Properties

Label 384.3.b.b.319.3
Level $384$
Weight $3$
Character 384.319
Analytic conductor $10.463$
Analytic rank $0$
Dimension $4$
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] = [384,3,Mod(319,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.319");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.b (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: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 319.3
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 384.319
Dual form 384.3.b.b.319.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205 q^{3} -6.92820i q^{5} +12.0000i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q+1.73205 q^{3} -6.92820i q^{5} +12.0000i q^{7} +3.00000 q^{9} +6.92820 q^{11} +13.8564i q^{13} -12.0000i q^{15} +14.0000 q^{17} +34.6410 q^{19} +20.7846i q^{21} -24.0000i q^{23} -23.0000 q^{25} +5.19615 q^{27} -34.6410i q^{29} -12.0000i q^{31} +12.0000 q^{33} +83.1384 q^{35} -27.7128i q^{37} +24.0000i q^{39} -14.0000 q^{41} +6.92820 q^{43} -20.7846i q^{45} +72.0000i q^{47} -95.0000 q^{49} +24.2487 q^{51} +62.3538i q^{53} -48.0000i q^{55} +60.0000 q^{57} -48.4974 q^{59} +55.4256i q^{61} +36.0000i q^{63} +96.0000 q^{65} +90.0666 q^{67} -41.5692i q^{69} +24.0000i q^{71} -50.0000 q^{73} -39.8372 q^{75} +83.1384i q^{77} -12.0000i q^{79} +9.00000 q^{81} -20.7846 q^{83} -96.9948i q^{85} -60.0000i q^{87} +62.0000 q^{89} -166.277 q^{91} -20.7846i q^{93} -240.000i q^{95} -146.000 q^{97} +20.7846 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{9} + 56 q^{17} - 92 q^{25} + 48 q^{33} - 56 q^{41} - 380 q^{49} + 240 q^{57} + 384 q^{65} - 200 q^{73} + 36 q^{81} + 248 q^{89} - 584 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 1.73205 0.577350
\(4\) 0 0
\(5\) − 6.92820i − 1.38564i −0.721110 0.692820i \(-0.756368\pi\)
0.721110 0.692820i \(-0.243632\pi\)
\(6\) 0 0
\(7\) 12.0000i 1.71429i 0.515079 + 0.857143i \(0.327763\pi\)
−0.515079 + 0.857143i \(0.672237\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) 6.92820 0.629837 0.314918 0.949119i \(-0.398023\pi\)
0.314918 + 0.949119i \(0.398023\pi\)
\(12\) 0 0
\(13\) 13.8564i 1.06588i 0.846154 + 0.532939i \(0.178912\pi\)
−0.846154 + 0.532939i \(0.821088\pi\)
\(14\) 0 0
\(15\) − 12.0000i − 0.800000i
\(16\) 0 0
\(17\) 14.0000 0.823529 0.411765 0.911290i \(-0.364913\pi\)
0.411765 + 0.911290i \(0.364913\pi\)
\(18\) 0 0
\(19\) 34.6410 1.82321 0.911606 0.411066i \(-0.134843\pi\)
0.911606 + 0.411066i \(0.134843\pi\)
\(20\) 0 0
\(21\) 20.7846i 0.989743i
\(22\) 0 0
\(23\) − 24.0000i − 1.04348i −0.853105 0.521739i \(-0.825283\pi\)
0.853105 0.521739i \(-0.174717\pi\)
\(24\) 0 0
\(25\) −23.0000 −0.920000
\(26\) 0 0
\(27\) 5.19615 0.192450
\(28\) 0 0
\(29\) − 34.6410i − 1.19452i −0.802049 0.597259i \(-0.796256\pi\)
0.802049 0.597259i \(-0.203744\pi\)
\(30\) 0 0
\(31\) − 12.0000i − 0.387097i −0.981091 0.193548i \(-0.938000\pi\)
0.981091 0.193548i \(-0.0619996\pi\)
\(32\) 0 0
\(33\) 12.0000 0.363636
\(34\) 0 0
\(35\) 83.1384 2.37538
\(36\) 0 0
\(37\) − 27.7128i − 0.748995i −0.927228 0.374497i \(-0.877815\pi\)
0.927228 0.374497i \(-0.122185\pi\)
\(38\) 0 0
\(39\) 24.0000i 0.615385i
\(40\) 0 0
\(41\) −14.0000 −0.341463 −0.170732 0.985318i \(-0.554613\pi\)
−0.170732 + 0.985318i \(0.554613\pi\)
\(42\) 0 0
\(43\) 6.92820 0.161121 0.0805605 0.996750i \(-0.474329\pi\)
0.0805605 + 0.996750i \(0.474329\pi\)
\(44\) 0 0
\(45\) − 20.7846i − 0.461880i
\(46\) 0 0
\(47\) 72.0000i 1.53191i 0.642891 + 0.765957i \(0.277735\pi\)
−0.642891 + 0.765957i \(0.722265\pi\)
\(48\) 0 0
\(49\) −95.0000 −1.93878
\(50\) 0 0
\(51\) 24.2487 0.475465
\(52\) 0 0
\(53\) 62.3538i 1.17649i 0.808684 + 0.588244i \(0.200180\pi\)
−0.808684 + 0.588244i \(0.799820\pi\)
\(54\) 0 0
\(55\) − 48.0000i − 0.872727i
\(56\) 0 0
\(57\) 60.0000 1.05263
\(58\) 0 0
\(59\) −48.4974 −0.821990 −0.410995 0.911638i \(-0.634819\pi\)
−0.410995 + 0.911638i \(0.634819\pi\)
\(60\) 0 0
\(61\) 55.4256i 0.908617i 0.890844 + 0.454308i \(0.150114\pi\)
−0.890844 + 0.454308i \(0.849886\pi\)
\(62\) 0 0
\(63\) 36.0000i 0.571429i
\(64\) 0 0
\(65\) 96.0000 1.47692
\(66\) 0 0
\(67\) 90.0666 1.34428 0.672139 0.740425i \(-0.265376\pi\)
0.672139 + 0.740425i \(0.265376\pi\)
\(68\) 0 0
\(69\) − 41.5692i − 0.602452i
\(70\) 0 0
\(71\) 24.0000i 0.338028i 0.985614 + 0.169014i \(0.0540583\pi\)
−0.985614 + 0.169014i \(0.945942\pi\)
\(72\) 0 0
\(73\) −50.0000 −0.684932 −0.342466 0.939530i \(-0.611262\pi\)
−0.342466 + 0.939530i \(0.611262\pi\)
\(74\) 0 0
\(75\) −39.8372 −0.531162
\(76\) 0 0
\(77\) 83.1384i 1.07972i
\(78\) 0 0
\(79\) − 12.0000i − 0.151899i −0.997112 0.0759494i \(-0.975801\pi\)
0.997112 0.0759494i \(-0.0241987\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) −20.7846 −0.250417 −0.125208 0.992130i \(-0.539960\pi\)
−0.125208 + 0.992130i \(0.539960\pi\)
\(84\) 0 0
\(85\) − 96.9948i − 1.14112i
\(86\) 0 0
\(87\) − 60.0000i − 0.689655i
\(88\) 0 0
\(89\) 62.0000 0.696629 0.348315 0.937378i \(-0.386754\pi\)
0.348315 + 0.937378i \(0.386754\pi\)
\(90\) 0 0
\(91\) −166.277 −1.82722
\(92\) 0 0
\(93\) − 20.7846i − 0.223490i
\(94\) 0 0
\(95\) − 240.000i − 2.52632i
\(96\) 0 0
\(97\) −146.000 −1.50515 −0.752577 0.658504i \(-0.771190\pi\)
−0.752577 + 0.658504i \(0.771190\pi\)
\(98\) 0 0
\(99\) 20.7846 0.209946
\(100\) 0 0
\(101\) − 20.7846i − 0.205788i −0.994692 0.102894i \(-0.967190\pi\)
0.994692 0.102894i \(-0.0328103\pi\)
\(102\) 0 0
\(103\) 84.0000i 0.815534i 0.913086 + 0.407767i \(0.133692\pi\)
−0.913086 + 0.407767i \(0.866308\pi\)
\(104\) 0 0
\(105\) 144.000 1.37143
\(106\) 0 0
\(107\) −131.636 −1.23024 −0.615121 0.788433i \(-0.710893\pi\)
−0.615121 + 0.788433i \(0.710893\pi\)
\(108\) 0 0
\(109\) − 180.133i − 1.65260i −0.563231 0.826299i \(-0.690442\pi\)
0.563231 0.826299i \(-0.309558\pi\)
\(110\) 0 0
\(111\) − 48.0000i − 0.432432i
\(112\) 0 0
\(113\) 130.000 1.15044 0.575221 0.817998i \(-0.304916\pi\)
0.575221 + 0.817998i \(0.304916\pi\)
\(114\) 0 0
\(115\) −166.277 −1.44589
\(116\) 0 0
\(117\) 41.5692i 0.355292i
\(118\) 0 0
\(119\) 168.000i 1.41176i
\(120\) 0 0
\(121\) −73.0000 −0.603306
\(122\) 0 0
\(123\) −24.2487 −0.197144
\(124\) 0 0
\(125\) − 13.8564i − 0.110851i
\(126\) 0 0
\(127\) 204.000i 1.60630i 0.595777 + 0.803150i \(0.296844\pi\)
−0.595777 + 0.803150i \(0.703156\pi\)
\(128\) 0 0
\(129\) 12.0000 0.0930233
\(130\) 0 0
\(131\) −20.7846 −0.158661 −0.0793306 0.996848i \(-0.525278\pi\)
−0.0793306 + 0.996848i \(0.525278\pi\)
\(132\) 0 0
\(133\) 415.692i 3.12551i
\(134\) 0 0
\(135\) − 36.0000i − 0.266667i
\(136\) 0 0
\(137\) −206.000 −1.50365 −0.751825 0.659363i \(-0.770826\pi\)
−0.751825 + 0.659363i \(0.770826\pi\)
\(138\) 0 0
\(139\) −48.4974 −0.348902 −0.174451 0.984666i \(-0.555815\pi\)
−0.174451 + 0.984666i \(0.555815\pi\)
\(140\) 0 0
\(141\) 124.708i 0.884451i
\(142\) 0 0
\(143\) 96.0000i 0.671329i
\(144\) 0 0
\(145\) −240.000 −1.65517
\(146\) 0 0
\(147\) −164.545 −1.11935
\(148\) 0 0
\(149\) − 200.918i − 1.34844i −0.738530 0.674221i \(-0.764479\pi\)
0.738530 0.674221i \(-0.235521\pi\)
\(150\) 0 0
\(151\) − 36.0000i − 0.238411i −0.992870 0.119205i \(-0.961965\pi\)
0.992870 0.119205i \(-0.0380347\pi\)
\(152\) 0 0
\(153\) 42.0000 0.274510
\(154\) 0 0
\(155\) −83.1384 −0.536377
\(156\) 0 0
\(157\) 27.7128i 0.176515i 0.996098 + 0.0882574i \(0.0281298\pi\)
−0.996098 + 0.0882574i \(0.971870\pi\)
\(158\) 0 0
\(159\) 108.000i 0.679245i
\(160\) 0 0
\(161\) 288.000 1.78882
\(162\) 0 0
\(163\) 62.3538 0.382539 0.191269 0.981538i \(-0.438740\pi\)
0.191269 + 0.981538i \(0.438740\pi\)
\(164\) 0 0
\(165\) − 83.1384i − 0.503869i
\(166\) 0 0
\(167\) − 240.000i − 1.43713i −0.695462 0.718563i \(-0.744800\pi\)
0.695462 0.718563i \(-0.255200\pi\)
\(168\) 0 0
\(169\) −23.0000 −0.136095
\(170\) 0 0
\(171\) 103.923 0.607737
\(172\) 0 0
\(173\) − 131.636i − 0.760901i −0.924801 0.380450i \(-0.875769\pi\)
0.924801 0.380450i \(-0.124231\pi\)
\(174\) 0 0
\(175\) − 276.000i − 1.57714i
\(176\) 0 0
\(177\) −84.0000 −0.474576
\(178\) 0 0
\(179\) −270.200 −1.50950 −0.754748 0.656014i \(-0.772241\pi\)
−0.754748 + 0.656014i \(0.772241\pi\)
\(180\) 0 0
\(181\) − 207.846i − 1.14832i −0.818743 0.574160i \(-0.805329\pi\)
0.818743 0.574160i \(-0.194671\pi\)
\(182\) 0 0
\(183\) 96.0000i 0.524590i
\(184\) 0 0
\(185\) −192.000 −1.03784
\(186\) 0 0
\(187\) 96.9948 0.518689
\(188\) 0 0
\(189\) 62.3538i 0.329914i
\(190\) 0 0
\(191\) 240.000i 1.25654i 0.777994 + 0.628272i \(0.216238\pi\)
−0.777994 + 0.628272i \(0.783762\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.0103627 −0.00518135 0.999987i \(-0.501649\pi\)
−0.00518135 + 0.999987i \(0.501649\pi\)
\(194\) 0 0
\(195\) 166.277 0.852702
\(196\) 0 0
\(197\) − 131.636i − 0.668202i −0.942537 0.334101i \(-0.891567\pi\)
0.942537 0.334101i \(-0.108433\pi\)
\(198\) 0 0
\(199\) − 300.000i − 1.50754i −0.657140 0.753769i \(-0.728234\pi\)
0.657140 0.753769i \(-0.271766\pi\)
\(200\) 0 0
\(201\) 156.000 0.776119
\(202\) 0 0
\(203\) 415.692 2.04774
\(204\) 0 0
\(205\) 96.9948i 0.473146i
\(206\) 0 0
\(207\) − 72.0000i − 0.347826i
\(208\) 0 0
\(209\) 240.000 1.14833
\(210\) 0 0
\(211\) −270.200 −1.28057 −0.640284 0.768138i \(-0.721183\pi\)
−0.640284 + 0.768138i \(0.721183\pi\)
\(212\) 0 0
\(213\) 41.5692i 0.195161i
\(214\) 0 0
\(215\) − 48.0000i − 0.223256i
\(216\) 0 0
\(217\) 144.000 0.663594
\(218\) 0 0
\(219\) −86.6025 −0.395445
\(220\) 0 0
\(221\) 193.990i 0.877781i
\(222\) 0 0
\(223\) − 132.000i − 0.591928i −0.955199 0.295964i \(-0.904359\pi\)
0.955199 0.295964i \(-0.0956409\pi\)
\(224\) 0 0
\(225\) −69.0000 −0.306667
\(226\) 0 0
\(227\) 200.918 0.885101 0.442550 0.896744i \(-0.354074\pi\)
0.442550 + 0.896744i \(0.354074\pi\)
\(228\) 0 0
\(229\) − 69.2820i − 0.302542i −0.988492 0.151271i \(-0.951663\pi\)
0.988492 0.151271i \(-0.0483366\pi\)
\(230\) 0 0
\(231\) 144.000i 0.623377i
\(232\) 0 0
\(233\) −130.000 −0.557940 −0.278970 0.960300i \(-0.589993\pi\)
−0.278970 + 0.960300i \(0.589993\pi\)
\(234\) 0 0
\(235\) 498.831 2.12268
\(236\) 0 0
\(237\) − 20.7846i − 0.0876988i
\(238\) 0 0
\(239\) 96.0000i 0.401674i 0.979625 + 0.200837i \(0.0643661\pi\)
−0.979625 + 0.200837i \(0.935634\pi\)
\(240\) 0 0
\(241\) 190.000 0.788382 0.394191 0.919029i \(-0.371025\pi\)
0.394191 + 0.919029i \(0.371025\pi\)
\(242\) 0 0
\(243\) 15.5885 0.0641500
\(244\) 0 0
\(245\) 658.179i 2.68645i
\(246\) 0 0
\(247\) 480.000i 1.94332i
\(248\) 0 0
\(249\) −36.0000 −0.144578
\(250\) 0 0
\(251\) 256.344 1.02129 0.510644 0.859792i \(-0.329407\pi\)
0.510644 + 0.859792i \(0.329407\pi\)
\(252\) 0 0
\(253\) − 166.277i − 0.657221i
\(254\) 0 0
\(255\) − 168.000i − 0.658824i
\(256\) 0 0
\(257\) −254.000 −0.988327 −0.494163 0.869369i \(-0.664526\pi\)
−0.494163 + 0.869369i \(0.664526\pi\)
\(258\) 0 0
\(259\) 332.554 1.28399
\(260\) 0 0
\(261\) − 103.923i − 0.398173i
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 432.000 1.63019
\(266\) 0 0
\(267\) 107.387 0.402199
\(268\) 0 0
\(269\) − 6.92820i − 0.0257554i −0.999917 0.0128777i \(-0.995901\pi\)
0.999917 0.0128777i \(-0.00409921\pi\)
\(270\) 0 0
\(271\) 348.000i 1.28413i 0.766649 + 0.642066i \(0.221923\pi\)
−0.766649 + 0.642066i \(0.778077\pi\)
\(272\) 0 0
\(273\) −288.000 −1.05495
\(274\) 0 0
\(275\) −159.349 −0.579450
\(276\) 0 0
\(277\) − 41.5692i − 0.150069i −0.997181 0.0750347i \(-0.976093\pi\)
0.997181 0.0750347i \(-0.0239068\pi\)
\(278\) 0 0
\(279\) − 36.0000i − 0.129032i
\(280\) 0 0
\(281\) −34.0000 −0.120996 −0.0604982 0.998168i \(-0.519269\pi\)
−0.0604982 + 0.998168i \(0.519269\pi\)
\(282\) 0 0
\(283\) 311.769 1.10166 0.550829 0.834618i \(-0.314312\pi\)
0.550829 + 0.834618i \(0.314312\pi\)
\(284\) 0 0
\(285\) − 415.692i − 1.45857i
\(286\) 0 0
\(287\) − 168.000i − 0.585366i
\(288\) 0 0
\(289\) −93.0000 −0.321799
\(290\) 0 0
\(291\) −252.879 −0.869001
\(292\) 0 0
\(293\) − 159.349i − 0.543852i −0.962318 0.271926i \(-0.912339\pi\)
0.962318 0.271926i \(-0.0876606\pi\)
\(294\) 0 0
\(295\) 336.000i 1.13898i
\(296\) 0 0
\(297\) 36.0000 0.121212
\(298\) 0 0
\(299\) 332.554 1.11222
\(300\) 0 0
\(301\) 83.1384i 0.276207i
\(302\) 0 0
\(303\) − 36.0000i − 0.118812i
\(304\) 0 0
\(305\) 384.000 1.25902
\(306\) 0 0
\(307\) −408.764 −1.33148 −0.665739 0.746184i \(-0.731884\pi\)
−0.665739 + 0.746184i \(0.731884\pi\)
\(308\) 0 0
\(309\) 145.492i 0.470849i
\(310\) 0 0
\(311\) − 48.0000i − 0.154341i −0.997018 0.0771704i \(-0.975411\pi\)
0.997018 0.0771704i \(-0.0245886\pi\)
\(312\) 0 0
\(313\) 98.0000 0.313099 0.156550 0.987670i \(-0.449963\pi\)
0.156550 + 0.987670i \(0.449963\pi\)
\(314\) 0 0
\(315\) 249.415 0.791795
\(316\) 0 0
\(317\) − 173.205i − 0.546388i −0.961959 0.273194i \(-0.911920\pi\)
0.961959 0.273194i \(-0.0880801\pi\)
\(318\) 0 0
\(319\) − 240.000i − 0.752351i
\(320\) 0 0
\(321\) −228.000 −0.710280
\(322\) 0 0
\(323\) 484.974 1.50147
\(324\) 0 0
\(325\) − 318.697i − 0.980607i
\(326\) 0 0
\(327\) − 312.000i − 0.954128i
\(328\) 0 0
\(329\) −864.000 −2.62614
\(330\) 0 0
\(331\) −20.7846 −0.0627934 −0.0313967 0.999507i \(-0.509996\pi\)
−0.0313967 + 0.999507i \(0.509996\pi\)
\(332\) 0 0
\(333\) − 83.1384i − 0.249665i
\(334\) 0 0
\(335\) − 624.000i − 1.86269i
\(336\) 0 0
\(337\) 50.0000 0.148368 0.0741840 0.997245i \(-0.476365\pi\)
0.0741840 + 0.997245i \(0.476365\pi\)
\(338\) 0 0
\(339\) 225.167 0.664208
\(340\) 0 0
\(341\) − 83.1384i − 0.243808i
\(342\) 0 0
\(343\) − 552.000i − 1.60933i
\(344\) 0 0
\(345\) −288.000 −0.834783
\(346\) 0 0
\(347\) −408.764 −1.17799 −0.588997 0.808135i \(-0.700477\pi\)
−0.588997 + 0.808135i \(0.700477\pi\)
\(348\) 0 0
\(349\) 498.831i 1.42931i 0.699475 + 0.714657i \(0.253417\pi\)
−0.699475 + 0.714657i \(0.746583\pi\)
\(350\) 0 0
\(351\) 72.0000i 0.205128i
\(352\) 0 0
\(353\) −542.000 −1.53541 −0.767705 0.640803i \(-0.778602\pi\)
−0.767705 + 0.640803i \(0.778602\pi\)
\(354\) 0 0
\(355\) 166.277 0.468386
\(356\) 0 0
\(357\) 290.985i 0.815083i
\(358\) 0 0
\(359\) 312.000i 0.869081i 0.900652 + 0.434540i \(0.143089\pi\)
−0.900652 + 0.434540i \(0.856911\pi\)
\(360\) 0 0
\(361\) 839.000 2.32410
\(362\) 0 0
\(363\) −126.440 −0.348319
\(364\) 0 0
\(365\) 346.410i 0.949069i
\(366\) 0 0
\(367\) 276.000i 0.752044i 0.926611 + 0.376022i \(0.122708\pi\)
−0.926611 + 0.376022i \(0.877292\pi\)
\(368\) 0 0
\(369\) −42.0000 −0.113821
\(370\) 0 0
\(371\) −748.246 −2.01684
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) − 24.0000i − 0.0640000i
\(376\) 0 0
\(377\) 480.000 1.27321
\(378\) 0 0
\(379\) −325.626 −0.859170 −0.429585 0.903026i \(-0.641340\pi\)
−0.429585 + 0.903026i \(0.641340\pi\)
\(380\) 0 0
\(381\) 353.338i 0.927397i
\(382\) 0 0
\(383\) 720.000i 1.87990i 0.341318 + 0.939948i \(0.389127\pi\)
−0.341318 + 0.939948i \(0.610873\pi\)
\(384\) 0 0
\(385\) 576.000 1.49610
\(386\) 0 0
\(387\) 20.7846 0.0537070
\(388\) 0 0
\(389\) 20.7846i 0.0534309i 0.999643 + 0.0267154i \(0.00850480\pi\)
−0.999643 + 0.0267154i \(0.991495\pi\)
\(390\) 0 0
\(391\) − 336.000i − 0.859335i
\(392\) 0 0
\(393\) −36.0000 −0.0916031
\(394\) 0 0
\(395\) −83.1384 −0.210477
\(396\) 0 0
\(397\) 221.703i 0.558445i 0.960226 + 0.279222i \(0.0900766\pi\)
−0.960226 + 0.279222i \(0.909923\pi\)
\(398\) 0 0
\(399\) 720.000i 1.80451i
\(400\) 0 0
\(401\) −178.000 −0.443890 −0.221945 0.975059i \(-0.571241\pi\)
−0.221945 + 0.975059i \(0.571241\pi\)
\(402\) 0 0
\(403\) 166.277 0.412598
\(404\) 0 0
\(405\) − 62.3538i − 0.153960i
\(406\) 0 0
\(407\) − 192.000i − 0.471744i
\(408\) 0 0
\(409\) −142.000 −0.347188 −0.173594 0.984817i \(-0.555538\pi\)
−0.173594 + 0.984817i \(0.555538\pi\)
\(410\) 0 0
\(411\) −356.802 −0.868133
\(412\) 0 0
\(413\) − 581.969i − 1.40913i
\(414\) 0 0
\(415\) 144.000i 0.346988i
\(416\) 0 0
\(417\) −84.0000 −0.201439
\(418\) 0 0
\(419\) −103.923 −0.248026 −0.124013 0.992281i \(-0.539577\pi\)
−0.124013 + 0.992281i \(0.539577\pi\)
\(420\) 0 0
\(421\) − 263.272i − 0.625349i −0.949860 0.312674i \(-0.898775\pi\)
0.949860 0.312674i \(-0.101225\pi\)
\(422\) 0 0
\(423\) 216.000i 0.510638i
\(424\) 0 0
\(425\) −322.000 −0.757647
\(426\) 0 0
\(427\) −665.108 −1.55763
\(428\) 0 0
\(429\) 166.277i 0.387592i
\(430\) 0 0
\(431\) − 168.000i − 0.389791i −0.980824 0.194896i \(-0.937563\pi\)
0.980824 0.194896i \(-0.0624368\pi\)
\(432\) 0 0
\(433\) 526.000 1.21478 0.607390 0.794404i \(-0.292216\pi\)
0.607390 + 0.794404i \(0.292216\pi\)
\(434\) 0 0
\(435\) −415.692 −0.955614
\(436\) 0 0
\(437\) − 831.384i − 1.90248i
\(438\) 0 0
\(439\) − 444.000i − 1.01139i −0.862712 0.505695i \(-0.831236\pi\)
0.862712 0.505695i \(-0.168764\pi\)
\(440\) 0 0
\(441\) −285.000 −0.646259
\(442\) 0 0
\(443\) −214.774 −0.484818 −0.242409 0.970174i \(-0.577938\pi\)
−0.242409 + 0.970174i \(0.577938\pi\)
\(444\) 0 0
\(445\) − 429.549i − 0.965278i
\(446\) 0 0
\(447\) − 348.000i − 0.778523i
\(448\) 0 0
\(449\) 494.000 1.10022 0.550111 0.835091i \(-0.314585\pi\)
0.550111 + 0.835091i \(0.314585\pi\)
\(450\) 0 0
\(451\) −96.9948 −0.215066
\(452\) 0 0
\(453\) − 62.3538i − 0.137646i
\(454\) 0 0
\(455\) 1152.00i 2.53187i
\(456\) 0 0
\(457\) −46.0000 −0.100656 −0.0503282 0.998733i \(-0.516027\pi\)
−0.0503282 + 0.998733i \(0.516027\pi\)
\(458\) 0 0
\(459\) 72.7461 0.158488
\(460\) 0 0
\(461\) − 769.031i − 1.66818i −0.551629 0.834090i \(-0.685994\pi\)
0.551629 0.834090i \(-0.314006\pi\)
\(462\) 0 0
\(463\) − 132.000i − 0.285097i −0.989788 0.142549i \(-0.954470\pi\)
0.989788 0.142549i \(-0.0455297\pi\)
\(464\) 0 0
\(465\) −144.000 −0.309677
\(466\) 0 0
\(467\) 117.779 0.252204 0.126102 0.992017i \(-0.459753\pi\)
0.126102 + 0.992017i \(0.459753\pi\)
\(468\) 0 0
\(469\) 1080.80i 2.30448i
\(470\) 0 0
\(471\) 48.0000i 0.101911i
\(472\) 0 0
\(473\) 48.0000 0.101480
\(474\) 0 0
\(475\) −796.743 −1.67735
\(476\) 0 0
\(477\) 187.061i 0.392162i
\(478\) 0 0
\(479\) − 408.000i − 0.851775i −0.904776 0.425887i \(-0.859962\pi\)
0.904776 0.425887i \(-0.140038\pi\)
\(480\) 0 0
\(481\) 384.000 0.798337
\(482\) 0 0
\(483\) 498.831 1.03278
\(484\) 0 0
\(485\) 1011.52i 2.08560i
\(486\) 0 0
\(487\) 444.000i 0.911704i 0.890056 + 0.455852i \(0.150665\pi\)
−0.890056 + 0.455852i \(0.849335\pi\)
\(488\) 0 0
\(489\) 108.000 0.220859
\(490\) 0 0
\(491\) −824.456 −1.67914 −0.839568 0.543254i \(-0.817192\pi\)
−0.839568 + 0.543254i \(0.817192\pi\)
\(492\) 0 0
\(493\) − 484.974i − 0.983721i
\(494\) 0 0
\(495\) − 144.000i − 0.290909i
\(496\) 0 0
\(497\) −288.000 −0.579477
\(498\) 0 0
\(499\) −381.051 −0.763630 −0.381815 0.924239i \(-0.624701\pi\)
−0.381815 + 0.924239i \(0.624701\pi\)
\(500\) 0 0
\(501\) − 415.692i − 0.829725i
\(502\) 0 0
\(503\) − 744.000i − 1.47913i −0.673088 0.739563i \(-0.735032\pi\)
0.673088 0.739563i \(-0.264968\pi\)
\(504\) 0 0
\(505\) −144.000 −0.285149
\(506\) 0 0
\(507\) −39.8372 −0.0785743
\(508\) 0 0
\(509\) 852.169i 1.67420i 0.547048 + 0.837101i \(0.315751\pi\)
−0.547048 + 0.837101i \(0.684249\pi\)
\(510\) 0 0
\(511\) − 600.000i − 1.17417i
\(512\) 0 0
\(513\) 180.000 0.350877
\(514\) 0 0
\(515\) 581.969 1.13004
\(516\) 0 0
\(517\) 498.831i 0.964856i
\(518\) 0 0
\(519\) − 228.000i − 0.439306i
\(520\) 0 0
\(521\) 82.0000 0.157390 0.0786948 0.996899i \(-0.474925\pi\)
0.0786948 + 0.996899i \(0.474925\pi\)
\(522\) 0 0
\(523\) 311.769 0.596117 0.298058 0.954548i \(-0.403661\pi\)
0.298058 + 0.954548i \(0.403661\pi\)
\(524\) 0 0
\(525\) − 478.046i − 0.910564i
\(526\) 0 0
\(527\) − 168.000i − 0.318786i
\(528\) 0 0
\(529\) −47.0000 −0.0888469
\(530\) 0 0
\(531\) −145.492 −0.273997
\(532\) 0 0
\(533\) − 193.990i − 0.363958i
\(534\) 0 0
\(535\) 912.000i 1.70467i
\(536\) 0 0
\(537\) −468.000 −0.871508
\(538\) 0 0
\(539\) −658.179 −1.22111
\(540\) 0 0
\(541\) 96.9948i 0.179288i 0.995974 + 0.0896440i \(0.0285729\pi\)
−0.995974 + 0.0896440i \(0.971427\pi\)
\(542\) 0 0
\(543\) − 360.000i − 0.662983i
\(544\) 0 0
\(545\) −1248.00 −2.28991
\(546\) 0 0
\(547\) 34.6410 0.0633291 0.0316645 0.999499i \(-0.489919\pi\)
0.0316645 + 0.999499i \(0.489919\pi\)
\(548\) 0 0
\(549\) 166.277i 0.302872i
\(550\) 0 0
\(551\) − 1200.00i − 2.17786i
\(552\) 0 0
\(553\) 144.000 0.260398
\(554\) 0 0
\(555\) −332.554 −0.599196
\(556\) 0 0
\(557\) 949.164i 1.70406i 0.523489 + 0.852032i \(0.324630\pi\)
−0.523489 + 0.852032i \(0.675370\pi\)
\(558\) 0 0
\(559\) 96.0000i 0.171735i
\(560\) 0 0
\(561\) 168.000 0.299465
\(562\) 0 0
\(563\) 949.164 1.68590 0.842952 0.537989i \(-0.180816\pi\)
0.842952 + 0.537989i \(0.180816\pi\)
\(564\) 0 0
\(565\) − 900.666i − 1.59410i
\(566\) 0 0
\(567\) 108.000i 0.190476i
\(568\) 0 0
\(569\) 658.000 1.15641 0.578207 0.815890i \(-0.303752\pi\)
0.578207 + 0.815890i \(0.303752\pi\)
\(570\) 0 0
\(571\) 256.344 0.448938 0.224469 0.974481i \(-0.427935\pi\)
0.224469 + 0.974481i \(0.427935\pi\)
\(572\) 0 0
\(573\) 415.692i 0.725466i
\(574\) 0 0
\(575\) 552.000i 0.960000i
\(576\) 0 0
\(577\) −526.000 −0.911612 −0.455806 0.890079i \(-0.650649\pi\)
−0.455806 + 0.890079i \(0.650649\pi\)
\(578\) 0 0
\(579\) −3.46410 −0.00598290
\(580\) 0 0
\(581\) − 249.415i − 0.429286i
\(582\) 0 0
\(583\) 432.000i 0.740995i
\(584\) 0 0
\(585\) 288.000 0.492308
\(586\) 0 0
\(587\) −464.190 −0.790783 −0.395391 0.918513i \(-0.629391\pi\)
−0.395391 + 0.918513i \(0.629391\pi\)
\(588\) 0 0
\(589\) − 415.692i − 0.705759i
\(590\) 0 0
\(591\) − 228.000i − 0.385787i
\(592\) 0 0
\(593\) 514.000 0.866779 0.433390 0.901207i \(-0.357317\pi\)
0.433390 + 0.901207i \(0.357317\pi\)
\(594\) 0 0
\(595\) 1163.94 1.95620
\(596\) 0 0
\(597\) − 519.615i − 0.870377i
\(598\) 0 0
\(599\) 408.000i 0.681135i 0.940220 + 0.340568i \(0.110619\pi\)
−0.940220 + 0.340568i \(0.889381\pi\)
\(600\) 0 0
\(601\) −818.000 −1.36106 −0.680532 0.732718i \(-0.738251\pi\)
−0.680532 + 0.732718i \(0.738251\pi\)
\(602\) 0 0
\(603\) 270.200 0.448093
\(604\) 0 0
\(605\) 505.759i 0.835965i
\(606\) 0 0
\(607\) 684.000i 1.12685i 0.826166 + 0.563427i \(0.190517\pi\)
−0.826166 + 0.563427i \(0.809483\pi\)
\(608\) 0 0
\(609\) 720.000 1.18227
\(610\) 0 0
\(611\) −997.661 −1.63283
\(612\) 0 0
\(613\) − 498.831i − 0.813753i −0.913483 0.406877i \(-0.866618\pi\)
0.913483 0.406877i \(-0.133382\pi\)
\(614\) 0 0
\(615\) 168.000i 0.273171i
\(616\) 0 0
\(617\) −322.000 −0.521880 −0.260940 0.965355i \(-0.584032\pi\)
−0.260940 + 0.965355i \(0.584032\pi\)
\(618\) 0 0
\(619\) 90.0666 0.145503 0.0727517 0.997350i \(-0.476822\pi\)
0.0727517 + 0.997350i \(0.476822\pi\)
\(620\) 0 0
\(621\) − 124.708i − 0.200817i
\(622\) 0 0
\(623\) 744.000i 1.19422i
\(624\) 0 0
\(625\) −671.000 −1.07360
\(626\) 0 0
\(627\) 415.692 0.662986
\(628\) 0 0
\(629\) − 387.979i − 0.616819i
\(630\) 0 0
\(631\) 252.000i 0.399366i 0.979861 + 0.199683i \(0.0639912\pi\)
−0.979861 + 0.199683i \(0.936009\pi\)
\(632\) 0 0
\(633\) −468.000 −0.739336
\(634\) 0 0
\(635\) 1413.35 2.22575
\(636\) 0 0
\(637\) − 1316.36i − 2.06650i
\(638\) 0 0
\(639\) 72.0000i 0.112676i
\(640\) 0 0
\(641\) −370.000 −0.577223 −0.288612 0.957446i \(-0.593194\pi\)
−0.288612 + 0.957446i \(0.593194\pi\)
\(642\) 0 0
\(643\) 1032.30 1.60545 0.802723 0.596352i \(-0.203384\pi\)
0.802723 + 0.596352i \(0.203384\pi\)
\(644\) 0 0
\(645\) − 83.1384i − 0.128897i
\(646\) 0 0
\(647\) − 216.000i − 0.333849i −0.985970 0.166924i \(-0.946616\pi\)
0.985970 0.166924i \(-0.0533835\pi\)
\(648\) 0 0
\(649\) −336.000 −0.517720
\(650\) 0 0
\(651\) 249.415 0.383126
\(652\) 0 0
\(653\) 145.492i 0.222806i 0.993775 + 0.111403i \(0.0355344\pi\)
−0.993775 + 0.111403i \(0.964466\pi\)
\(654\) 0 0
\(655\) 144.000i 0.219847i
\(656\) 0 0
\(657\) −150.000 −0.228311
\(658\) 0 0
\(659\) 1060.02 1.60852 0.804260 0.594277i \(-0.202562\pi\)
0.804260 + 0.594277i \(0.202562\pi\)
\(660\) 0 0
\(661\) 1302.50i 1.97050i 0.171114 + 0.985251i \(0.445263\pi\)
−0.171114 + 0.985251i \(0.554737\pi\)
\(662\) 0 0
\(663\) 336.000i 0.506787i
\(664\) 0 0
\(665\) 2880.00 4.33083
\(666\) 0 0
\(667\) −831.384 −1.24645
\(668\) 0 0
\(669\) − 228.631i − 0.341750i
\(670\) 0 0
\(671\) 384.000i 0.572280i
\(672\) 0 0
\(673\) 1006.00 1.49480 0.747400 0.664375i \(-0.231302\pi\)
0.747400 + 0.664375i \(0.231302\pi\)
\(674\) 0 0
\(675\) −119.512 −0.177054
\(676\) 0 0
\(677\) 270.200i 0.399114i 0.979886 + 0.199557i \(0.0639502\pi\)
−0.979886 + 0.199557i \(0.936050\pi\)
\(678\) 0 0
\(679\) − 1752.00i − 2.58027i
\(680\) 0 0
\(681\) 348.000 0.511013
\(682\) 0 0
\(683\) 1198.58 1.75487 0.877437 0.479692i \(-0.159251\pi\)
0.877437 + 0.479692i \(0.159251\pi\)
\(684\) 0 0
\(685\) 1427.21i 2.08352i
\(686\) 0 0
\(687\) − 120.000i − 0.174672i
\(688\) 0 0
\(689\) −864.000 −1.25399
\(690\) 0 0
\(691\) −436.477 −0.631660 −0.315830 0.948816i \(-0.602283\pi\)
−0.315830 + 0.948816i \(0.602283\pi\)
\(692\) 0 0
\(693\) 249.415i 0.359907i
\(694\) 0 0
\(695\) 336.000i 0.483453i
\(696\) 0 0
\(697\) −196.000 −0.281205
\(698\) 0 0
\(699\) −225.167 −0.322127
\(700\) 0 0
\(701\) − 588.897i − 0.840082i −0.907505 0.420041i \(-0.862016\pi\)
0.907505 0.420041i \(-0.137984\pi\)
\(702\) 0 0
\(703\) − 960.000i − 1.36558i
\(704\) 0 0
\(705\) 864.000 1.22553
\(706\) 0 0
\(707\) 249.415 0.352780
\(708\) 0 0
\(709\) − 568.113i − 0.801287i −0.916234 0.400644i \(-0.868787\pi\)
0.916234 0.400644i \(-0.131213\pi\)
\(710\) 0 0
\(711\) − 36.0000i − 0.0506329i
\(712\) 0 0
\(713\) −288.000 −0.403927
\(714\) 0 0
\(715\) 665.108 0.930220
\(716\) 0 0
\(717\) 166.277i 0.231906i
\(718\) 0 0
\(719\) − 360.000i − 0.500695i −0.968156 0.250348i \(-0.919455\pi\)
0.968156 0.250348i \(-0.0805449\pi\)
\(720\) 0 0
\(721\) −1008.00 −1.39806
\(722\) 0 0
\(723\) 329.090 0.455172
\(724\) 0 0
\(725\) 796.743i 1.09896i
\(726\) 0 0
\(727\) − 660.000i − 0.907840i −0.891042 0.453920i \(-0.850025\pi\)
0.891042 0.453920i \(-0.149975\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) 96.9948 0.132688
\(732\) 0 0
\(733\) 956.092i 1.30435i 0.758066 + 0.652177i \(0.226144\pi\)
−0.758066 + 0.652177i \(0.773856\pi\)
\(734\) 0 0
\(735\) 1140.00i 1.55102i
\(736\) 0 0
\(737\) 624.000 0.846676
\(738\) 0 0
\(739\) −547.328 −0.740633 −0.370317 0.928906i \(-0.620751\pi\)
−0.370317 + 0.928906i \(0.620751\pi\)
\(740\) 0 0
\(741\) 831.384i 1.12198i
\(742\) 0 0
\(743\) − 864.000i − 1.16285i −0.813599 0.581427i \(-0.802495\pi\)
0.813599 0.581427i \(-0.197505\pi\)
\(744\) 0 0
\(745\) −1392.00 −1.86846
\(746\) 0 0
\(747\) −62.3538 −0.0834723
\(748\) 0 0
\(749\) − 1579.63i − 2.10899i
\(750\) 0 0
\(751\) − 444.000i − 0.591212i −0.955310 0.295606i \(-0.904478\pi\)
0.955310 0.295606i \(-0.0955215\pi\)
\(752\) 0 0
\(753\) 444.000 0.589641
\(754\) 0 0
\(755\) −249.415 −0.330351
\(756\) 0 0
\(757\) 872.954i 1.15318i 0.817035 + 0.576588i \(0.195616\pi\)
−0.817035 + 0.576588i \(0.804384\pi\)
\(758\) 0 0
\(759\) − 288.000i − 0.379447i
\(760\) 0 0
\(761\) −14.0000 −0.0183968 −0.00919842 0.999958i \(-0.502928\pi\)
−0.00919842 + 0.999958i \(0.502928\pi\)
\(762\) 0 0
\(763\) 2161.60 2.83303
\(764\) 0 0
\(765\) − 290.985i − 0.380372i
\(766\) 0 0
\(767\) − 672.000i − 0.876141i
\(768\) 0 0
\(769\) −194.000 −0.252276 −0.126138 0.992013i \(-0.540258\pi\)
−0.126138 + 0.992013i \(0.540258\pi\)
\(770\) 0 0
\(771\) −439.941 −0.570611
\(772\) 0 0
\(773\) − 713.605i − 0.923163i −0.887098 0.461581i \(-0.847282\pi\)
0.887098 0.461581i \(-0.152718\pi\)
\(774\) 0 0
\(775\) 276.000i 0.356129i
\(776\) 0 0
\(777\) 576.000 0.741313
\(778\) 0 0
\(779\) −484.974 −0.622560
\(780\) 0 0
\(781\) 166.277i 0.212903i
\(782\) 0 0
\(783\) − 180.000i − 0.229885i
\(784\) 0 0
\(785\) 192.000 0.244586
\(786\) 0 0
\(787\) 1060.02 1.34691 0.673453 0.739230i \(-0.264810\pi\)
0.673453 + 0.739230i \(0.264810\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1560.00i 1.97219i
\(792\) 0 0
\(793\) −768.000 −0.968474
\(794\) 0 0
\(795\) 748.246 0.941190
\(796\) 0 0
\(797\) − 1461.85i − 1.83419i −0.398667 0.917096i \(-0.630527\pi\)
0.398667 0.917096i \(-0.369473\pi\)
\(798\) 0 0
\(799\) 1008.00i 1.26158i
\(800\) 0 0
\(801\) 186.000 0.232210
\(802\) 0 0
\(803\) −346.410 −0.431395
\(804\) 0 0
\(805\) − 1995.32i − 2.47866i
\(806\) 0 0
\(807\) − 12.0000i − 0.0148699i
\(808\) 0 0
\(809\) 1138.00 1.40667 0.703337 0.710856i \(-0.251692\pi\)
0.703337 + 0.710856i \(0.251692\pi\)
\(810\) 0 0
\(811\) −630.466 −0.777394 −0.388697 0.921366i \(-0.627075\pi\)
−0.388697 + 0.921366i \(0.627075\pi\)
\(812\) 0 0
\(813\) 602.754i 0.741394i
\(814\) 0 0
\(815\) − 432.000i − 0.530061i
\(816\) 0 0
\(817\) 240.000 0.293758
\(818\) 0 0
\(819\) −498.831 −0.609073
\(820\) 0 0
\(821\) 256.344i 0.312233i 0.987739 + 0.156117i \(0.0498976\pi\)
−0.987739 + 0.156117i \(0.950102\pi\)
\(822\) 0 0
\(823\) 396.000i 0.481166i 0.970629 + 0.240583i \(0.0773387\pi\)
−0.970629 + 0.240583i \(0.922661\pi\)
\(824\) 0 0
\(825\) −276.000 −0.334545
\(826\) 0 0
\(827\) 145.492 0.175928 0.0879639 0.996124i \(-0.471964\pi\)
0.0879639 + 0.996124i \(0.471964\pi\)
\(828\) 0 0
\(829\) 41.5692i 0.0501438i 0.999686 + 0.0250719i \(0.00798147\pi\)
−0.999686 + 0.0250719i \(0.992019\pi\)
\(830\) 0 0
\(831\) − 72.0000i − 0.0866426i
\(832\) 0 0
\(833\) −1330.00 −1.59664
\(834\) 0 0
\(835\) −1662.77 −1.99134
\(836\) 0 0
\(837\) − 62.3538i − 0.0744968i
\(838\) 0 0
\(839\) 1512.00i 1.80215i 0.433668 + 0.901073i \(0.357219\pi\)
−0.433668 + 0.901073i \(0.642781\pi\)
\(840\) 0 0
\(841\) −359.000 −0.426873
\(842\) 0 0
\(843\) −58.8897 −0.0698573
\(844\) 0 0
\(845\) 159.349i 0.188578i
\(846\) 0 0
\(847\) − 876.000i − 1.03424i
\(848\) 0 0
\(849\) 540.000 0.636042
\(850\) 0 0
\(851\) −665.108 −0.781560
\(852\) 0 0
\(853\) − 748.246i − 0.877193i −0.898684 0.438597i \(-0.855476\pi\)
0.898684 0.438597i \(-0.144524\pi\)
\(854\) 0 0
\(855\) − 720.000i − 0.842105i
\(856\) 0 0
\(857\) 1042.00 1.21587 0.607935 0.793987i \(-0.291998\pi\)
0.607935 + 0.793987i \(0.291998\pi\)
\(858\) 0 0
\(859\) −159.349 −0.185505 −0.0927524 0.995689i \(-0.529567\pi\)
−0.0927524 + 0.995689i \(0.529567\pi\)
\(860\) 0 0
\(861\) − 290.985i − 0.337961i
\(862\) 0 0
\(863\) − 1104.00i − 1.27926i −0.768684 0.639629i \(-0.779088\pi\)
0.768684 0.639629i \(-0.220912\pi\)
\(864\) 0 0
\(865\) −912.000 −1.05434
\(866\) 0 0
\(867\) −161.081 −0.185791
\(868\) 0 0
\(869\) − 83.1384i − 0.0956714i
\(870\) 0 0
\(871\) 1248.00i 1.43284i
\(872\) 0 0
\(873\) −438.000 −0.501718
\(874\) 0 0
\(875\) 166.277 0.190031
\(876\) 0 0
\(877\) 554.256i 0.631991i 0.948761 + 0.315996i \(0.102338\pi\)
−0.948761 + 0.315996i \(0.897662\pi\)
\(878\) 0 0
\(879\) − 276.000i − 0.313993i
\(880\) 0 0
\(881\) 610.000 0.692395 0.346198 0.938162i \(-0.387473\pi\)
0.346198 + 0.938162i \(0.387473\pi\)
\(882\) 0 0
\(883\) −76.2102 −0.0863083 −0.0431542 0.999068i \(-0.513741\pi\)
−0.0431542 + 0.999068i \(0.513741\pi\)
\(884\) 0 0
\(885\) 581.969i 0.657592i
\(886\) 0 0
\(887\) − 384.000i − 0.432920i −0.976291 0.216460i \(-0.930549\pi\)
0.976291 0.216460i \(-0.0694511\pi\)
\(888\) 0 0
\(889\) −2448.00 −2.75366
\(890\) 0 0
\(891\) 62.3538 0.0699819
\(892\) 0 0
\(893\) 2494.15i 2.79300i
\(894\) 0 0
\(895\) 1872.00i 2.09162i
\(896\) 0 0
\(897\) 576.000 0.642140
\(898\) 0 0
\(899\) −415.692 −0.462394
\(900\) 0 0
\(901\) 872.954i 0.968872i
\(902\) 0 0
\(903\) 144.000i 0.159468i
\(904\) 0 0
\(905\) −1440.00 −1.59116
\(906\) 0 0
\(907\) −713.605 −0.786775 −0.393388 0.919373i \(-0.628697\pi\)
−0.393388 + 0.919373i \(0.628697\pi\)
\(908\) 0 0
\(909\) − 62.3538i − 0.0685961i
\(910\) 0 0
\(911\) 864.000i 0.948408i 0.880415 + 0.474204i \(0.157264\pi\)
−0.880415 + 0.474204i \(0.842736\pi\)
\(912\) 0 0
\(913\) −144.000 −0.157722
\(914\) 0 0
\(915\) 665.108 0.726893
\(916\) 0 0
\(917\) − 249.415i − 0.271991i
\(918\) 0 0
\(919\) 1068.00i 1.16213i 0.813856 + 0.581066i \(0.197364\pi\)
−0.813856 + 0.581066i \(0.802636\pi\)
\(920\) 0 0
\(921\) −708.000 −0.768730
\(922\) 0 0
\(923\) −332.554 −0.360297
\(924\) 0 0
\(925\) 637.395i 0.689075i
\(926\) 0 0
\(927\) 252.000i 0.271845i
\(928\) 0 0
\(929\) 110.000 0.118407 0.0592034 0.998246i \(-0.481144\pi\)
0.0592034 + 0.998246i \(0.481144\pi\)
\(930\) 0 0
\(931\) −3290.90 −3.53480
\(932\) 0 0
\(933\) − 83.1384i − 0.0891087i
\(934\) 0 0
\(935\) − 672.000i − 0.718717i
\(936\) 0 0
\(937\) 1630.00 1.73959 0.869797 0.493409i \(-0.164250\pi\)
0.869797 + 0.493409i \(0.164250\pi\)
\(938\) 0 0
\(939\) 169.741 0.180768
\(940\) 0 0
\(941\) − 48.4974i − 0.0515382i −0.999668 0.0257691i \(-0.991797\pi\)
0.999668 0.0257691i \(-0.00820346\pi\)
\(942\) 0 0
\(943\) 336.000i 0.356310i
\(944\) 0 0
\(945\) 432.000 0.457143
\(946\) 0 0
\(947\) −1738.98 −1.83630 −0.918152 0.396229i \(-0.870318\pi\)
−0.918152 + 0.396229i \(0.870318\pi\)
\(948\) 0 0
\(949\) − 692.820i − 0.730053i
\(950\) 0 0
\(951\) − 300.000i − 0.315457i
\(952\) 0 0
\(953\) 1714.00 1.79853 0.899265 0.437403i \(-0.144102\pi\)
0.899265 + 0.437403i \(0.144102\pi\)
\(954\) 0 0
\(955\) 1662.77 1.74112
\(956\) 0 0
\(957\) − 415.692i − 0.434370i
\(958\) 0 0
\(959\) − 2472.00i − 2.57769i
\(960\) 0 0
\(961\) 817.000 0.850156
\(962\) 0 0
\(963\) −394.908 −0.410081
\(964\) 0 0
\(965\) 13.8564i 0.0143590i
\(966\) 0 0
\(967\) − 1212.00i − 1.25336i −0.779276 0.626680i \(-0.784413\pi\)
0.779276 0.626680i \(-0.215587\pi\)
\(968\) 0 0
\(969\) 840.000 0.866873
\(970\) 0 0
\(971\) 311.769 0.321080 0.160540 0.987029i \(-0.448676\pi\)
0.160540 + 0.987029i \(0.448676\pi\)
\(972\) 0 0
\(973\) − 581.969i − 0.598118i
\(974\) 0 0
\(975\) − 552.000i − 0.566154i
\(976\) 0 0
\(977\) 14.0000 0.0143296 0.00716479 0.999974i \(-0.497719\pi\)
0.00716479 + 0.999974i \(0.497719\pi\)
\(978\) 0 0
\(979\) 429.549 0.438763
\(980\) 0 0
\(981\) − 540.400i − 0.550866i
\(982\) 0 0
\(983\) − 384.000i − 0.390641i −0.980739 0.195320i \(-0.937425\pi\)
0.980739 0.195320i \(-0.0625747\pi\)
\(984\) 0 0
\(985\) −912.000 −0.925888
\(986\) 0 0
\(987\) −1496.49 −1.51620
\(988\) 0 0
\(989\) − 166.277i − 0.168126i
\(990\) 0 0
\(991\) − 636.000i − 0.641776i −0.947117 0.320888i \(-0.896019\pi\)
0.947117 0.320888i \(-0.103981\pi\)
\(992\) 0 0
\(993\) −36.0000 −0.0362538
\(994\) 0 0
\(995\) −2078.46 −2.08891
\(996\) 0 0
\(997\) 1441.07i 1.44540i 0.691161 + 0.722701i \(0.257100\pi\)
−0.691161 + 0.722701i \(0.742900\pi\)
\(998\) 0 0
\(999\) − 144.000i − 0.144144i
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 384.3.b.b.319.3 yes 4
3.2 odd 2 1152.3.b.e.703.4 4
4.3 odd 2 inner 384.3.b.b.319.1 4
8.3 odd 2 inner 384.3.b.b.319.4 yes 4
8.5 even 2 inner 384.3.b.b.319.2 yes 4
12.11 even 2 1152.3.b.e.703.3 4
16.3 odd 4 768.3.g.e.511.1 4
16.5 even 4 768.3.g.e.511.2 4
16.11 odd 4 768.3.g.e.511.4 4
16.13 even 4 768.3.g.e.511.3 4
24.5 odd 2 1152.3.b.e.703.2 4
24.11 even 2 1152.3.b.e.703.1 4
48.5 odd 4 2304.3.g.r.1279.1 4
48.11 even 4 2304.3.g.r.1279.2 4
48.29 odd 4 2304.3.g.r.1279.3 4
48.35 even 4 2304.3.g.r.1279.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.b.b.319.1 4 4.3 odd 2 inner
384.3.b.b.319.2 yes 4 8.5 even 2 inner
384.3.b.b.319.3 yes 4 1.1 even 1 trivial
384.3.b.b.319.4 yes 4 8.3 odd 2 inner
768.3.g.e.511.1 4 16.3 odd 4
768.3.g.e.511.2 4 16.5 even 4
768.3.g.e.511.3 4 16.13 even 4
768.3.g.e.511.4 4 16.11 odd 4
1152.3.b.e.703.1 4 24.11 even 2
1152.3.b.e.703.2 4 24.5 odd 2
1152.3.b.e.703.3 4 12.11 even 2
1152.3.b.e.703.4 4 3.2 odd 2
2304.3.g.r.1279.1 4 48.5 odd 4
2304.3.g.r.1279.2 4 48.11 even 4
2304.3.g.r.1279.3 4 48.29 odd 4
2304.3.g.r.1279.4 4 48.35 even 4