Properties

Label 1440.2.o.b.1439.12
Level $1440$
Weight $2$
Character 1440.1439
Analytic conductor $11.498$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1440,2,Mod(1439,1440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1440, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1440.1439");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.o (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.426337261060096.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1439.12
Root \(1.41127 - 0.0912546i\) of defining polynomial
Character \(\chi\) \(=\) 1440.1439
Dual form 1440.2.o.b.1439.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.20963 + 0.342849i) q^{5} -2.64002 q^{7} +O(q^{10})\) \(q+(2.20963 + 0.342849i) q^{5} -2.64002 q^{7} -3.00504 q^{11} +0.640023i q^{13} +0.685698 q^{17} +5.28005i q^{19} +2.27653i q^{23} +(4.76491 + 1.51514i) q^{25} +8.15281i q^{29} +2.96972i q^{31} +(-5.83347 - 0.905130i) q^{35} -1.60975i q^{37} +7.42430i q^{41} +11.2195 q^{43} +4.19982i q^{47} -0.0302761 q^{49} -9.60984 q^{53} +(-6.64002 - 1.03028i) q^{55} -7.20487 q^{59} +10.4995 q^{61} +(-0.219432 + 1.41421i) q^{65} +8.49954 q^{67} -13.1240 q^{71} -14.2498i q^{73} +7.93338 q^{77} +11.4693i q^{79} -13.1240i q^{83} +(1.51514 + 0.235091i) q^{85} +10.2527i q^{89} -1.68968i q^{91} +(-1.81026 + 11.6669i) q^{95} +8.31032i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{25} + 64 q^{43} - 4 q^{49} - 48 q^{55} - 8 q^{61} - 32 q^{67} + 20 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.20963 + 0.342849i 0.988176 + 0.153327i
\(6\) 0 0
\(7\) −2.64002 −0.997835 −0.498918 0.866649i \(-0.666269\pi\)
−0.498918 + 0.866649i \(0.666269\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.00504 −0.906054 −0.453027 0.891497i \(-0.649656\pi\)
−0.453027 + 0.891497i \(0.649656\pi\)
\(12\) 0 0
\(13\) 0.640023i 0.177511i 0.996053 + 0.0887553i \(0.0282889\pi\)
−0.996053 + 0.0887553i \(0.971711\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.685698 0.166306 0.0831531 0.996537i \(-0.473501\pi\)
0.0831531 + 0.996537i \(0.473501\pi\)
\(18\) 0 0
\(19\) 5.28005i 1.21133i 0.795721 + 0.605663i \(0.207092\pi\)
−0.795721 + 0.605663i \(0.792908\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.27653i 0.474689i 0.971426 + 0.237344i \(0.0762770\pi\)
−0.971426 + 0.237344i \(0.923723\pi\)
\(24\) 0 0
\(25\) 4.76491 + 1.51514i 0.952982 + 0.303028i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.15281i 1.51394i 0.653450 + 0.756970i \(0.273321\pi\)
−0.653450 + 0.756970i \(0.726679\pi\)
\(30\) 0 0
\(31\) 2.96972i 0.533378i 0.963783 + 0.266689i \(0.0859297\pi\)
−0.963783 + 0.266689i \(0.914070\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.83347 0.905130i −0.986036 0.152995i
\(36\) 0 0
\(37\) 1.60975i 0.264641i −0.991207 0.132320i \(-0.957757\pi\)
0.991207 0.132320i \(-0.0422428\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.42430i 1.15948i 0.814801 + 0.579740i \(0.196846\pi\)
−0.814801 + 0.579740i \(0.803154\pi\)
\(42\) 0 0
\(43\) 11.2195 1.71096 0.855478 0.517838i \(-0.173263\pi\)
0.855478 + 0.517838i \(0.173263\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.19982i 0.612607i 0.951934 + 0.306304i \(0.0990923\pi\)
−0.951934 + 0.306304i \(0.900908\pi\)
\(48\) 0 0
\(49\) −0.0302761 −0.00432516
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.60984 −1.32001 −0.660007 0.751260i \(-0.729447\pi\)
−0.660007 + 0.751260i \(0.729447\pi\)
\(54\) 0 0
\(55\) −6.64002 1.03028i −0.895341 0.138922i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.20487 −0.937994 −0.468997 0.883200i \(-0.655384\pi\)
−0.468997 + 0.883200i \(0.655384\pi\)
\(60\) 0 0
\(61\) 10.4995 1.34433 0.672164 0.740402i \(-0.265365\pi\)
0.672164 + 0.740402i \(0.265365\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.219432 + 1.41421i −0.0272171 + 0.175412i
\(66\) 0 0
\(67\) 8.49954 1.03838 0.519192 0.854658i \(-0.326233\pi\)
0.519192 + 0.854658i \(0.326233\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.1240 −1.55753 −0.778764 0.627317i \(-0.784153\pi\)
−0.778764 + 0.627317i \(0.784153\pi\)
\(72\) 0 0
\(73\) 14.2498i 1.66781i −0.551908 0.833905i \(-0.686100\pi\)
0.551908 0.833905i \(-0.313900\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.93338 0.904093
\(78\) 0 0
\(79\) 11.4693i 1.29039i 0.764017 + 0.645197i \(0.223225\pi\)
−0.764017 + 0.645197i \(0.776775\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.1240i 1.44054i −0.693692 0.720271i \(-0.744017\pi\)
0.693692 0.720271i \(-0.255983\pi\)
\(84\) 0 0
\(85\) 1.51514 + 0.235091i 0.164340 + 0.0254992i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.2527i 1.08679i 0.839478 + 0.543393i \(0.182861\pi\)
−0.839478 + 0.543393i \(0.817139\pi\)
\(90\) 0 0
\(91\) 1.68968i 0.177126i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.81026 + 11.6669i −0.185729 + 1.19700i
\(96\) 0 0
\(97\) 8.31032i 0.843785i 0.906646 + 0.421893i \(0.138634\pi\)
−0.906646 + 0.421893i \(0.861366\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.51413i 0.349669i 0.984598 + 0.174834i \(0.0559390\pi\)
−0.984598 + 0.174834i \(0.944061\pi\)
\(102\) 0 0
\(103\) −3.13957 −0.309351 −0.154675 0.987965i \(-0.549433\pi\)
−0.154675 + 0.987965i \(0.549433\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.46711i 0.721873i −0.932590 0.360937i \(-0.882457\pi\)
0.932590 0.360937i \(-0.117543\pi\)
\(108\) 0 0
\(109\) 2.96972 0.284448 0.142224 0.989835i \(-0.454575\pi\)
0.142224 + 0.989835i \(0.454575\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.67761 0.534105 0.267053 0.963682i \(-0.413950\pi\)
0.267053 + 0.963682i \(0.413950\pi\)
\(114\) 0 0
\(115\) −0.780505 + 5.03028i −0.0727825 + 0.469076i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.81026 −0.165946
\(120\) 0 0
\(121\) −1.96972 −0.179066
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10.0092 + 4.98154i 0.895251 + 0.445562i
\(126\) 0 0
\(127\) −19.1396 −1.69836 −0.849181 0.528102i \(-0.822904\pi\)
−0.849181 + 0.528102i \(0.822904\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 11.5760 1.01140 0.505698 0.862711i \(-0.331235\pi\)
0.505698 + 0.862711i \(0.331235\pi\)
\(132\) 0 0
\(133\) 13.9394i 1.20870i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.49596 −0.213244 −0.106622 0.994300i \(-0.534003\pi\)
−0.106622 + 0.994300i \(0.534003\pi\)
\(138\) 0 0
\(139\) 7.46927i 0.633535i −0.948503 0.316767i \(-0.897403\pi\)
0.948503 0.316767i \(-0.102597\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.92330i 0.160834i
\(144\) 0 0
\(145\) −2.79518 + 18.0147i −0.232127 + 1.49604i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.60984i 0.787269i −0.919267 0.393634i \(-0.871218\pi\)
0.919267 0.393634i \(-0.128782\pi\)
\(150\) 0 0
\(151\) 19.4693i 1.58439i 0.610270 + 0.792193i \(0.291061\pi\)
−0.610270 + 0.792193i \(0.708939\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.01817 + 6.56198i −0.0817812 + 0.527071i
\(156\) 0 0
\(157\) 12.8292i 1.02388i 0.859020 + 0.511942i \(0.171074\pi\)
−0.859020 + 0.511942i \(0.828926\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.01008i 0.473661i
\(162\) 0 0
\(163\) 21.2800 1.66678 0.833391 0.552684i \(-0.186396\pi\)
0.833391 + 0.552684i \(0.186396\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 22.8676i 1.76955i 0.466020 + 0.884774i \(0.345688\pi\)
−0.466020 + 0.884774i \(0.654312\pi\)
\(168\) 0 0
\(169\) 12.5904 0.968490
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −5.98932 −0.455360 −0.227680 0.973736i \(-0.573114\pi\)
−0.227680 + 0.973736i \(0.573114\pi\)
\(174\) 0 0
\(175\) −12.5795 4.00000i −0.950919 0.302372i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.65181 −0.198206 −0.0991029 0.995077i \(-0.531597\pi\)
−0.0991029 + 0.995077i \(0.531597\pi\)
\(180\) 0 0
\(181\) −11.4693 −0.852504 −0.426252 0.904605i \(-0.640166\pi\)
−0.426252 + 0.904605i \(0.640166\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.551901 3.55694i 0.0405765 0.261512i
\(186\) 0 0
\(187\) −2.06055 −0.150682
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.30362 −0.383757 −0.191878 0.981419i \(-0.561458\pi\)
−0.191878 + 0.981419i \(0.561458\pi\)
\(192\) 0 0
\(193\) 10.0606i 0.724174i −0.932144 0.362087i \(-0.882064\pi\)
0.932144 0.362087i \(-0.117936\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.79065 −0.412567 −0.206283 0.978492i \(-0.566137\pi\)
−0.206283 + 0.978492i \(0.566137\pi\)
\(198\) 0 0
\(199\) 24.0899i 1.70769i −0.520529 0.853844i \(-0.674265\pi\)
0.520529 0.853844i \(-0.325735\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 21.5236i 1.51066i
\(204\) 0 0
\(205\) −2.54541 + 16.4049i −0.177779 + 1.14577i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 15.8668i 1.09753i
\(210\) 0 0
\(211\) 23.4693i 1.61569i −0.589394 0.807845i \(-0.700634\pi\)
0.589394 0.807845i \(-0.299366\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 24.7909 + 3.84659i 1.69073 + 0.262336i
\(216\) 0 0
\(217\) 7.84014i 0.532223i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.438863i 0.0295211i
\(222\) 0 0
\(223\) 13.3600 0.894650 0.447325 0.894371i \(-0.352377\pi\)
0.447325 + 0.894371i \(0.352377\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.57092i 0.568872i 0.958695 + 0.284436i \(0.0918063\pi\)
−0.958695 + 0.284436i \(0.908194\pi\)
\(228\) 0 0
\(229\) −8.90917 −0.588735 −0.294367 0.955692i \(-0.595109\pi\)
−0.294367 + 0.955692i \(0.595109\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0196 −1.57357 −0.786787 0.617224i \(-0.788257\pi\)
−0.786787 + 0.617224i \(0.788257\pi\)
\(234\) 0 0
\(235\) −1.43991 + 9.28005i −0.0939291 + 0.605364i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 14.4097 0.932088 0.466044 0.884762i \(-0.345679\pi\)
0.466044 + 0.884762i \(0.345679\pi\)
\(240\) 0 0
\(241\) 3.87890 0.249862 0.124931 0.992165i \(-0.460129\pi\)
0.124931 + 0.992165i \(0.460129\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.0668989 0.0103801i −0.00427401 0.000663162i
\(246\) 0 0
\(247\) −3.37935 −0.215023
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.1754 1.52594 0.762970 0.646434i \(-0.223741\pi\)
0.762970 + 0.646434i \(0.223741\pi\)
\(252\) 0 0
\(253\) 6.84106i 0.430094i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.4383 0.775878 0.387939 0.921685i \(-0.373187\pi\)
0.387939 + 0.921685i \(0.373187\pi\)
\(258\) 0 0
\(259\) 4.24977i 0.264068i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 19.2471i 1.18683i −0.804898 0.593413i \(-0.797780\pi\)
0.804898 0.593413i \(-0.202220\pi\)
\(264\) 0 0
\(265\) −21.2342 3.29473i −1.30440 0.202393i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 15.8875i 0.968679i −0.874880 0.484340i \(-0.839060\pi\)
0.874880 0.484340i \(-0.160940\pi\)
\(270\) 0 0
\(271\) 6.02936i 0.366258i −0.983089 0.183129i \(-0.941377\pi\)
0.983089 0.183129i \(-0.0586225\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −14.3188 4.55305i −0.863453 0.274559i
\(276\) 0 0
\(277\) 12.3591i 0.742584i −0.928516 0.371292i \(-0.878915\pi\)
0.928516 0.371292i \(-0.121085\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 17.7198i 1.05708i −0.848909 0.528538i \(-0.822740\pi\)
0.848909 0.528538i \(-0.177260\pi\)
\(282\) 0 0
\(283\) 20.6206 1.22577 0.612885 0.790172i \(-0.290009\pi\)
0.612885 + 0.790172i \(0.290009\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 19.6003i 1.15697i
\(288\) 0 0
\(289\) −16.5298 −0.972342
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −21.6574 −1.26524 −0.632620 0.774463i \(-0.718020\pi\)
−0.632620 + 0.774463i \(0.718020\pi\)
\(294\) 0 0
\(295\) −15.9201 2.47018i −0.926902 0.143820i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.45703 −0.0842622
\(300\) 0 0
\(301\) −29.6197 −1.70725
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 23.2001 + 3.59976i 1.32843 + 0.206122i
\(306\) 0 0
\(307\) −3.87890 −0.221380 −0.110690 0.993855i \(-0.535306\pi\)
−0.110690 + 0.993855i \(0.535306\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.71654 0.380860 0.190430 0.981701i \(-0.439012\pi\)
0.190430 + 0.981701i \(0.439012\pi\)
\(312\) 0 0
\(313\) 21.9394i 1.24009i −0.784566 0.620045i \(-0.787114\pi\)
0.784566 0.620045i \(-0.212886\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.3616 0.806626 0.403313 0.915062i \(-0.367859\pi\)
0.403313 + 0.915062i \(0.367859\pi\)
\(318\) 0 0
\(319\) 24.4995i 1.37171i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.62052i 0.201451i
\(324\) 0 0
\(325\) −0.969724 + 3.04965i −0.0537906 + 0.169164i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11.0876i 0.611281i
\(330\) 0 0
\(331\) 4.78051i 0.262760i −0.991332 0.131380i \(-0.958059\pi\)
0.991332 0.131380i \(-0.0419409\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 18.7808 + 2.91406i 1.02611 + 0.159212i
\(336\) 0 0
\(337\) 33.1883i 1.80788i 0.427657 + 0.903941i \(0.359339\pi\)
−0.427657 + 0.903941i \(0.640661\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.92414i 0.483270i
\(342\) 0 0
\(343\) 18.5601 1.00215
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13.8304i 0.742456i −0.928542 0.371228i \(-0.878937\pi\)
0.928542 0.371228i \(-0.121063\pi\)
\(348\) 0 0
\(349\) 20.4390 1.09407 0.547037 0.837108i \(-0.315756\pi\)
0.547037 + 0.837108i \(0.315756\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −20.8379 −1.10909 −0.554545 0.832154i \(-0.687108\pi\)
−0.554545 + 0.832154i \(0.687108\pi\)
\(354\) 0 0
\(355\) −28.9991 4.49954i −1.53911 0.238811i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 22.0481 1.16365 0.581827 0.813312i \(-0.302338\pi\)
0.581827 + 0.813312i \(0.302338\pi\)
\(360\) 0 0
\(361\) −8.87890 −0.467310
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.88552 31.4867i 0.255720 1.64809i
\(366\) 0 0
\(367\) −4.70058 −0.245368 −0.122684 0.992446i \(-0.539150\pi\)
−0.122684 + 0.992446i \(0.539150\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 25.3702 1.31716
\(372\) 0 0
\(373\) 35.5104i 1.83866i −0.393486 0.919330i \(-0.628731\pi\)
0.393486 0.919330i \(-0.371269\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5.21799 −0.268740
\(378\) 0 0
\(379\) 1.52982i 0.0785815i −0.999228 0.0392907i \(-0.987490\pi\)
0.999228 0.0392907i \(-0.0125098\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.16349i 0.110549i 0.998471 + 0.0552746i \(0.0176034\pi\)
−0.998471 + 0.0552746i \(0.982397\pi\)
\(384\) 0 0
\(385\) 17.5298 + 2.71995i 0.893402 + 0.138622i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0.0207603i 0.00105259i 1.00000 0.000526294i \(0.000167524\pi\)
−1.00000 0.000526294i \(0.999832\pi\)
\(390\) 0 0
\(391\) 1.56101i 0.0789437i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.93223 + 25.3428i −0.197852 + 1.27513i
\(396\) 0 0
\(397\) 36.6694i 1.84038i 0.391468 + 0.920192i \(0.371967\pi\)
−0.391468 + 0.920192i \(0.628033\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.59556i 0.379304i 0.981851 + 0.189652i \(0.0607360\pi\)
−0.981851 + 0.189652i \(0.939264\pi\)
\(402\) 0 0
\(403\) −1.90069 −0.0946803
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.83736i 0.239779i
\(408\) 0 0
\(409\) −26.5289 −1.31177 −0.655885 0.754861i \(-0.727704\pi\)
−0.655885 + 0.754861i \(0.727704\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 19.0210 0.935963
\(414\) 0 0
\(415\) 4.49954 28.9991i 0.220874 1.42351i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −0.841553 −0.0411125 −0.0205563 0.999789i \(-0.506544\pi\)
−0.0205563 + 0.999789i \(0.506544\pi\)
\(420\) 0 0
\(421\) 26.1505 1.27450 0.637248 0.770659i \(-0.280073\pi\)
0.637248 + 0.770659i \(0.280073\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.26729 + 1.03893i 0.158487 + 0.0503954i
\(426\) 0 0
\(427\) −27.7190 −1.34142
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −38.8474 −1.87121 −0.935607 0.353044i \(-0.885147\pi\)
−0.935607 + 0.353044i \(0.885147\pi\)
\(432\) 0 0
\(433\) 30.7787i 1.47913i 0.673086 + 0.739564i \(0.264968\pi\)
−0.673086 + 0.739564i \(0.735032\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12.0202 −0.575003
\(438\) 0 0
\(439\) 26.1505i 1.24809i 0.781387 + 0.624047i \(0.214513\pi\)
−0.781387 + 0.624047i \(0.785487\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 13.8304i 0.657103i 0.944486 + 0.328552i \(0.106561\pi\)
−0.944486 + 0.328552i \(0.893439\pi\)
\(444\) 0 0
\(445\) −3.51514 + 22.6547i −0.166634 + 1.07394i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.07915i 0.0981212i −0.998796 0.0490606i \(-0.984377\pi\)
0.998796 0.0490606i \(-0.0156228\pi\)
\(450\) 0 0
\(451\) 22.3103i 1.05055i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.579304 3.73356i 0.0271582 0.175032i
\(456\) 0 0
\(457\) 18.6282i 0.871391i −0.900094 0.435695i \(-0.856502\pi\)
0.900094 0.435695i \(-0.143498\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 23.0014i 1.07128i −0.844446 0.535641i \(-0.820070\pi\)
0.844446 0.535641i \(-0.179930\pi\)
\(462\) 0 0
\(463\) 25.9806 1.20742 0.603711 0.797203i \(-0.293688\pi\)
0.603711 + 0.797203i \(0.293688\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 33.3177i 1.54176i 0.636981 + 0.770880i \(0.280183\pi\)
−0.636981 + 0.770880i \(0.719817\pi\)
\(468\) 0 0
\(469\) −22.4390 −1.03614
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −33.7151 −1.55022
\(474\) 0 0
\(475\) −8.00000 + 25.1589i −0.367065 + 1.15437i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 16.0380 0.732796 0.366398 0.930458i \(-0.380591\pi\)
0.366398 + 0.930458i \(0.380591\pi\)
\(480\) 0 0
\(481\) 1.03028 0.0469765
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.84919 + 18.3627i −0.129375 + 0.833808i
\(486\) 0 0
\(487\) 19.1396 0.867296 0.433648 0.901082i \(-0.357226\pi\)
0.433648 + 0.901082i \(0.357226\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 36.3669 1.64121 0.820607 0.571493i \(-0.193636\pi\)
0.820607 + 0.571493i \(0.193636\pi\)
\(492\) 0 0
\(493\) 5.59037i 0.251778i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 34.6476 1.55416
\(498\) 0 0
\(499\) 8.65940i 0.387648i 0.981036 + 0.193824i \(0.0620891\pi\)
−0.981036 + 0.193824i \(0.937911\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 31.1542i 1.38910i −0.719445 0.694549i \(-0.755604\pi\)
0.719445 0.694549i \(-0.244396\pi\)
\(504\) 0 0
\(505\) −1.20482 + 7.76491i −0.0536136 + 0.345534i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 28.3907i 1.25839i 0.777246 + 0.629197i \(0.216616\pi\)
−0.777246 + 0.629197i \(0.783384\pi\)
\(510\) 0 0
\(511\) 37.6197i 1.66420i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.93727 1.07640i −0.305693 0.0474317i
\(516\) 0 0
\(517\) 12.6206i 0.555055i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 16.4341i 0.719990i −0.932954 0.359995i \(-0.882778\pi\)
0.932954 0.359995i \(-0.117222\pi\)
\(522\) 0 0
\(523\) −14.2791 −0.624383 −0.312191 0.950019i \(-0.601063\pi\)
−0.312191 + 0.950019i \(0.601063\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.03633i 0.0887041i
\(528\) 0 0
\(529\) 17.8174 0.774671
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.75172 −0.205820
\(534\) 0 0
\(535\) 2.56009 16.4995i 0.110683 0.713337i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.0909809 0.00391883
\(540\) 0 0
\(541\) 1.40871 0.0605653 0.0302827 0.999541i \(-0.490359\pi\)
0.0302827 + 0.999541i \(0.490359\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.56198 + 1.01817i 0.281085 + 0.0436135i
\(546\) 0 0
\(547\) −15.3406 −0.655917 −0.327958 0.944692i \(-0.606361\pi\)
−0.327958 + 0.944692i \(0.606361\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −43.0472 −1.83387
\(552\) 0 0
\(553\) 30.2791i 1.28760i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7.53199 −0.319141 −0.159570 0.987187i \(-0.551011\pi\)
−0.159570 + 0.987187i \(0.551011\pi\)
\(558\) 0 0
\(559\) 7.18074i 0.303713i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 40.0784i 1.68910i −0.535475 0.844551i \(-0.679868\pi\)
0.535475 0.844551i \(-0.320132\pi\)
\(564\) 0 0
\(565\) 12.5454 + 1.94657i 0.527790 + 0.0818926i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 28.6803i 1.20234i 0.799121 + 0.601171i \(0.205299\pi\)
−0.799121 + 0.601171i \(0.794701\pi\)
\(570\) 0 0
\(571\) 33.9007i 1.41870i −0.704857 0.709350i \(-0.748989\pi\)
0.704857 0.709350i \(-0.251011\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3.44925 + 10.8474i −0.143844 + 0.452370i
\(576\) 0 0
\(577\) 18.4002i 0.766012i −0.923746 0.383006i \(-0.874889\pi\)
0.923746 0.383006i \(-0.125111\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 34.6476i 1.43742i
\(582\) 0 0
\(583\) 28.8780 1.19600
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 27.8869i 1.15102i 0.817796 + 0.575508i \(0.195196\pi\)
−0.817796 + 0.575508i \(0.804804\pi\)
\(588\) 0 0
\(589\) −15.6803 −0.646095
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 34.2295 1.40564 0.702818 0.711370i \(-0.251925\pi\)
0.702818 + 0.711370i \(0.251925\pi\)
\(594\) 0 0
\(595\) −4.00000 0.620646i −0.163984 0.0254440i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −22.7546 −0.929727 −0.464863 0.885382i \(-0.653897\pi\)
−0.464863 + 0.885382i \(0.653897\pi\)
\(600\) 0 0
\(601\) 11.5298 0.470311 0.235156 0.971958i \(-0.424440\pi\)
0.235156 + 0.971958i \(0.424440\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.35236 0.675318i −0.176948 0.0274556i
\(606\) 0 0
\(607\) 17.0790 0.693216 0.346608 0.938010i \(-0.387333\pi\)
0.346608 + 0.938010i \(0.387333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.68799 −0.108744
\(612\) 0 0
\(613\) 43.1084i 1.74113i −0.492053 0.870565i \(-0.663753\pi\)
0.492053 0.870565i \(-0.336247\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −23.7935 −0.957890 −0.478945 0.877845i \(-0.658981\pi\)
−0.478945 + 0.877845i \(0.658981\pi\)
\(618\) 0 0
\(619\) 32.4683i 1.30501i 0.757783 + 0.652507i \(0.226283\pi\)
−0.757783 + 0.652507i \(0.773717\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 27.0674i 1.08443i
\(624\) 0 0
\(625\) 20.4087 + 14.4390i 0.816349 + 0.577560i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.10380i 0.0440114i
\(630\) 0 0
\(631\) 13.0303i 0.518727i 0.965780 + 0.259364i \(0.0835128\pi\)
−0.965780 + 0.259364i \(0.916487\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −42.2913 6.56198i −1.67828 0.260404i
\(636\) 0 0
\(637\) 0.0193774i 0.000767761i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 38.1397i 1.50643i 0.657777 + 0.753213i \(0.271497\pi\)
−0.657777 + 0.753213i \(0.728503\pi\)
\(642\) 0 0
\(643\) −17.1589 −0.676683 −0.338341 0.941023i \(-0.609866\pi\)
−0.338341 + 0.941023i \(0.609866\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 30.4478i 1.19702i −0.801113 0.598512i \(-0.795759\pi\)
0.801113 0.598512i \(-0.204241\pi\)
\(648\) 0 0
\(649\) 21.6509 0.849873
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 11.7151 0.458447 0.229224 0.973374i \(-0.426381\pi\)
0.229224 + 0.973374i \(0.426381\pi\)
\(654\) 0 0
\(655\) 25.5786 + 3.96881i 0.999437 + 0.155074i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −10.1189 −0.394177 −0.197089 0.980386i \(-0.563149\pi\)
−0.197089 + 0.980386i \(0.563149\pi\)
\(660\) 0 0
\(661\) 11.1202 0.432525 0.216263 0.976335i \(-0.430613\pi\)
0.216263 + 0.976335i \(0.430613\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.77913 30.8010i 0.185327 1.19441i
\(666\) 0 0
\(667\) −18.5601 −0.718650
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −31.5516 −1.21803
\(672\) 0 0
\(673\) 15.0984i 0.582000i −0.956723 0.291000i \(-0.906012\pi\)
0.956723 0.291000i \(-0.0939880\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 7.82699 0.300816 0.150408 0.988624i \(-0.451941\pi\)
0.150408 + 0.988624i \(0.451941\pi\)
\(678\) 0 0
\(679\) 21.9394i 0.841959i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 34.4763i 1.31920i 0.751617 + 0.659600i \(0.229274\pi\)
−0.751617 + 0.659600i \(0.770726\pi\)
\(684\) 0 0
\(685\) −5.51514 0.855737i −0.210723 0.0326960i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 6.15052i 0.234316i
\(690\) 0 0
\(691\) 36.7181i 1.39682i 0.715696 + 0.698412i \(0.246109\pi\)
−0.715696 + 0.698412i \(0.753891\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.56083 16.5043i 0.0971379 0.626044i
\(696\) 0 0
\(697\) 5.09083i 0.192829i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.47779i 0.0558154i 0.999611 + 0.0279077i \(0.00888445\pi\)
−0.999611 + 0.0279077i \(0.991116\pi\)
\(702\) 0 0
\(703\) 8.49954 0.320566
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.27737i 0.348912i
\(708\) 0 0
\(709\) 6.02936 0.226437 0.113219 0.993570i \(-0.463884\pi\)
0.113219 + 0.993570i \(0.463884\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6.76066 −0.253189
\(714\) 0 0
\(715\) 0.659401 4.24977i 0.0246602 0.158932i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −13.3059 −0.496227 −0.248114 0.968731i \(-0.579811\pi\)
−0.248114 + 0.968731i \(0.579811\pi\)
\(720\) 0 0
\(721\) 8.28853 0.308681
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −12.3526 + 38.8474i −0.458765 + 1.44276i
\(726\) 0 0
\(727\) 17.7384 0.657881 0.328941 0.944351i \(-0.393308\pi\)
0.328941 + 0.944351i \(0.393308\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7.69319 0.284543
\(732\) 0 0
\(733\) 29.6391i 1.09475i 0.836889 + 0.547373i \(0.184372\pi\)
−0.836889 + 0.547373i \(0.815628\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −25.5415 −0.940832
\(738\) 0 0
\(739\) 19.2195i 0.707001i 0.935434 + 0.353500i \(0.115009\pi\)
−0.935434 + 0.353500i \(0.884991\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 21.8768i 0.802584i 0.915950 + 0.401292i \(0.131439\pi\)
−0.915950 + 0.401292i \(0.868561\pi\)
\(744\) 0 0
\(745\) 3.29473 21.2342i 0.120709 0.777960i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 19.7134i 0.720310i
\(750\) 0 0
\(751\) 14.5913i 0.532444i 0.963912 + 0.266222i \(0.0857754\pi\)
−0.963912 + 0.266222i \(0.914225\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −6.67502 + 43.0198i −0.242929 + 1.56565i
\(756\) 0 0
\(757\) 5.26067i 0.191202i 0.995420 + 0.0956011i \(0.0304773\pi\)
−0.995420 + 0.0956011i \(0.969523\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 22.6261i 0.820196i 0.912041 + 0.410098i \(0.134506\pi\)
−0.912041 + 0.410098i \(0.865494\pi\)
\(762\) 0 0
\(763\) −7.84014 −0.283832
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.61128i 0.166504i
\(768\) 0 0
\(769\) 0.150464 0.00542586 0.00271293 0.999996i \(-0.499136\pi\)
0.00271293 + 0.999996i \(0.499136\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.23760 0.0445133 0.0222567 0.999752i \(-0.492915\pi\)
0.0222567 + 0.999752i \(0.492915\pi\)
\(774\) 0 0
\(775\) −4.49954 + 14.1505i −0.161628 + 0.508300i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −39.2006 −1.40451
\(780\) 0 0
\(781\) 39.4381 1.41121
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −4.39850 + 28.3478i −0.156989 + 1.01178i
\(786\) 0 0
\(787\) 6.84106 0.243857 0.121929 0.992539i \(-0.461092\pi\)
0.121929 + 0.992539i \(0.461092\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −14.9890 −0.532949
\(792\) 0 0
\(793\) 6.71995i 0.238633i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 32.9437 1.16693 0.583463 0.812140i \(-0.301697\pi\)
0.583463 + 0.812140i \(0.301697\pi\)
\(798\) 0 0
\(799\) 2.87981i 0.101880i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 42.8212i 1.51113i
\(804\) 0 0
\(805\) 2.06055 13.2800i 0.0726249 0.468060i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 53.6532i 1.88635i 0.332303 + 0.943173i \(0.392174\pi\)
−0.332303 + 0.943173i \(0.607826\pi\)
\(810\) 0 0
\(811\) 19.5904i 0.687911i −0.938986 0.343955i \(-0.888233\pi\)
0.938986 0.343955i \(-0.111767\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 47.0210 + 7.29585i 1.64707 + 0.255562i
\(816\) 0 0
\(817\) 59.2395i 2.07253i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.49596i 0.0871095i 0.999051 + 0.0435548i \(0.0138683\pi\)
−0.999051 + 0.0435548i \(0.986132\pi\)
\(822\) 0 0
\(823\) −4.04117 −0.140866 −0.0704332 0.997516i \(-0.522438\pi\)
−0.0704332 + 0.997516i \(0.522438\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6.76066i 0.235091i −0.993067 0.117546i \(-0.962497\pi\)
0.993067 0.117546i \(-0.0375026\pi\)
\(828\) 0 0
\(829\) 29.5298 1.02561 0.512806 0.858504i \(-0.328606\pi\)
0.512806 + 0.858504i \(0.328606\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −0.0207603 −0.000719301
\(834\) 0 0
\(835\) −7.84014 + 50.5289i −0.271319 + 1.74862i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 10.7344 0.370593 0.185296 0.982683i \(-0.440675\pi\)
0.185296 + 0.982683i \(0.440675\pi\)
\(840\) 0 0
\(841\) −37.4683 −1.29201
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 27.8200 + 4.31660i 0.957038 + 0.148495i
\(846\) 0 0
\(847\) 5.20012 0.178678
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3.66463 0.125622
\(852\) 0 0
\(853\) 19.0109i 0.650921i 0.945556 + 0.325460i \(0.105519\pi\)
−0.945556 + 0.325460i \(0.894481\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.4584 0.835484 0.417742 0.908566i \(-0.362822\pi\)
0.417742 + 0.908566i \(0.362822\pi\)
\(858\) 0 0
\(859\) 23.4693i 0.800761i 0.916349 + 0.400381i \(0.131122\pi\)
−0.916349 + 0.400381i \(0.868878\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 11.5539i 0.393299i −0.980474 0.196650i \(-0.936994\pi\)
0.980474 0.196650i \(-0.0630062\pi\)
\(864\) 0 0
\(865\) −13.2342 2.05343i −0.449975 0.0698189i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 34.4656i 1.16917i
\(870\) 0 0
\(871\) 5.43991i 0.184324i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −26.4246 13.1514i −0.893313 0.444598i
\(876\) 0 0
\(877\) 7.38934i 0.249520i −0.992187 0.124760i \(-0.960184\pi\)
0.992187 0.124760i \(-0.0398161\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 3.05321i 0.102865i −0.998676 0.0514326i \(-0.983621\pi\)
0.998676 0.0514326i \(-0.0163787\pi\)
\(882\) 0 0
\(883\) −21.2800 −0.716131 −0.358065 0.933697i \(-0.616563\pi\)
−0.358065 + 0.933697i \(0.616563\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 31.9737i 1.07357i 0.843718 + 0.536786i \(0.180362\pi\)
−0.843718 + 0.536786i \(0.819638\pi\)
\(888\) 0 0
\(889\) 50.5289 1.69468
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −22.1753 −0.742067
\(894\) 0 0
\(895\) −5.85952 0.909172i −0.195862 0.0303903i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −24.2116 −0.807502
\(900\) 0 0
\(901\) −6.58945 −0.219527
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −25.3428 3.93223i −0.842423 0.130712i
\(906\) 0 0
\(907\) 16.6594 0.553166 0.276583 0.960990i \(-0.410798\pi\)
0.276583 + 0.960990i \(0.410798\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −23.3339 −0.773086 −0.386543 0.922271i \(-0.626331\pi\)
−0.386543 + 0.922271i \(0.626331\pi\)
\(912\) 0 0
\(913\) 39.4381i 1.30521i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −30.5608 −1.00921
\(918\) 0 0
\(919\) 21.0303i 0.693725i −0.937916 0.346862i \(-0.887247\pi\)
0.937916 0.346862i \(-0.112753\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 8.39965i 0.276478i
\(924\) 0 0
\(925\) 2.43899 7.67030i 0.0801935 0.252198i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 39.8536i 1.30755i −0.756687 0.653777i \(-0.773184\pi\)
0.756687 0.653777i \(-0.226816\pi\)
\(930\) 0 0
\(931\) 0.159859i 0.00523917i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −4.55305 0.706459i −0.148901 0.0231037i
\(936\) 0 0
\(937\) 42.5601i 1.39038i −0.718827 0.695189i \(-0.755321\pi\)
0.718827 0.695189i \(-0.244679\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.05710i 0.0670594i −0.999438 0.0335297i \(-0.989325\pi\)
0.999438 0.0335297i \(-0.0106748\pi\)
\(942\) 0 0
\(943\) −16.9016 −0.550392
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 26.4740i 0.860290i 0.902760 + 0.430145i \(0.141538\pi\)
−0.902760 + 0.430145i \(0.858462\pi\)
\(948\) 0 0
\(949\) 9.12019 0.296054
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 42.9717 1.39199 0.695994 0.718047i \(-0.254964\pi\)
0.695994 + 0.718047i \(0.254964\pi\)
\(954\) 0 0
\(955\) −11.7190 1.81834i −0.379219 0.0588402i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 6.58939 0.212782
\(960\) 0 0
\(961\) 22.1807 0.715508
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 3.44925 22.2301i 0.111035 0.715611i
\(966\) 0 0
\(967\) −11.7990 −0.379429 −0.189715 0.981839i \(-0.560756\pi\)
−0.189715 + 0.981839i \(0.560756\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 3.23112 0.103691 0.0518457 0.998655i \(-0.483490\pi\)
0.0518457 + 0.998655i \(0.483490\pi\)
\(972\) 0 0
\(973\) 19.7190i 0.632163i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 47.6210 1.52353 0.761766 0.647852i \(-0.224333\pi\)
0.761766 + 0.647852i \(0.224333\pi\)
\(978\) 0 0
\(979\) 30.8099i 0.984688i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 13.4772i 0.429856i 0.976630 + 0.214928i \(0.0689517\pi\)
−0.976630 + 0.214928i \(0.931048\pi\)
\(984\) 0 0
\(985\) −12.7952 1.98532i −0.407688 0.0632576i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 25.5415i 0.812172i
\(990\) 0 0
\(991\) 47.0284i 1.49391i 0.664876 + 0.746954i \(0.268484\pi\)
−0.664876 + 0.746954i \(0.731516\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8.25921 53.2297i 0.261834 1.68750i
\(996\) 0 0
\(997\) 31.2295i 0.989047i −0.869164 0.494524i \(-0.835343\pi\)
0.869164 0.494524i \(-0.164657\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1440.2.o.b.1439.12 yes 12
3.2 odd 2 inner 1440.2.o.b.1439.1 yes 12
4.3 odd 2 1440.2.o.a.1439.12 yes 12
5.2 odd 4 7200.2.h.l.1151.3 12
5.3 odd 4 7200.2.h.m.1151.9 12
5.4 even 2 1440.2.o.a.1439.2 yes 12
8.3 odd 2 2880.2.o.f.2879.1 12
8.5 even 2 2880.2.o.e.2879.1 12
12.11 even 2 1440.2.o.a.1439.1 12
15.2 even 4 7200.2.h.l.1151.4 12
15.8 even 4 7200.2.h.m.1151.10 12
15.14 odd 2 1440.2.o.a.1439.11 yes 12
20.3 even 4 7200.2.h.m.1151.4 12
20.7 even 4 7200.2.h.l.1151.10 12
20.19 odd 2 inner 1440.2.o.b.1439.2 yes 12
24.5 odd 2 2880.2.o.e.2879.12 12
24.11 even 2 2880.2.o.f.2879.12 12
40.19 odd 2 2880.2.o.e.2879.11 12
40.29 even 2 2880.2.o.f.2879.11 12
60.23 odd 4 7200.2.h.m.1151.3 12
60.47 odd 4 7200.2.h.l.1151.9 12
60.59 even 2 inner 1440.2.o.b.1439.11 yes 12
120.29 odd 2 2880.2.o.f.2879.2 12
120.59 even 2 2880.2.o.e.2879.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1440.2.o.a.1439.1 12 12.11 even 2
1440.2.o.a.1439.2 yes 12 5.4 even 2
1440.2.o.a.1439.11 yes 12 15.14 odd 2
1440.2.o.a.1439.12 yes 12 4.3 odd 2
1440.2.o.b.1439.1 yes 12 3.2 odd 2 inner
1440.2.o.b.1439.2 yes 12 20.19 odd 2 inner
1440.2.o.b.1439.11 yes 12 60.59 even 2 inner
1440.2.o.b.1439.12 yes 12 1.1 even 1 trivial
2880.2.o.e.2879.1 12 8.5 even 2
2880.2.o.e.2879.2 12 120.59 even 2
2880.2.o.e.2879.11 12 40.19 odd 2
2880.2.o.e.2879.12 12 24.5 odd 2
2880.2.o.f.2879.1 12 8.3 odd 2
2880.2.o.f.2879.2 12 120.29 odd 2
2880.2.o.f.2879.11 12 40.29 even 2
2880.2.o.f.2879.12 12 24.11 even 2
7200.2.h.l.1151.3 12 5.2 odd 4
7200.2.h.l.1151.4 12 15.2 even 4
7200.2.h.l.1151.9 12 60.47 odd 4
7200.2.h.l.1151.10 12 20.7 even 4
7200.2.h.m.1151.3 12 60.23 odd 4
7200.2.h.m.1151.4 12 20.3 even 4
7200.2.h.m.1151.9 12 5.3 odd 4
7200.2.h.m.1151.10 12 15.8 even 4