2.a.d.1.1 1 5.3 odd 4
368.2.a.a.1.1 1 20.3 even 4
1472.2.a.a.1.1 1 40.13 odd 4
1472.2.a.m.1.1 1 40.3 even 4
1656.2.a.c.1.1 1 15.8 even 4
3312.2.a.i.1.1 1 60.23 odd 4
4232.2.a.j.1.1 1 115.68 even 4
4600.2.a.a.1.1 1 5.2 odd 4
4600.2.e.a.4049.1 2 1.1 even 1 trivial
4600.2.e.a.4049.2 2 5.4 even 2 inner
8464.2.a.b.1.1 1 460.183 odd 4
9016.2.a.b.1.1 1 35.13 even 4
9200.2.a.bj.1.1 1 20.7 even 4