Properties

Label 225.2.b.a.199.2
Level $225$
Weight $2$
Character 225.199
Analytic conductor $1.797$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 225.199
Dual form 225.2.b.a.199.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{2} -2.00000 q^{4} +3.00000i q^{7} +O(q^{10})\) \(q+2.00000i q^{2} -2.00000 q^{4} +3.00000i q^{7} -2.00000 q^{11} +1.00000i q^{13} -6.00000 q^{14} -4.00000 q^{16} +2.00000i q^{17} +5.00000 q^{19} -4.00000i q^{22} -6.00000i q^{23} -2.00000 q^{26} -6.00000i q^{28} +10.0000 q^{29} -3.00000 q^{31} -8.00000i q^{32} -4.00000 q^{34} -2.00000i q^{37} +10.0000i q^{38} +8.00000 q^{41} +1.00000i q^{43} +4.00000 q^{44} +12.0000 q^{46} +2.00000i q^{47} -2.00000 q^{49} -2.00000i q^{52} +4.00000i q^{53} +20.0000i q^{58} -10.0000 q^{59} +7.00000 q^{61} -6.00000i q^{62} +8.00000 q^{64} +3.00000i q^{67} -4.00000i q^{68} +8.00000 q^{71} -14.0000i q^{73} +4.00000 q^{74} -10.0000 q^{76} -6.00000i q^{77} +16.0000i q^{82} -6.00000i q^{83} -2.00000 q^{86} -3.00000 q^{91} +12.0000i q^{92} -4.00000 q^{94} -17.0000i q^{97} -4.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} + O(q^{10}) \) \( 2q - 4q^{4} - 4q^{11} - 12q^{14} - 8q^{16} + 10q^{19} - 4q^{26} + 20q^{29} - 6q^{31} - 8q^{34} + 16q^{41} + 8q^{44} + 24q^{46} - 4q^{49} - 20q^{59} + 14q^{61} + 16q^{64} + 16q^{71} + 8q^{74} - 20q^{76} - 4q^{86} - 6q^{91} - 8q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) −6.00000 −1.60357
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 4.00000i − 0.852803i
\(23\) − 6.00000i − 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) − 6.00000i − 1.13389i
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) − 8.00000i − 1.41421i
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 10.0000i 1.62221i
\(39\) 0 0
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 1.00000i 0.152499i 0.997089 + 0.0762493i \(0.0242945\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 12.0000 1.76930
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) − 2.00000i − 0.277350i
\(53\) 4.00000i 0.549442i 0.961524 + 0.274721i \(0.0885855\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 20.0000i 2.62613i
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) − 6.00000i − 0.762001i
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 3.00000i 0.366508i 0.983066 + 0.183254i \(0.0586631\pi\)
−0.983066 + 0.183254i \(0.941337\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) − 14.0000i − 1.63858i −0.573382 0.819288i \(-0.694369\pi\)
0.573382 0.819288i \(-0.305631\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) −10.0000 −1.14708
\(77\) − 6.00000i − 0.683763i
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 16.0000i 1.76690i
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) 12.0000i 1.25109i
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) 0 0
\(97\) − 17.0000i − 1.72609i −0.505128 0.863044i \(-0.668555\pi\)
0.505128 0.863044i \(-0.331445\pi\)
\(98\) − 4.00000i − 0.404061i
\(99\) 0 0
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −8.00000 −0.777029
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 12.0000i − 1.13389i
\(113\) 4.00000i 0.376288i 0.982141 + 0.188144i \(0.0602472\pi\)
−0.982141 + 0.188144i \(0.939753\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −20.0000 −1.85695
\(117\) 0 0
\(118\) − 20.0000i − 1.84115i
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 14.0000i 1.26750i
\(123\) 0 0
\(124\) 6.00000 0.538816
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 15.0000i 1.30066i
\(134\) −6.00000 −0.518321
\(135\) 0 0
\(136\) 0 0
\(137\) − 18.0000i − 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 16.0000i 1.34269i
\(143\) − 2.00000i − 0.167248i
\(144\) 0 0
\(145\) 0 0
\(146\) 28.0000 2.31730
\(147\) 0 0
\(148\) 4.00000i 0.328798i
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 7.00000 0.569652 0.284826 0.958579i \(-0.408064\pi\)
0.284826 + 0.958579i \(0.408064\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 12.0000 0.966988
\(155\) 0 0
\(156\) 0 0
\(157\) 13.0000i 1.03751i 0.854922 + 0.518756i \(0.173605\pi\)
−0.854922 + 0.518756i \(0.826395\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 18.0000 1.41860
\(162\) 0 0
\(163\) 11.0000i 0.861586i 0.902451 + 0.430793i \(0.141766\pi\)
−0.902451 + 0.430793i \(0.858234\pi\)
\(164\) −16.0000 −1.24939
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) − 2.00000i − 0.152499i
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 8.00000 0.603023
\(177\) 0 0
\(178\) 0 0
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) 17.0000 1.26360 0.631800 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) − 6.00000i − 0.444750i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 4.00000i − 0.292509i
\(188\) − 4.00000i − 0.291730i
\(189\) 0 0
\(190\) 0 0
\(191\) −22.0000 −1.59186 −0.795932 0.605386i \(-0.793019\pi\)
−0.795932 + 0.605386i \(0.793019\pi\)
\(192\) 0 0
\(193\) 11.0000i 0.791797i 0.918294 + 0.395899i \(0.129567\pi\)
−0.918294 + 0.395899i \(0.870433\pi\)
\(194\) 34.0000 2.44106
\(195\) 0 0
\(196\) 4.00000 0.285714
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 0 0
\(199\) 5.00000 0.354441 0.177220 0.984171i \(-0.443289\pi\)
0.177220 + 0.984171i \(0.443289\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 24.0000i − 1.68863i
\(203\) 30.0000i 2.10559i
\(204\) 0 0
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) − 4.00000i − 0.277350i
\(209\) −10.0000 −0.691714
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) − 8.00000i − 0.549442i
\(213\) 0 0
\(214\) −24.0000 −1.64061
\(215\) 0 0
\(216\) 0 0
\(217\) − 9.00000i − 0.610960i
\(218\) − 10.0000i − 0.677285i
\(219\) 0 0
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) − 19.0000i − 1.27233i −0.771551 0.636167i \(-0.780519\pi\)
0.771551 0.636167i \(-0.219481\pi\)
\(224\) 24.0000 1.60357
\(225\) 0 0
\(226\) −8.00000 −0.532152
\(227\) − 8.00000i − 0.530979i −0.964114 0.265489i \(-0.914466\pi\)
0.964114 0.265489i \(-0.0855335\pi\)
\(228\) 0 0
\(229\) 15.0000 0.991228 0.495614 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 24.0000i 1.57229i 0.618041 + 0.786146i \(0.287927\pi\)
−0.618041 + 0.786146i \(0.712073\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 20.0000 1.30189
\(237\) 0 0
\(238\) − 12.0000i − 0.777844i
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) −23.0000 −1.48156 −0.740780 0.671748i \(-0.765544\pi\)
−0.740780 + 0.671748i \(0.765544\pi\)
\(242\) − 14.0000i − 0.899954i
\(243\) 0 0
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) 0 0
\(247\) 5.00000i 0.318142i
\(248\) 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
\(253\) 12.0000i 0.754434i
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 12.0000i 0.748539i 0.927320 + 0.374270i \(0.122107\pi\)
−0.927320 + 0.374270i \(0.877893\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) 0 0
\(262\) − 24.0000i − 1.48272i
\(263\) − 16.0000i − 0.986602i −0.869859 0.493301i \(-0.835790\pi\)
0.869859 0.493301i \(-0.164210\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −30.0000 −1.83942
\(267\) 0 0
\(268\) − 6.00000i − 0.366508i
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) − 8.00000i − 0.485071i
\(273\) 0 0
\(274\) 36.0000 2.17484
\(275\) 0 0
\(276\) 0 0
\(277\) 3.00000i 0.180253i 0.995930 + 0.0901263i \(0.0287271\pi\)
−0.995930 + 0.0901263i \(0.971273\pi\)
\(278\) − 40.0000i − 2.39904i
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) − 9.00000i − 0.534994i −0.963559 0.267497i \(-0.913803\pi\)
0.963559 0.267497i \(-0.0861966\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 24.0000i 1.41668i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 28.0000i 1.63858i
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 20.0000i 1.15857i
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) 14.0000i 0.805609i
\(303\) 0 0
\(304\) −20.0000 −1.14708
\(305\) 0 0
\(306\) 0 0
\(307\) − 7.00000i − 0.399511i −0.979846 0.199756i \(-0.935985\pi\)
0.979846 0.199756i \(-0.0640148\pi\)
\(308\) 12.0000i 0.683763i
\(309\) 0 0
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) 11.0000i 0.621757i 0.950450 + 0.310878i \(0.100623\pi\)
−0.950450 + 0.310878i \(0.899377\pi\)
\(314\) −26.0000 −1.46726
\(315\) 0 0
\(316\) 0 0
\(317\) − 8.00000i − 0.449325i −0.974437 0.224662i \(-0.927872\pi\)
0.974437 0.224662i \(-0.0721279\pi\)
\(318\) 0 0
\(319\) −20.0000 −1.11979
\(320\) 0 0
\(321\) 0 0
\(322\) 36.0000i 2.00620i
\(323\) 10.0000i 0.556415i
\(324\) 0 0
\(325\) 0 0
\(326\) −22.0000 −1.21847
\(327\) 0 0
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 12.0000i 0.658586i
\(333\) 0 0
\(334\) −24.0000 −1.31322
\(335\) 0 0
\(336\) 0 0
\(337\) 23.0000i 1.25289i 0.779466 + 0.626445i \(0.215491\pi\)
−0.779466 + 0.626445i \(0.784509\pi\)
\(338\) 24.0000i 1.30543i
\(339\) 0 0
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) 15.0000i 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) 12.0000 0.645124
\(347\) 2.00000i 0.107366i 0.998558 + 0.0536828i \(0.0170960\pi\)
−0.998558 + 0.0536828i \(0.982904\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 16.0000i 0.852803i
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) − 20.0000i − 1.05703i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 34.0000i 1.78700i
\(363\) 0 0
\(364\) 6.00000 0.314485
\(365\) 0 0
\(366\) 0 0
\(367\) − 27.0000i − 1.40939i −0.709511 0.704694i \(-0.751084\pi\)
0.709511 0.704694i \(-0.248916\pi\)
\(368\) 24.0000i 1.25109i
\(369\) 0 0
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) − 29.0000i − 1.50156i −0.660551 0.750782i \(-0.729677\pi\)
0.660551 0.750782i \(-0.270323\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) 10.0000i 0.515026i
\(378\) 0 0
\(379\) −25.0000 −1.28416 −0.642082 0.766636i \(-0.721929\pi\)
−0.642082 + 0.766636i \(0.721929\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 44.0000i − 2.25124i
\(383\) − 36.0000i − 1.83951i −0.392488 0.919757i \(-0.628386\pi\)
0.392488 0.919757i \(-0.371614\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −22.0000 −1.11977
\(387\) 0 0
\(388\) 34.0000i 1.72609i
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) 0 0
\(394\) 36.0000 1.81365
\(395\) 0 0
\(396\) 0 0
\(397\) − 7.00000i − 0.351320i −0.984451 0.175660i \(-0.943794\pi\)
0.984451 0.175660i \(-0.0562059\pi\)
\(398\) 10.0000i 0.501255i
\(399\) 0 0
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) − 3.00000i − 0.149441i
\(404\) 24.0000 1.19404
\(405\) 0 0
\(406\) −60.0000 −2.97775
\(407\) 4.00000i 0.198273i
\(408\) 0 0
\(409\) −5.00000 −0.247234 −0.123617 0.992330i \(-0.539449\pi\)
−0.123617 + 0.992330i \(0.539449\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8.00000i 0.394132i
\(413\) − 30.0000i − 1.47620i
\(414\) 0 0
\(415\) 0 0
\(416\) 8.00000 0.392232
\(417\) 0 0
\(418\) − 20.0000i − 0.978232i
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) − 26.0000i − 1.26566i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 21.0000i 1.01626i
\(428\) − 24.0000i − 1.16008i
\(429\) 0 0
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) − 29.0000i − 1.39365i −0.717241 0.696826i \(-0.754595\pi\)
0.717241 0.696826i \(-0.245405\pi\)
\(434\) 18.0000 0.864028
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) − 30.0000i − 1.43509i
\(438\) 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
\(442\) − 4.00000i − 0.190261i
\(443\) 24.0000i 1.14027i 0.821549 + 0.570137i \(0.193110\pi\)
−0.821549 + 0.570137i \(0.806890\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 38.0000 1.79935
\(447\) 0 0
\(448\) 24.0000i 1.13389i
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) −16.0000 −0.753411
\(452\) − 8.00000i − 0.376288i
\(453\) 0 0
\(454\) 16.0000 0.750917
\(455\) 0 0
\(456\) 0 0
\(457\) − 22.0000i − 1.02912i −0.857455 0.514558i \(-0.827956\pi\)
0.857455 0.514558i \(-0.172044\pi\)
\(458\) 30.0000i 1.40181i
\(459\) 0 0
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) − 24.0000i − 1.11537i −0.830051 0.557687i \(-0.811689\pi\)
0.830051 0.557687i \(-0.188311\pi\)
\(464\) −40.0000 −1.85695
\(465\) 0 0
\(466\) −48.0000 −2.22356
\(467\) − 38.0000i − 1.75843i −0.476425 0.879215i \(-0.658068\pi\)
0.476425 0.879215i \(-0.341932\pi\)
\(468\) 0 0
\(469\) −9.00000 −0.415581
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 2.00000i − 0.0919601i
\(474\) 0 0
\(475\) 0 0
\(476\) 12.0000 0.550019
\(477\) 0 0
\(478\) 40.0000i 1.82956i
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) − 46.0000i − 2.09524i
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) 0 0
\(487\) 13.0000i 0.589086i 0.955638 + 0.294543i \(0.0951675\pi\)
−0.955638 + 0.294543i \(0.904833\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8.00000 0.361035 0.180517 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 0 0
\(493\) 20.0000i 0.900755i
\(494\) −10.0000 −0.449921
\(495\) 0 0
\(496\) 12.0000 0.538816
\(497\) 24.0000i 1.07655i
\(498\) 0 0
\(499\) 5.00000 0.223831 0.111915 0.993718i \(-0.464301\pi\)
0.111915 + 0.993718i \(0.464301\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 24.0000i − 1.07117i
\(503\) − 16.0000i − 0.713405i −0.934218 0.356702i \(-0.883901\pi\)
0.934218 0.356702i \(-0.116099\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −24.0000 −1.06693
\(507\) 0 0
\(508\) − 16.0000i − 0.709885i
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 42.0000 1.85797
\(512\) 32.0000i 1.41421i
\(513\) 0 0
\(514\) −24.0000 −1.05859
\(515\) 0 0
\(516\) 0 0
\(517\) − 4.00000i − 0.175920i
\(518\) 12.0000i 0.527250i
\(519\) 0 0
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 0 0
\(523\) 31.0000i 1.35554i 0.735276 + 0.677768i \(0.237052\pi\)
−0.735276 + 0.677768i \(0.762948\pi\)
\(524\) 24.0000 1.04844
\(525\) 0 0
\(526\) 32.0000 1.39527
\(527\) − 6.00000i − 0.261364i
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) − 30.0000i − 1.30066i
\(533\) 8.00000i 0.346518i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) − 20.0000i − 0.862261i
\(539\) 4.00000 0.172292
\(540\) 0 0
\(541\) −3.00000 −0.128980 −0.0644900 0.997918i \(-0.520542\pi\)
−0.0644900 + 0.997918i \(0.520542\pi\)
\(542\) − 16.0000i − 0.687259i
\(543\) 0 0
\(544\) 16.0000 0.685994
\(545\) 0 0
\(546\) 0 0
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) 36.0000i 1.53784i
\(549\) 0 0
\(550\) 0 0
\(551\) 50.0000 2.13007
\(552\) 0 0
\(553\) 0 0
\(554\) −6.00000 −0.254916
\(555\) 0 0
\(556\) 40.0000 1.69638
\(557\) 42.0000i 1.77960i 0.456354 + 0.889799i \(0.349155\pi\)
−0.456354 + 0.889799i \(0.650845\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) 0 0
\(562\) 36.0000i 1.51857i
\(563\) − 6.00000i − 0.252870i −0.991975 0.126435i \(-0.959647\pi\)
0.991975 0.126435i \(-0.0403535\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 18.0000 0.756596
\(567\) 0 0
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −13.0000 −0.544033 −0.272017 0.962293i \(-0.587691\pi\)
−0.272017 + 0.962293i \(0.587691\pi\)
\(572\) 4.00000i 0.167248i
\(573\) 0 0
\(574\) −48.0000 −2.00348
\(575\) 0 0
\(576\) 0 0
\(577\) 13.0000i 0.541197i 0.962692 + 0.270599i \(0.0872216\pi\)
−0.962692 + 0.270599i \(0.912778\pi\)
\(578\) 26.0000i 1.08146i
\(579\) 0 0
\(580\) 0 0
\(581\) 18.0000 0.746766
\(582\) 0 0
\(583\) − 8.00000i − 0.331326i
\(584\) 0 0
\(585\) 0 0
\(586\) 12.0000 0.495715
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) 0 0
\(589\) −15.0000 −0.618064
\(590\) 0 0
\(591\) 0 0
\(592\) 8.00000i 0.328798i
\(593\) − 16.0000i − 0.657041i −0.944497 0.328521i \(-0.893450\pi\)
0.944497 0.328521i \(-0.106550\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −20.0000 −0.819232
\(597\) 0 0
\(598\) 12.0000i 0.490716i
\(599\) −20.0000 −0.817178 −0.408589 0.912719i \(-0.633979\pi\)
−0.408589 + 0.912719i \(0.633979\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) − 6.00000i − 0.244542i
\(603\) 0 0
\(604\) −14.0000 −0.569652
\(605\) 0 0
\(606\) 0 0
\(607\) 8.00000i 0.324710i 0.986732 + 0.162355i \(0.0519090\pi\)
−0.986732 + 0.162355i \(0.948091\pi\)
\(608\) − 40.0000i − 1.62221i
\(609\) 0 0
\(610\) 0 0
\(611\) −2.00000 −0.0809113
\(612\) 0 0
\(613\) − 14.0000i − 0.565455i −0.959200 0.282727i \(-0.908761\pi\)
0.959200 0.282727i \(-0.0912392\pi\)
\(614\) 14.0000 0.564994
\(615\) 0 0
\(616\) 0 0
\(617\) 12.0000i 0.483102i 0.970388 + 0.241551i \(0.0776561\pi\)
−0.970388 + 0.241551i \(0.922344\pi\)
\(618\) 0 0
\(619\) 25.0000 1.00483 0.502417 0.864625i \(-0.332444\pi\)
0.502417 + 0.864625i \(0.332444\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 36.0000i 1.44347i
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) − 26.0000i − 1.03751i
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) −23.0000 −0.915616 −0.457808 0.889051i \(-0.651365\pi\)
−0.457808 + 0.889051i \(0.651365\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 16.0000 0.635441
\(635\) 0 0
\(636\) 0 0
\(637\) − 2.00000i − 0.0792429i
\(638\) − 40.0000i − 1.58362i
\(639\) 0 0
\(640\) 0 0
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 0 0
\(643\) 36.0000i 1.41970i 0.704352 + 0.709851i \(0.251238\pi\)
−0.704352 + 0.709851i \(0.748762\pi\)
\(644\) −36.0000 −1.41860
\(645\) 0 0
\(646\) −20.0000 −0.786889
\(647\) − 28.0000i − 1.10079i −0.834903 0.550397i \(-0.814476\pi\)
0.834903 0.550397i \(-0.185524\pi\)
\(648\) 0 0
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) − 22.0000i − 0.861586i
\(653\) 14.0000i 0.547862i 0.961749 + 0.273931i \(0.0883240\pi\)
−0.961749 + 0.273931i \(0.911676\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −32.0000 −1.24939
\(657\) 0 0
\(658\) − 12.0000i − 0.467809i
\(659\) −40.0000 −1.55818 −0.779089 0.626913i \(-0.784318\pi\)
−0.779089 + 0.626913i \(0.784318\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 24.0000i 0.932786i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 60.0000i − 2.32321i
\(668\) − 24.0000i − 0.928588i
\(669\) 0 0
\(670\) 0 0
\(671\) −14.0000 −0.540464
\(672\) 0 0
\(673\) 6.00000i 0.231283i 0.993291 + 0.115642i \(0.0368924\pi\)
−0.993291 + 0.115642i \(0.963108\pi\)
\(674\) −46.0000 −1.77185
\(675\) 0 0
\(676\) −24.0000 −0.923077
\(677\) 42.0000i 1.61419i 0.590421 + 0.807096i \(0.298962\pi\)
−0.590421 + 0.807096i \(0.701038\pi\)
\(678\) 0 0
\(679\) 51.0000 1.95720
\(680\) 0 0
\(681\) 0 0
\(682\) 12.0000i 0.459504i
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −30.0000 −1.14541
\(687\) 0 0
\(688\) − 4.00000i − 0.152499i
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 12.0000i 0.456172i
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) 0 0
\(697\) 16.0000i 0.606043i
\(698\) − 20.0000i − 0.757011i
\(699\) 0 0
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) − 10.0000i − 0.377157i
\(704\) −16.0000 −0.603023
\(705\) 0 0
\(706\) 12.0000 0.451626
\(707\) − 36.0000i − 1.35392i
\(708\) 0 0
\(709\) −25.0000 −0.938895 −0.469447 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 18.0000i 0.674105i
\(714\) 0 0
\(715\) 0 0
\(716\) 20.0000 0.747435
\(717\) 0 0
\(718\) 0 0
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 12.0000i 0.446594i
\(723\) 0 0
\(724\) −34.0000 −1.26360
\(725\) 0 0
\(726\) 0 0
\(727\) 43.0000i 1.59478i 0.603463 + 0.797391i \(0.293787\pi\)
−0.603463 + 0.797391i \(0.706213\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.00000 −0.0739727
\(732\) 0 0
\(733\) − 34.0000i − 1.25582i −0.778287 0.627909i \(-0.783911\pi\)
0.778287 0.627909i \(-0.216089\pi\)
\(734\) 54.0000 1.99318
\(735\) 0 0
\(736\) −48.0000 −1.76930
\(737\) − 6.00000i − 0.221013i
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 24.0000i − 0.881068i
\(743\) 4.00000i 0.146746i 0.997305 + 0.0733729i \(0.0233763\pi\)
−0.997305 + 0.0733729i \(0.976624\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 58.0000 2.12353
\(747\) 0 0
\(748\) 8.00000i 0.292509i
\(749\) −36.0000 −1.31541
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) − 8.00000i − 0.291730i
\(753\) 0 0
\(754\) −20.0000 −0.728357
\(755\) 0 0
\(756\) 0 0
\(757\) 23.0000i 0.835949i 0.908459 + 0.417975i \(0.137260\pi\)
−0.908459 + 0.417975i \(0.862740\pi\)
\(758\) − 50.0000i − 1.81608i
\(759\) 0 0
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 0 0
\(763\) − 15.0000i − 0.543036i
\(764\) 44.0000 1.59186
\(765\) 0 0
\(766\) 72.0000 2.60147
\(767\) − 10.0000i − 0.361079i
\(768\) 0 0
\(769\) −35.0000 −1.26213 −0.631066 0.775729i \(-0.717382\pi\)
−0.631066 + 0.775729i \(0.717382\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 22.0000i − 0.791797i
\(773\) 24.0000i 0.863220i 0.902060 + 0.431610i \(0.142054\pi\)
−0.902060 + 0.431610i \(0.857946\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 40.0000 1.43315
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 24.0000i 0.858238i
\(783\) 0 0
\(784\) 8.00000 0.285714
\(785\) 0 0
\(786\) 0 0
\(787\) − 7.00000i − 0.249523i −0.992187 0.124762i \(-0.960183\pi\)
0.992187 0.124762i \(-0.0398166\pi\)
\(788\) 36.0000i 1.28245i
\(789\) 0 0
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 7.00000i 0.248577i
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) 52.0000i 1.84193i 0.389640 + 0.920967i \(0.372599\pi\)
−0.389640 + 0.920967i \(0.627401\pi\)
\(798\) 0 0
\(799\) −4.00000 −0.141510
\(800\) 0 0
\(801\) 0 0
\(802\) − 24.0000i − 0.847469i
\(803\) 28.0000i 0.988099i
\(804\) 0 0
\(805\) 0 0
\(806\) 6.00000 0.211341
\(807\) 0 0
\(808\) 0 0
\(809\) −20.0000 −0.703163 −0.351581 0.936157i \(-0.614356\pi\)
−0.351581 + 0.936157i \(0.614356\pi\)
\(810\) 0 0
\(811\) 27.0000 0.948098 0.474049 0.880498i \(-0.342792\pi\)
0.474049 + 0.880498i \(0.342792\pi\)
\(812\) − 60.0000i − 2.10559i
\(813\) 0 0
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) 0 0
\(817\) 5.00000i 0.174928i
\(818\) − 10.0000i − 0.349642i
\(819\) 0 0
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 41.0000i 1.42917i 0.699549 + 0.714585i \(0.253384\pi\)
−0.699549 + 0.714585i \(0.746616\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 60.0000 2.08767
\(827\) − 28.0000i − 0.973655i −0.873498 0.486828i \(-0.838154\pi\)
0.873498 0.486828i \(-0.161846\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 8.00000i 0.277350i
\(833\) − 4.00000i − 0.138592i
\(834\) 0 0
\(835\) 0 0
\(836\) 20.0000 0.691714
\(837\) 0 0
\(838\) − 40.0000i − 1.38178i
\(839\) 10.0000 0.345238 0.172619 0.984989i \(-0.444777\pi\)
0.172619 + 0.984989i \(0.444777\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 44.0000i 1.51634i
\(843\) 0 0
\(844\) 26.0000 0.894957
\(845\) 0 0
\(846\) 0 0
\(847\) − 21.0000i − 0.721569i
\(848\) − 16.0000i − 0.549442i
\(849\) 0 0
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) 51.0000i 1.74621i 0.487535 + 0.873103i \(0.337896\pi\)
−0.487535 + 0.873103i \(0.662104\pi\)
\(854\) −42.0000 −1.43721
\(855\) 0 0
\(856\) 0 0
\(857\) − 28.0000i − 0.956462i −0.878234 0.478231i \(-0.841278\pi\)
0.878234 0.478231i \(-0.158722\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 36.0000i 1.22616i
\(863\) − 16.0000i − 0.544646i −0.962206 0.272323i \(-0.912208\pi\)
0.962206 0.272323i \(-0.0877920\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 58.0000 1.97092
\(867\) 0 0
\(868\) 18.0000i 0.610960i
\(869\) 0 0
\(870\) 0 0
\(871\) −3.00000 −0.101651
\(872\) 0 0
\(873\) 0 0
\(874\) 60.0000 2.02953
\(875\) 0 0
\(876\) 0 0
\(877\) − 27.0000i − 0.911725i −0.890050 0.455863i \(-0.849331\pi\)
0.890050 0.455863i \(-0.150669\pi\)
\(878\) 70.0000i 2.36239i
\(879\) 0 0
\(880\) 0 0
\(881\) −32.0000 −1.07811 −0.539054 0.842271i \(-0.681218\pi\)
−0.539054 + 0.842271i \(0.681218\pi\)
\(882\) 0 0
\(883\) 41.0000i 1.37976i 0.723924 + 0.689880i \(0.242337\pi\)
−0.723924 + 0.689880i \(0.757663\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) −48.0000 −1.61259
\(887\) − 18.0000i − 0.604381i −0.953248 0.302190i \(-0.902282\pi\)
0.953248 0.302190i \(-0.0977178\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) 0 0
\(892\) 38.0000i 1.27233i
\(893\) 10.0000i 0.334637i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 40.0000i 1.33482i
\(899\) −30.0000 −1.00056
\(900\) 0 0
\(901\) −8.00000 −0.266519
\(902\) − 32.0000i − 1.06548i
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 12.0000i − 0.398453i −0.979953 0.199227i \(-0.936157\pi\)
0.979953 0.199227i \(-0.0638430\pi\)
\(908\) 16.0000i 0.530979i
\(909\) 0 0
\(910\) 0 0
\(911\) 58.0000 1.92163 0.960813 0.277198i \(-0.0894057\pi\)
0.960813 + 0.277198i \(0.0894057\pi\)
\(912\) 0 0
\(913\) 12.0000i 0.397142i
\(914\) 44.0000 1.45539
\(915\) 0 0
\(916\) −30.0000 −0.991228
\(917\) − 36.0000i − 1.18882i
\(918\) 0 0
\(919\) −55.0000 −1.81428 −0.907141 0.420826i \(-0.861740\pi\)
−0.907141 + 0.420826i \(0.861740\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 24.0000i − 0.790398i
\(923\) 8.00000i 0.263323i
\(924\) 0 0
\(925\) 0 0
\(926\) 48.0000 1.57738
\(927\) 0 0
\(928\) − 80.0000i − 2.62613i
\(929\) −50.0000 −1.64045 −0.820223 0.572043i \(-0.806151\pi\)
−0.820223 + 0.572043i \(0.806151\pi\)
\(930\) 0 0
\(931\) −10.0000 −0.327737
\(932\) − 48.0000i − 1.57229i
\(933\) 0 0
\(934\) 76.0000 2.48680
\(935\) 0 0
\(936\) 0 0
\(937\) 33.0000i 1.07806i 0.842286 + 0.539032i \(0.181210\pi\)
−0.842286 + 0.539032i \(0.818790\pi\)
\(938\) − 18.0000i − 0.587721i
\(939\) 0 0
\(940\) 0 0
\(941\) −22.0000 −0.717180 −0.358590 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) 0 0
\(943\) − 48.0000i − 1.56310i
\(944\) 40.0000 1.30189
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) − 18.0000i − 0.584921i −0.956278 0.292461i \(-0.905526\pi\)
0.956278 0.292461i \(-0.0944741\pi\)
\(948\) 0 0
\(949\) 14.0000 0.454459
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 56.0000i − 1.81402i −0.421111 0.907009i \(-0.638360\pi\)
0.421111 0.907009i \(-0.361640\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −40.0000 −1.29369
\(957\) 0 0
\(958\) 60.0000i 1.93851i
\(959\) 54.0000 1.74375
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 4.00000i 0.128965i
\(963\) 0 0
\(964\) 46.0000 1.48156
\(965\) 0 0
\(966\) 0 0
\(967\) − 32.0000i − 1.02905i −0.857475 0.514525i \(-0.827968\pi\)
0.857475 0.514525i \(-0.172032\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −42.0000 −1.34784 −0.673922 0.738802i \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) 0 0
\(973\) − 60.0000i − 1.92351i
\(974\) −26.0000 −0.833094
\(975\) 0 0
\(976\) −28.0000 −0.896258
\(977\) 2.00000i 0.0639857i 0.999488 + 0.0319928i \(0.0101854\pi\)
−0.999488 + 0.0319928i \(0.989815\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 16.0000i 0.510581i
\(983\) − 36.0000i − 1.14822i −0.818778 0.574111i \(-0.805348\pi\)
0.818778 0.574111i \(-0.194652\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −40.0000 −1.27386
\(987\) 0 0
\(988\) − 10.0000i − 0.318142i
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) 17.0000 0.540023 0.270011 0.962857i \(-0.412973\pi\)
0.270011 + 0.962857i \(0.412973\pi\)
\(992\) 24.0000i 0.762001i
\(993\) 0 0
\(994\) −48.0000 −1.52247
\(995\) 0 0
\(996\) 0 0
\(997\) − 42.0000i − 1.33015i −0.746775 0.665077i \(-0.768399\pi\)
0.746775 0.665077i \(-0.231601\pi\)
\(998\) 10.0000i 0.316544i
\(999\) 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 225.2.b.a.199.2 2
3.2 odd 2 75.2.b.a.49.1 2
4.3 odd 2 3600.2.f.p.2449.1 2
5.2 odd 4 225.2.a.a.1.1 1
5.3 odd 4 225.2.a.e.1.1 1
5.4 even 2 inner 225.2.b.a.199.1 2
12.11 even 2 1200.2.f.d.49.2 2
15.2 even 4 75.2.a.c.1.1 yes 1
15.8 even 4 75.2.a.a.1.1 1
15.14 odd 2 75.2.b.a.49.2 2
20.3 even 4 3600.2.a.j.1.1 1
20.7 even 4 3600.2.a.bk.1.1 1
20.19 odd 2 3600.2.f.p.2449.2 2
24.5 odd 2 4800.2.f.l.3649.2 2
24.11 even 2 4800.2.f.y.3649.1 2
60.23 odd 4 1200.2.a.c.1.1 1
60.47 odd 4 1200.2.a.p.1.1 1
60.59 even 2 1200.2.f.d.49.1 2
105.62 odd 4 3675.2.a.q.1.1 1
105.83 odd 4 3675.2.a.b.1.1 1
120.29 odd 2 4800.2.f.l.3649.1 2
120.53 even 4 4800.2.a.bb.1.1 1
120.59 even 2 4800.2.f.y.3649.2 2
120.77 even 4 4800.2.a.bq.1.1 1
120.83 odd 4 4800.2.a.br.1.1 1
120.107 odd 4 4800.2.a.be.1.1 1
165.32 odd 4 9075.2.a.a.1.1 1
165.98 odd 4 9075.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.a.a.1.1 1 15.8 even 4
75.2.a.c.1.1 yes 1 15.2 even 4
75.2.b.a.49.1 2 3.2 odd 2
75.2.b.a.49.2 2 15.14 odd 2
225.2.a.a.1.1 1 5.2 odd 4
225.2.a.e.1.1 1 5.3 odd 4
225.2.b.a.199.1 2 5.4 even 2 inner
225.2.b.a.199.2 2 1.1 even 1 trivial
1200.2.a.c.1.1 1 60.23 odd 4
1200.2.a.p.1.1 1 60.47 odd 4
1200.2.f.d.49.1 2 60.59 even 2
1200.2.f.d.49.2 2 12.11 even 2
3600.2.a.j.1.1 1 20.3 even 4
3600.2.a.bk.1.1 1 20.7 even 4
3600.2.f.p.2449.1 2 4.3 odd 2
3600.2.f.p.2449.2 2 20.19 odd 2
3675.2.a.b.1.1 1 105.83 odd 4
3675.2.a.q.1.1 1 105.62 odd 4
4800.2.a.bb.1.1 1 120.53 even 4
4800.2.a.be.1.1 1 120.107 odd 4
4800.2.a.bq.1.1 1 120.77 even 4
4800.2.a.br.1.1 1 120.83 odd 4
4800.2.f.l.3649.1 2 120.29 odd 2
4800.2.f.l.3649.2 2 24.5 odd 2
4800.2.f.y.3649.1 2 24.11 even 2
4800.2.f.y.3649.2 2 120.59 even 2
9075.2.a.a.1.1 1 165.32 odd 4
9075.2.a.s.1.1 1 165.98 odd 4