Properties

Label 192.3.e.d.65.2
Level $192$
Weight $3$
Character 192.65
Analytic conductor $5.232$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,3,Mod(65,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 192.e (of order \(2\), degree \(1\), not 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: \(5.23162107572\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 24)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 65.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 192.65
Dual form 192.3.e.d.65.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 + 2.82843i) q^{3} +5.65685i q^{5} +6.00000 q^{7} +(-7.00000 + 5.65685i) q^{9} +O(q^{10})\) \(q+(1.00000 + 2.82843i) q^{3} +5.65685i q^{5} +6.00000 q^{7} +(-7.00000 + 5.65685i) q^{9} -5.65685i q^{11} -10.0000 q^{13} +(-16.0000 + 5.65685i) q^{15} +22.6274i q^{17} +2.00000 q^{19} +(6.00000 + 16.9706i) q^{21} -11.3137i q^{23} -7.00000 q^{25} +(-23.0000 - 14.1421i) q^{27} +16.9706i q^{29} +22.0000 q^{31} +(16.0000 - 5.65685i) q^{33} +33.9411i q^{35} +6.00000 q^{37} +(-10.0000 - 28.2843i) q^{39} -33.9411i q^{41} +82.0000 q^{43} +(-32.0000 - 39.5980i) q^{45} +67.8823i q^{47} -13.0000 q^{49} +(-64.0000 + 22.6274i) q^{51} -62.2254i q^{53} +32.0000 q^{55} +(2.00000 + 5.65685i) q^{57} -73.5391i q^{59} +86.0000 q^{61} +(-42.0000 + 33.9411i) q^{63} -56.5685i q^{65} +2.00000 q^{67} +(32.0000 - 11.3137i) q^{69} -124.451i q^{71} +82.0000 q^{73} +(-7.00000 - 19.7990i) q^{75} -33.9411i q^{77} -10.0000 q^{79} +(17.0000 - 79.1960i) q^{81} +73.5391i q^{83} -128.000 q^{85} +(-48.0000 + 16.9706i) q^{87} +33.9411i q^{89} -60.0000 q^{91} +(22.0000 + 62.2254i) q^{93} +11.3137i q^{95} -94.0000 q^{97} +(32.0000 + 39.5980i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 12 q^{7} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 12 q^{7} - 14 q^{9} - 20 q^{13} - 32 q^{15} + 4 q^{19} + 12 q^{21} - 14 q^{25} - 46 q^{27} + 44 q^{31} + 32 q^{33} + 12 q^{37} - 20 q^{39} + 164 q^{43} - 64 q^{45} - 26 q^{49} - 128 q^{51} + 64 q^{55} + 4 q^{57} + 172 q^{61} - 84 q^{63} + 4 q^{67} + 64 q^{69} + 164 q^{73} - 14 q^{75} - 20 q^{79} + 34 q^{81} - 256 q^{85} - 96 q^{87} - 120 q^{91} + 44 q^{93} - 188 q^{97} + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 1.00000 + 2.82843i 0.333333 + 0.942809i
\(4\) 0 0
\(5\) 5.65685i 1.13137i 0.824621 + 0.565685i \(0.191388\pi\)
−0.824621 + 0.565685i \(0.808612\pi\)
\(6\) 0 0
\(7\) 6.00000 0.857143 0.428571 0.903508i \(-0.359017\pi\)
0.428571 + 0.903508i \(0.359017\pi\)
\(8\) 0 0
\(9\) −7.00000 + 5.65685i −0.777778 + 0.628539i
\(10\) 0 0
\(11\) 5.65685i 0.514259i −0.966377 0.257130i \(-0.917223\pi\)
0.966377 0.257130i \(-0.0827768\pi\)
\(12\) 0 0
\(13\) −10.0000 −0.769231 −0.384615 0.923077i \(-0.625666\pi\)
−0.384615 + 0.923077i \(0.625666\pi\)
\(14\) 0 0
\(15\) −16.0000 + 5.65685i −1.06667 + 0.377124i
\(16\) 0 0
\(17\) 22.6274i 1.33102i 0.746387 + 0.665512i \(0.231787\pi\)
−0.746387 + 0.665512i \(0.768213\pi\)
\(18\) 0 0
\(19\) 2.00000 0.105263 0.0526316 0.998614i \(-0.483239\pi\)
0.0526316 + 0.998614i \(0.483239\pi\)
\(20\) 0 0
\(21\) 6.00000 + 16.9706i 0.285714 + 0.808122i
\(22\) 0 0
\(23\) 11.3137i 0.491900i −0.969282 0.245950i \(-0.920900\pi\)
0.969282 0.245950i \(-0.0791000\pi\)
\(24\) 0 0
\(25\) −7.00000 −0.280000
\(26\) 0 0
\(27\) −23.0000 14.1421i −0.851852 0.523783i
\(28\) 0 0
\(29\) 16.9706i 0.585192i 0.956236 + 0.292596i \(0.0945191\pi\)
−0.956236 + 0.292596i \(0.905481\pi\)
\(30\) 0 0
\(31\) 22.0000 0.709677 0.354839 0.934928i \(-0.384536\pi\)
0.354839 + 0.934928i \(0.384536\pi\)
\(32\) 0 0
\(33\) 16.0000 5.65685i 0.484848 0.171420i
\(34\) 0 0
\(35\) 33.9411i 0.969746i
\(36\) 0 0
\(37\) 6.00000 0.162162 0.0810811 0.996708i \(-0.474163\pi\)
0.0810811 + 0.996708i \(0.474163\pi\)
\(38\) 0 0
\(39\) −10.0000 28.2843i −0.256410 0.725238i
\(40\) 0 0
\(41\) 33.9411i 0.827832i −0.910315 0.413916i \(-0.864161\pi\)
0.910315 0.413916i \(-0.135839\pi\)
\(42\) 0 0
\(43\) 82.0000 1.90698 0.953488 0.301430i \(-0.0974639\pi\)
0.953488 + 0.301430i \(0.0974639\pi\)
\(44\) 0 0
\(45\) −32.0000 39.5980i −0.711111 0.879955i
\(46\) 0 0
\(47\) 67.8823i 1.44430i 0.691735 + 0.722152i \(0.256847\pi\)
−0.691735 + 0.722152i \(0.743153\pi\)
\(48\) 0 0
\(49\) −13.0000 −0.265306
\(50\) 0 0
\(51\) −64.0000 + 22.6274i −1.25490 + 0.443675i
\(52\) 0 0
\(53\) 62.2254i 1.17406i −0.809564 0.587032i \(-0.800296\pi\)
0.809564 0.587032i \(-0.199704\pi\)
\(54\) 0 0
\(55\) 32.0000 0.581818
\(56\) 0 0
\(57\) 2.00000 + 5.65685i 0.0350877 + 0.0992431i
\(58\) 0 0
\(59\) 73.5391i 1.24643i −0.782052 0.623213i \(-0.785827\pi\)
0.782052 0.623213i \(-0.214173\pi\)
\(60\) 0 0
\(61\) 86.0000 1.40984 0.704918 0.709289i \(-0.250984\pi\)
0.704918 + 0.709289i \(0.250984\pi\)
\(62\) 0 0
\(63\) −42.0000 + 33.9411i −0.666667 + 0.538748i
\(64\) 0 0
\(65\) 56.5685i 0.870285i
\(66\) 0 0
\(67\) 2.00000 0.0298507 0.0149254 0.999889i \(-0.495249\pi\)
0.0149254 + 0.999889i \(0.495249\pi\)
\(68\) 0 0
\(69\) 32.0000 11.3137i 0.463768 0.163967i
\(70\) 0 0
\(71\) 124.451i 1.75283i −0.481558 0.876414i \(-0.659929\pi\)
0.481558 0.876414i \(-0.340071\pi\)
\(72\) 0 0
\(73\) 82.0000 1.12329 0.561644 0.827379i \(-0.310169\pi\)
0.561644 + 0.827379i \(0.310169\pi\)
\(74\) 0 0
\(75\) −7.00000 19.7990i −0.0933333 0.263987i
\(76\) 0 0
\(77\) 33.9411i 0.440794i
\(78\) 0 0
\(79\) −10.0000 −0.126582 −0.0632911 0.997995i \(-0.520160\pi\)
−0.0632911 + 0.997995i \(0.520160\pi\)
\(80\) 0 0
\(81\) 17.0000 79.1960i 0.209877 0.977728i
\(82\) 0 0
\(83\) 73.5391i 0.886013i 0.896518 + 0.443007i \(0.146088\pi\)
−0.896518 + 0.443007i \(0.853912\pi\)
\(84\) 0 0
\(85\) −128.000 −1.50588
\(86\) 0 0
\(87\) −48.0000 + 16.9706i −0.551724 + 0.195064i
\(88\) 0 0
\(89\) 33.9411i 0.381361i 0.981652 + 0.190680i \(0.0610694\pi\)
−0.981652 + 0.190680i \(0.938931\pi\)
\(90\) 0 0
\(91\) −60.0000 −0.659341
\(92\) 0 0
\(93\) 22.0000 + 62.2254i 0.236559 + 0.669090i
\(94\) 0 0
\(95\) 11.3137i 0.119092i
\(96\) 0 0
\(97\) −94.0000 −0.969072 −0.484536 0.874771i \(-0.661012\pi\)
−0.484536 + 0.874771i \(0.661012\pi\)
\(98\) 0 0
\(99\) 32.0000 + 39.5980i 0.323232 + 0.399980i
\(100\) 0 0
\(101\) 50.9117i 0.504076i 0.967717 + 0.252038i \(0.0811008\pi\)
−0.967717 + 0.252038i \(0.918899\pi\)
\(102\) 0 0
\(103\) 134.000 1.30097 0.650485 0.759519i \(-0.274565\pi\)
0.650485 + 0.759519i \(0.274565\pi\)
\(104\) 0 0
\(105\) −96.0000 + 33.9411i −0.914286 + 0.323249i
\(106\) 0 0
\(107\) 50.9117i 0.475810i −0.971288 0.237905i \(-0.923539\pi\)
0.971288 0.237905i \(-0.0764607\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.0917431 −0.0458716 0.998947i \(-0.514606\pi\)
−0.0458716 + 0.998947i \(0.514606\pi\)
\(110\) 0 0
\(111\) 6.00000 + 16.9706i 0.0540541 + 0.152888i
\(112\) 0 0
\(113\) 67.8823i 0.600728i 0.953825 + 0.300364i \(0.0971081\pi\)
−0.953825 + 0.300364i \(0.902892\pi\)
\(114\) 0 0
\(115\) 64.0000 0.556522
\(116\) 0 0
\(117\) 70.0000 56.5685i 0.598291 0.483492i
\(118\) 0 0
\(119\) 135.765i 1.14088i
\(120\) 0 0
\(121\) 89.0000 0.735537
\(122\) 0 0
\(123\) 96.0000 33.9411i 0.780488 0.275944i
\(124\) 0 0
\(125\) 101.823i 0.814587i
\(126\) 0 0
\(127\) −106.000 −0.834646 −0.417323 0.908758i \(-0.637032\pi\)
−0.417323 + 0.908758i \(0.637032\pi\)
\(128\) 0 0
\(129\) 82.0000 + 231.931i 0.635659 + 1.79791i
\(130\) 0 0
\(131\) 5.65685i 0.0431821i 0.999767 + 0.0215910i \(0.00687318\pi\)
−0.999767 + 0.0215910i \(0.993127\pi\)
\(132\) 0 0
\(133\) 12.0000 0.0902256
\(134\) 0 0
\(135\) 80.0000 130.108i 0.592593 0.963760i
\(136\) 0 0
\(137\) 101.823i 0.743236i 0.928386 + 0.371618i \(0.121197\pi\)
−0.928386 + 0.371618i \(0.878803\pi\)
\(138\) 0 0
\(139\) −78.0000 −0.561151 −0.280576 0.959832i \(-0.590525\pi\)
−0.280576 + 0.959832i \(0.590525\pi\)
\(140\) 0 0
\(141\) −192.000 + 67.8823i −1.36170 + 0.481434i
\(142\) 0 0
\(143\) 56.5685i 0.395584i
\(144\) 0 0
\(145\) −96.0000 −0.662069
\(146\) 0 0
\(147\) −13.0000 36.7696i −0.0884354 0.250133i
\(148\) 0 0
\(149\) 164.049i 1.10100i 0.834836 + 0.550499i \(0.185563\pi\)
−0.834836 + 0.550499i \(0.814437\pi\)
\(150\) 0 0
\(151\) −218.000 −1.44371 −0.721854 0.692045i \(-0.756710\pi\)
−0.721854 + 0.692045i \(0.756710\pi\)
\(152\) 0 0
\(153\) −128.000 158.392i −0.836601 1.03524i
\(154\) 0 0
\(155\) 124.451i 0.802908i
\(156\) 0 0
\(157\) 86.0000 0.547771 0.273885 0.961762i \(-0.411691\pi\)
0.273885 + 0.961762i \(0.411691\pi\)
\(158\) 0 0
\(159\) 176.000 62.2254i 1.10692 0.391355i
\(160\) 0 0
\(161\) 67.8823i 0.421629i
\(162\) 0 0
\(163\) −222.000 −1.36196 −0.680982 0.732301i \(-0.738447\pi\)
−0.680982 + 0.732301i \(0.738447\pi\)
\(164\) 0 0
\(165\) 32.0000 + 90.5097i 0.193939 + 0.548543i
\(166\) 0 0
\(167\) 169.706i 1.01620i −0.861298 0.508101i \(-0.830348\pi\)
0.861298 0.508101i \(-0.169652\pi\)
\(168\) 0 0
\(169\) −69.0000 −0.408284
\(170\) 0 0
\(171\) −14.0000 + 11.3137i −0.0818713 + 0.0661620i
\(172\) 0 0
\(173\) 186.676i 1.07905i −0.841969 0.539527i \(-0.818603\pi\)
0.841969 0.539527i \(-0.181397\pi\)
\(174\) 0 0
\(175\) −42.0000 −0.240000
\(176\) 0 0
\(177\) 208.000 73.5391i 1.17514 0.415475i
\(178\) 0 0
\(179\) 152.735i 0.853269i −0.904424 0.426634i \(-0.859699\pi\)
0.904424 0.426634i \(-0.140301\pi\)
\(180\) 0 0
\(181\) −90.0000 −0.497238 −0.248619 0.968601i \(-0.579977\pi\)
−0.248619 + 0.968601i \(0.579977\pi\)
\(182\) 0 0
\(183\) 86.0000 + 243.245i 0.469945 + 1.32921i
\(184\) 0 0
\(185\) 33.9411i 0.183466i
\(186\) 0 0
\(187\) 128.000 0.684492
\(188\) 0 0
\(189\) −138.000 84.8528i −0.730159 0.448957i
\(190\) 0 0
\(191\) 271.529i 1.42162i −0.703385 0.710809i \(-0.748329\pi\)
0.703385 0.710809i \(-0.251671\pi\)
\(192\) 0 0
\(193\) 2.00000 0.0103627 0.00518135 0.999987i \(-0.498351\pi\)
0.00518135 + 0.999987i \(0.498351\pi\)
\(194\) 0 0
\(195\) 160.000 56.5685i 0.820513 0.290095i
\(196\) 0 0
\(197\) 84.8528i 0.430725i −0.976534 0.215362i \(-0.930907\pi\)
0.976534 0.215362i \(-0.0690933\pi\)
\(198\) 0 0
\(199\) −250.000 −1.25628 −0.628141 0.778100i \(-0.716184\pi\)
−0.628141 + 0.778100i \(0.716184\pi\)
\(200\) 0 0
\(201\) 2.00000 + 5.65685i 0.00995025 + 0.0281436i
\(202\) 0 0
\(203\) 101.823i 0.501593i
\(204\) 0 0
\(205\) 192.000 0.936585
\(206\) 0 0
\(207\) 64.0000 + 79.1960i 0.309179 + 0.382589i
\(208\) 0 0
\(209\) 11.3137i 0.0541326i
\(210\) 0 0
\(211\) 34.0000 0.161137 0.0805687 0.996749i \(-0.474326\pi\)
0.0805687 + 0.996749i \(0.474326\pi\)
\(212\) 0 0
\(213\) 352.000 124.451i 1.65258 0.584276i
\(214\) 0 0
\(215\) 463.862i 2.15750i
\(216\) 0 0
\(217\) 132.000 0.608295
\(218\) 0 0
\(219\) 82.0000 + 231.931i 0.374429 + 1.05905i
\(220\) 0 0
\(221\) 226.274i 1.02387i
\(222\) 0 0
\(223\) 278.000 1.24664 0.623318 0.781968i \(-0.285784\pi\)
0.623318 + 0.781968i \(0.285784\pi\)
\(224\) 0 0
\(225\) 49.0000 39.5980i 0.217778 0.175991i
\(226\) 0 0
\(227\) 220.617i 0.971882i −0.873991 0.485941i \(-0.838477\pi\)
0.873991 0.485941i \(-0.161523\pi\)
\(228\) 0 0
\(229\) −58.0000 −0.253275 −0.126638 0.991949i \(-0.540419\pi\)
−0.126638 + 0.991949i \(0.540419\pi\)
\(230\) 0 0
\(231\) 96.0000 33.9411i 0.415584 0.146931i
\(232\) 0 0
\(233\) 395.980i 1.69948i −0.527199 0.849742i \(-0.676758\pi\)
0.527199 0.849742i \(-0.323242\pi\)
\(234\) 0 0
\(235\) −384.000 −1.63404
\(236\) 0 0
\(237\) −10.0000 28.2843i −0.0421941 0.119343i
\(238\) 0 0
\(239\) 22.6274i 0.0946754i −0.998879 0.0473377i \(-0.984926\pi\)
0.998879 0.0473377i \(-0.0150737\pi\)
\(240\) 0 0
\(241\) −30.0000 −0.124481 −0.0622407 0.998061i \(-0.519825\pi\)
−0.0622407 + 0.998061i \(0.519825\pi\)
\(242\) 0 0
\(243\) 241.000 31.1127i 0.991770 0.128036i
\(244\) 0 0
\(245\) 73.5391i 0.300160i
\(246\) 0 0
\(247\) −20.0000 −0.0809717
\(248\) 0 0
\(249\) −208.000 + 73.5391i −0.835341 + 0.295338i
\(250\) 0 0
\(251\) 107.480i 0.428208i 0.976811 + 0.214104i \(0.0686832\pi\)
−0.976811 + 0.214104i \(0.931317\pi\)
\(252\) 0 0
\(253\) −64.0000 −0.252964
\(254\) 0 0
\(255\) −128.000 362.039i −0.501961 1.41976i
\(256\) 0 0
\(257\) 181.019i 0.704355i −0.935933 0.352178i \(-0.885441\pi\)
0.935933 0.352178i \(-0.114559\pi\)
\(258\) 0 0
\(259\) 36.0000 0.138996
\(260\) 0 0
\(261\) −96.0000 118.794i −0.367816 0.455149i
\(262\) 0 0
\(263\) 214.960i 0.817340i −0.912682 0.408670i \(-0.865993\pi\)
0.912682 0.408670i \(-0.134007\pi\)
\(264\) 0 0
\(265\) 352.000 1.32830
\(266\) 0 0
\(267\) −96.0000 + 33.9411i −0.359551 + 0.127120i
\(268\) 0 0
\(269\) 401.637i 1.49307i 0.665344 + 0.746537i \(0.268285\pi\)
−0.665344 + 0.746537i \(0.731715\pi\)
\(270\) 0 0
\(271\) −266.000 −0.981550 −0.490775 0.871286i \(-0.663286\pi\)
−0.490775 + 0.871286i \(0.663286\pi\)
\(272\) 0 0
\(273\) −60.0000 169.706i −0.219780 0.621632i
\(274\) 0 0
\(275\) 39.5980i 0.143993i
\(276\) 0 0
\(277\) −346.000 −1.24910 −0.624549 0.780986i \(-0.714717\pi\)
−0.624549 + 0.780986i \(0.714717\pi\)
\(278\) 0 0
\(279\) −154.000 + 124.451i −0.551971 + 0.446060i
\(280\) 0 0
\(281\) 124.451i 0.442885i 0.975173 + 0.221443i \(0.0710766\pi\)
−0.975173 + 0.221443i \(0.928923\pi\)
\(282\) 0 0
\(283\) −46.0000 −0.162544 −0.0812721 0.996692i \(-0.525898\pi\)
−0.0812721 + 0.996692i \(0.525898\pi\)
\(284\) 0 0
\(285\) −32.0000 + 11.3137i −0.112281 + 0.0396972i
\(286\) 0 0
\(287\) 203.647i 0.709571i
\(288\) 0 0
\(289\) −223.000 −0.771626
\(290\) 0 0
\(291\) −94.0000 265.872i −0.323024 0.913650i
\(292\) 0 0
\(293\) 220.617i 0.752960i −0.926425 0.376480i \(-0.877134\pi\)
0.926425 0.376480i \(-0.122866\pi\)
\(294\) 0 0
\(295\) 416.000 1.41017
\(296\) 0 0
\(297\) −80.0000 + 130.108i −0.269360 + 0.438073i
\(298\) 0 0
\(299\) 113.137i 0.378385i
\(300\) 0 0
\(301\) 492.000 1.63455
\(302\) 0 0
\(303\) −144.000 + 50.9117i −0.475248 + 0.168025i
\(304\) 0 0
\(305\) 486.489i 1.59505i
\(306\) 0 0
\(307\) −30.0000 −0.0977199 −0.0488599 0.998806i \(-0.515559\pi\)
−0.0488599 + 0.998806i \(0.515559\pi\)
\(308\) 0 0
\(309\) 134.000 + 379.009i 0.433657 + 1.22657i
\(310\) 0 0
\(311\) 576.999i 1.85530i 0.373447 + 0.927651i \(0.378176\pi\)
−0.373447 + 0.927651i \(0.621824\pi\)
\(312\) 0 0
\(313\) 210.000 0.670927 0.335463 0.942053i \(-0.391107\pi\)
0.335463 + 0.942053i \(0.391107\pi\)
\(314\) 0 0
\(315\) −192.000 237.588i −0.609524 0.754247i
\(316\) 0 0
\(317\) 152.735i 0.481814i 0.970548 + 0.240907i \(0.0774449\pi\)
−0.970548 + 0.240907i \(0.922555\pi\)
\(318\) 0 0
\(319\) 96.0000 0.300940
\(320\) 0 0
\(321\) 144.000 50.9117i 0.448598 0.158603i
\(322\) 0 0
\(323\) 45.2548i 0.140108i
\(324\) 0 0
\(325\) 70.0000 0.215385
\(326\) 0 0
\(327\) −10.0000 28.2843i −0.0305810 0.0864962i
\(328\) 0 0
\(329\) 407.294i 1.23797i
\(330\) 0 0
\(331\) 434.000 1.31118 0.655589 0.755118i \(-0.272420\pi\)
0.655589 + 0.755118i \(0.272420\pi\)
\(332\) 0 0
\(333\) −42.0000 + 33.9411i −0.126126 + 0.101925i
\(334\) 0 0
\(335\) 11.3137i 0.0337723i
\(336\) 0 0
\(337\) −510.000 −1.51335 −0.756677 0.653789i \(-0.773178\pi\)
−0.756677 + 0.653789i \(0.773178\pi\)
\(338\) 0 0
\(339\) −192.000 + 67.8823i −0.566372 + 0.200243i
\(340\) 0 0
\(341\) 124.451i 0.364958i
\(342\) 0 0
\(343\) −372.000 −1.08455
\(344\) 0 0
\(345\) 64.0000 + 181.019i 0.185507 + 0.524694i
\(346\) 0 0
\(347\) 152.735i 0.440159i 0.975482 + 0.220079i \(0.0706316\pi\)
−0.975482 + 0.220079i \(0.929368\pi\)
\(348\) 0 0
\(349\) −426.000 −1.22063 −0.610315 0.792159i \(-0.708957\pi\)
−0.610315 + 0.792159i \(0.708957\pi\)
\(350\) 0 0
\(351\) 230.000 + 141.421i 0.655271 + 0.402910i
\(352\) 0 0
\(353\) 45.2548i 0.128201i −0.997943 0.0641003i \(-0.979582\pi\)
0.997943 0.0641003i \(-0.0204178\pi\)
\(354\) 0 0
\(355\) 704.000 1.98310
\(356\) 0 0
\(357\) −384.000 + 135.765i −1.07563 + 0.380293i
\(358\) 0 0
\(359\) 441.235i 1.22907i −0.788891 0.614533i \(-0.789345\pi\)
0.788891 0.614533i \(-0.210655\pi\)
\(360\) 0 0
\(361\) −357.000 −0.988920
\(362\) 0 0
\(363\) 89.0000 + 251.730i 0.245179 + 0.693471i
\(364\) 0 0
\(365\) 463.862i 1.27085i
\(366\) 0 0
\(367\) 566.000 1.54223 0.771117 0.636693i \(-0.219698\pi\)
0.771117 + 0.636693i \(0.219698\pi\)
\(368\) 0 0
\(369\) 192.000 + 237.588i 0.520325 + 0.643870i
\(370\) 0 0
\(371\) 373.352i 1.00634i
\(372\) 0 0
\(373\) −218.000 −0.584450 −0.292225 0.956350i \(-0.594396\pi\)
−0.292225 + 0.956350i \(0.594396\pi\)
\(374\) 0 0
\(375\) −288.000 + 101.823i −0.768000 + 0.271529i
\(376\) 0 0
\(377\) 169.706i 0.450148i
\(378\) 0 0
\(379\) −142.000 −0.374670 −0.187335 0.982296i \(-0.559985\pi\)
−0.187335 + 0.982296i \(0.559985\pi\)
\(380\) 0 0
\(381\) −106.000 299.813i −0.278215 0.786911i
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 192.000 0.498701
\(386\) 0 0
\(387\) −574.000 + 463.862i −1.48320 + 1.19861i
\(388\) 0 0
\(389\) 548.715i 1.41058i 0.708920 + 0.705289i \(0.249183\pi\)
−0.708920 + 0.705289i \(0.750817\pi\)
\(390\) 0 0
\(391\) 256.000 0.654731
\(392\) 0 0
\(393\) −16.0000 + 5.65685i −0.0407125 + 0.0143940i
\(394\) 0 0
\(395\) 56.5685i 0.143211i
\(396\) 0 0
\(397\) 310.000 0.780856 0.390428 0.920633i \(-0.372327\pi\)
0.390428 + 0.920633i \(0.372327\pi\)
\(398\) 0 0
\(399\) 12.0000 + 33.9411i 0.0300752 + 0.0850655i
\(400\) 0 0
\(401\) 339.411i 0.846412i −0.906033 0.423206i \(-0.860905\pi\)
0.906033 0.423206i \(-0.139095\pi\)
\(402\) 0 0
\(403\) −220.000 −0.545906
\(404\) 0 0
\(405\) 448.000 + 96.1665i 1.10617 + 0.237448i
\(406\) 0 0
\(407\) 33.9411i 0.0833934i
\(408\) 0 0
\(409\) −270.000 −0.660147 −0.330073 0.943955i \(-0.607073\pi\)
−0.330073 + 0.943955i \(0.607073\pi\)
\(410\) 0 0
\(411\) −288.000 + 101.823i −0.700730 + 0.247745i
\(412\) 0 0
\(413\) 441.235i 1.06836i
\(414\) 0 0
\(415\) −416.000 −1.00241
\(416\) 0 0
\(417\) −78.0000 220.617i −0.187050 0.529058i
\(418\) 0 0
\(419\) 50.9117i 0.121508i 0.998153 + 0.0607538i \(0.0193505\pi\)
−0.998153 + 0.0607538i \(0.980650\pi\)
\(420\) 0 0
\(421\) 454.000 1.07838 0.539192 0.842183i \(-0.318730\pi\)
0.539192 + 0.842183i \(0.318730\pi\)
\(422\) 0 0
\(423\) −384.000 475.176i −0.907801 1.12335i
\(424\) 0 0
\(425\) 158.392i 0.372687i
\(426\) 0 0
\(427\) 516.000 1.20843
\(428\) 0 0
\(429\) −160.000 + 56.5685i −0.372960 + 0.131861i
\(430\) 0 0
\(431\) 248.902i 0.577498i 0.957405 + 0.288749i \(0.0932393\pi\)
−0.957405 + 0.288749i \(0.906761\pi\)
\(432\) 0 0
\(433\) 706.000 1.63048 0.815242 0.579120i \(-0.196604\pi\)
0.815242 + 0.579120i \(0.196604\pi\)
\(434\) 0 0
\(435\) −96.0000 271.529i −0.220690 0.624205i
\(436\) 0 0
\(437\) 22.6274i 0.0517790i
\(438\) 0 0
\(439\) 486.000 1.10706 0.553531 0.832829i \(-0.313280\pi\)
0.553531 + 0.832829i \(0.313280\pi\)
\(440\) 0 0
\(441\) 91.0000 73.5391i 0.206349 0.166755i
\(442\) 0 0
\(443\) 707.107i 1.59618i −0.602540 0.798089i \(-0.705844\pi\)
0.602540 0.798089i \(-0.294156\pi\)
\(444\) 0 0
\(445\) −192.000 −0.431461
\(446\) 0 0
\(447\) −464.000 + 164.049i −1.03803 + 0.366999i
\(448\) 0 0
\(449\) 724.077i 1.61264i 0.591477 + 0.806322i \(0.298545\pi\)
−0.591477 + 0.806322i \(0.701455\pi\)
\(450\) 0 0
\(451\) −192.000 −0.425721
\(452\) 0 0
\(453\) −218.000 616.597i −0.481236 1.36114i
\(454\) 0 0
\(455\) 339.411i 0.745959i
\(456\) 0 0
\(457\) 338.000 0.739606 0.369803 0.929110i \(-0.379425\pi\)
0.369803 + 0.929110i \(0.379425\pi\)
\(458\) 0 0
\(459\) 320.000 520.431i 0.697168 1.13384i
\(460\) 0 0
\(461\) 774.989i 1.68110i −0.541731 0.840552i \(-0.682231\pi\)
0.541731 0.840552i \(-0.317769\pi\)
\(462\) 0 0
\(463\) −74.0000 −0.159827 −0.0799136 0.996802i \(-0.525464\pi\)
−0.0799136 + 0.996802i \(0.525464\pi\)
\(464\) 0 0
\(465\) −352.000 + 124.451i −0.756989 + 0.267636i
\(466\) 0 0
\(467\) 797.616i 1.70796i 0.520307 + 0.853979i \(0.325817\pi\)
−0.520307 + 0.853979i \(0.674183\pi\)
\(468\) 0 0
\(469\) 12.0000 0.0255864
\(470\) 0 0
\(471\) 86.0000 + 243.245i 0.182590 + 0.516443i
\(472\) 0 0
\(473\) 463.862i 0.980681i
\(474\) 0 0
\(475\) −14.0000 −0.0294737
\(476\) 0 0
\(477\) 352.000 + 435.578i 0.737945 + 0.913161i
\(478\) 0 0
\(479\) 316.784i 0.661344i 0.943746 + 0.330672i \(0.107275\pi\)
−0.943746 + 0.330672i \(0.892725\pi\)
\(480\) 0 0
\(481\) −60.0000 −0.124740
\(482\) 0 0
\(483\) 192.000 67.8823i 0.397516 0.140543i
\(484\) 0 0
\(485\) 531.744i 1.09638i
\(486\) 0 0
\(487\) 134.000 0.275154 0.137577 0.990491i \(-0.456069\pi\)
0.137577 + 0.990491i \(0.456069\pi\)
\(488\) 0 0
\(489\) −222.000 627.911i −0.453988 1.28407i
\(490\) 0 0
\(491\) 50.9117i 0.103690i −0.998655 0.0518449i \(-0.983490\pi\)
0.998655 0.0518449i \(-0.0165101\pi\)
\(492\) 0 0
\(493\) −384.000 −0.778905
\(494\) 0 0
\(495\) −224.000 + 181.019i −0.452525 + 0.365696i
\(496\) 0 0
\(497\) 746.705i 1.50242i
\(498\) 0 0
\(499\) −30.0000 −0.0601202 −0.0300601 0.999548i \(-0.509570\pi\)
−0.0300601 + 0.999548i \(0.509570\pi\)
\(500\) 0 0
\(501\) 480.000 169.706i 0.958084 0.338734i
\(502\) 0 0
\(503\) 237.588i 0.472342i −0.971712 0.236171i \(-0.924107\pi\)
0.971712 0.236171i \(-0.0758925\pi\)
\(504\) 0 0
\(505\) −288.000 −0.570297
\(506\) 0 0
\(507\) −69.0000 195.161i −0.136095 0.384934i
\(508\) 0 0
\(509\) 118.794i 0.233387i −0.993168 0.116693i \(-0.962770\pi\)
0.993168 0.116693i \(-0.0372295\pi\)
\(510\) 0 0
\(511\) 492.000 0.962818
\(512\) 0 0
\(513\) −46.0000 28.2843i −0.0896686 0.0551350i
\(514\) 0 0
\(515\) 758.018i 1.47188i
\(516\) 0 0
\(517\) 384.000 0.742747
\(518\) 0 0
\(519\) 528.000 186.676i 1.01734 0.359684i
\(520\) 0 0
\(521\) 79.1960i 0.152008i −0.997108 0.0760038i \(-0.975784\pi\)
0.997108 0.0760038i \(-0.0242161\pi\)
\(522\) 0 0
\(523\) −494.000 −0.944551 −0.472275 0.881451i \(-0.656567\pi\)
−0.472275 + 0.881451i \(0.656567\pi\)
\(524\) 0 0
\(525\) −42.0000 118.794i −0.0800000 0.226274i
\(526\) 0 0
\(527\) 497.803i 0.944598i
\(528\) 0 0
\(529\) 401.000 0.758034
\(530\) 0 0
\(531\) 416.000 + 514.774i 0.783427 + 0.969442i
\(532\) 0 0
\(533\) 339.411i 0.636794i
\(534\) 0 0
\(535\) 288.000 0.538318
\(536\) 0 0
\(537\) 432.000 152.735i 0.804469 0.284423i
\(538\) 0 0
\(539\) 73.5391i 0.136436i
\(540\) 0 0
\(541\) −234.000 −0.432532 −0.216266 0.976334i \(-0.569388\pi\)
−0.216266 + 0.976334i \(0.569388\pi\)
\(542\) 0 0
\(543\) −90.0000 254.558i −0.165746 0.468800i
\(544\) 0 0
\(545\) 56.5685i 0.103795i
\(546\) 0 0
\(547\) 290.000 0.530165 0.265082 0.964226i \(-0.414601\pi\)
0.265082 + 0.964226i \(0.414601\pi\)
\(548\) 0 0
\(549\) −602.000 + 486.489i −1.09654 + 0.886137i
\(550\) 0 0
\(551\) 33.9411i 0.0615991i
\(552\) 0 0
\(553\) −60.0000 −0.108499
\(554\) 0 0
\(555\) −96.0000 + 33.9411i −0.172973 + 0.0611552i
\(556\) 0 0
\(557\) 175.362i 0.314834i 0.987532 + 0.157417i \(0.0503167\pi\)
−0.987532 + 0.157417i \(0.949683\pi\)
\(558\) 0 0
\(559\) −820.000 −1.46691
\(560\) 0 0
\(561\) 128.000 + 362.039i 0.228164 + 0.645345i
\(562\) 0 0
\(563\) 243.245i 0.432051i −0.976388 0.216026i \(-0.930691\pi\)
0.976388 0.216026i \(-0.0693094\pi\)
\(564\) 0 0
\(565\) −384.000 −0.679646
\(566\) 0 0
\(567\) 102.000 475.176i 0.179894 0.838052i
\(568\) 0 0
\(569\) 622.254i 1.09359i 0.837266 + 0.546796i \(0.184153\pi\)
−0.837266 + 0.546796i \(0.815847\pi\)
\(570\) 0 0
\(571\) 402.000 0.704028 0.352014 0.935995i \(-0.385497\pi\)
0.352014 + 0.935995i \(0.385497\pi\)
\(572\) 0 0
\(573\) 768.000 271.529i 1.34031 0.473873i
\(574\) 0 0
\(575\) 79.1960i 0.137732i
\(576\) 0 0
\(577\) 98.0000 0.169844 0.0849220 0.996388i \(-0.472936\pi\)
0.0849220 + 0.996388i \(0.472936\pi\)
\(578\) 0 0
\(579\) 2.00000 + 5.65685i 0.00345423 + 0.00977004i
\(580\) 0 0
\(581\) 441.235i 0.759440i
\(582\) 0 0
\(583\) −352.000 −0.603774
\(584\) 0 0
\(585\) 320.000 + 395.980i 0.547009 + 0.676889i
\(586\) 0 0
\(587\) 548.715i 0.934778i −0.884052 0.467389i \(-0.845195\pi\)
0.884052 0.467389i \(-0.154805\pi\)
\(588\) 0 0
\(589\) 44.0000 0.0747029
\(590\) 0 0
\(591\) 240.000 84.8528i 0.406091 0.143575i
\(592\) 0 0
\(593\) 701.450i 1.18288i −0.806348 0.591442i \(-0.798559\pi\)
0.806348 0.591442i \(-0.201441\pi\)
\(594\) 0 0
\(595\) −768.000 −1.29076
\(596\) 0 0
\(597\) −250.000 707.107i −0.418760 1.18443i
\(598\) 0 0
\(599\) 644.881i 1.07660i −0.842754 0.538298i \(-0.819067\pi\)
0.842754 0.538298i \(-0.180933\pi\)
\(600\) 0 0
\(601\) −398.000 −0.662230 −0.331115 0.943590i \(-0.607425\pi\)
−0.331115 + 0.943590i \(0.607425\pi\)
\(602\) 0 0
\(603\) −14.0000 + 11.3137i −0.0232172 + 0.0187624i
\(604\) 0 0
\(605\) 503.460i 0.832165i
\(606\) 0 0
\(607\) −170.000 −0.280066 −0.140033 0.990147i \(-0.544721\pi\)
−0.140033 + 0.990147i \(0.544721\pi\)
\(608\) 0 0
\(609\) −288.000 + 101.823i −0.472906 + 0.167198i
\(610\) 0 0
\(611\) 678.823i 1.11100i
\(612\) 0 0
\(613\) 1030.00 1.68026 0.840131 0.542384i \(-0.182478\pi\)
0.840131 + 0.542384i \(0.182478\pi\)
\(614\) 0 0
\(615\) 192.000 + 543.058i 0.312195 + 0.883021i
\(616\) 0 0
\(617\) 1052.17i 1.70531i 0.522476 + 0.852654i \(0.325008\pi\)
−0.522476 + 0.852654i \(0.674992\pi\)
\(618\) 0 0
\(619\) −14.0000 −0.0226171 −0.0113086 0.999936i \(-0.503600\pi\)
−0.0113086 + 0.999936i \(0.503600\pi\)
\(620\) 0 0
\(621\) −160.000 + 260.215i −0.257649 + 0.419026i
\(622\) 0 0
\(623\) 203.647i 0.326881i
\(624\) 0 0
\(625\) −751.000 −1.20160
\(626\) 0 0
\(627\) 32.0000 11.3137i 0.0510367 0.0180442i
\(628\) 0 0
\(629\) 135.765i 0.215842i
\(630\) 0 0
\(631\) −1114.00 −1.76545 −0.882726 0.469888i \(-0.844294\pi\)
−0.882726 + 0.469888i \(0.844294\pi\)
\(632\) 0 0
\(633\) 34.0000 + 96.1665i 0.0537125 + 0.151922i
\(634\) 0 0
\(635\) 599.627i 0.944294i
\(636\) 0 0
\(637\) 130.000 0.204082
\(638\) 0 0
\(639\) 704.000 + 871.156i 1.10172 + 1.36331i
\(640\) 0 0
\(641\) 452.548i 0.706004i 0.935623 + 0.353002i \(0.114839\pi\)
−0.935623 + 0.353002i \(0.885161\pi\)
\(642\) 0 0
\(643\) −798.000 −1.24106 −0.620529 0.784184i \(-0.713082\pi\)
−0.620529 + 0.784184i \(0.713082\pi\)
\(644\) 0 0
\(645\) −1312.00 + 463.862i −2.03411 + 0.719166i
\(646\) 0 0
\(647\) 147.078i 0.227323i 0.993520 + 0.113662i \(0.0362580\pi\)
−0.993520 + 0.113662i \(0.963742\pi\)
\(648\) 0 0
\(649\) −416.000 −0.640986
\(650\) 0 0
\(651\) 132.000 + 373.352i 0.202765 + 0.573506i
\(652\) 0 0
\(653\) 322.441i 0.493784i −0.969043 0.246892i \(-0.920591\pi\)
0.969043 0.246892i \(-0.0794092\pi\)
\(654\) 0 0
\(655\) −32.0000 −0.0488550
\(656\) 0 0
\(657\) −574.000 + 463.862i −0.873668 + 0.706031i
\(658\) 0 0
\(659\) 797.616i 1.21034i 0.796095 + 0.605172i \(0.206896\pi\)
−0.796095 + 0.605172i \(0.793104\pi\)
\(660\) 0 0
\(661\) −986.000 −1.49168 −0.745840 0.666126i \(-0.767951\pi\)
−0.745840 + 0.666126i \(0.767951\pi\)
\(662\) 0 0
\(663\) 640.000 226.274i 0.965309 0.341288i
\(664\) 0 0
\(665\) 67.8823i 0.102079i
\(666\) 0 0
\(667\) 192.000 0.287856
\(668\) 0 0
\(669\) 278.000 + 786.303i 0.415546 + 1.17534i
\(670\) 0 0
\(671\) 486.489i 0.725022i
\(672\) 0 0
\(673\) 34.0000 0.0505201 0.0252600 0.999681i \(-0.491959\pi\)
0.0252600 + 0.999681i \(0.491959\pi\)
\(674\) 0 0
\(675\) 161.000 + 98.9949i 0.238519 + 0.146659i
\(676\) 0 0
\(677\) 401.637i 0.593259i −0.954993 0.296630i \(-0.904137\pi\)
0.954993 0.296630i \(-0.0958627\pi\)
\(678\) 0 0
\(679\) −564.000 −0.830633
\(680\) 0 0
\(681\) 624.000 220.617i 0.916300 0.323961i
\(682\) 0 0
\(683\) 130.108i 0.190494i 0.995454 + 0.0952472i \(0.0303641\pi\)
−0.995454 + 0.0952472i \(0.969636\pi\)
\(684\) 0 0
\(685\) −576.000 −0.840876
\(686\) 0 0
\(687\) −58.0000 164.049i −0.0844250 0.238790i
\(688\) 0 0
\(689\) 622.254i 0.903126i
\(690\) 0 0
\(691\) 578.000 0.836469 0.418234 0.908339i \(-0.362649\pi\)
0.418234 + 0.908339i \(0.362649\pi\)
\(692\) 0 0
\(693\) 192.000 + 237.588i 0.277056 + 0.342840i
\(694\) 0 0
\(695\) 441.235i 0.634870i
\(696\) 0 0
\(697\) 768.000 1.10187
\(698\) 0 0
\(699\) 1120.00 395.980i 1.60229 0.566495i
\(700\) 0 0
\(701\) 1238.85i 1.76726i 0.468183 + 0.883631i \(0.344909\pi\)
−0.468183 + 0.883631i \(0.655091\pi\)
\(702\) 0 0
\(703\) 12.0000 0.0170697
\(704\) 0 0
\(705\) −384.000 1086.12i −0.544681 1.54059i
\(706\) 0 0
\(707\) 305.470i 0.432065i
\(708\) 0 0
\(709\) 1222.00 1.72355 0.861777 0.507287i \(-0.169352\pi\)
0.861777 + 0.507287i \(0.169352\pi\)
\(710\) 0 0
\(711\) 70.0000 56.5685i 0.0984529 0.0795619i
\(712\) 0 0
\(713\) 248.902i 0.349091i
\(714\) 0 0
\(715\) −320.000 −0.447552
\(716\) 0 0
\(717\) 64.0000 22.6274i 0.0892608 0.0315585i
\(718\) 0 0
\(719\) 248.902i 0.346177i −0.984906 0.173089i \(-0.944625\pi\)
0.984906 0.173089i \(-0.0553747\pi\)
\(720\) 0 0
\(721\) 804.000 1.11512
\(722\) 0 0
\(723\) −30.0000 84.8528i −0.0414938 0.117362i
\(724\) 0 0
\(725\) 118.794i 0.163854i
\(726\) 0 0
\(727\) 870.000 1.19670 0.598349 0.801235i \(-0.295823\pi\)
0.598349 + 0.801235i \(0.295823\pi\)
\(728\) 0 0
\(729\) 329.000 + 650.538i 0.451303 + 0.892371i
\(730\) 0 0
\(731\) 1855.45i 2.53823i
\(732\) 0 0
\(733\) 214.000 0.291951 0.145975 0.989288i \(-0.453368\pi\)
0.145975 + 0.989288i \(0.453368\pi\)
\(734\) 0 0
\(735\) 208.000 73.5391i 0.282993 0.100053i
\(736\) 0 0
\(737\) 11.3137i 0.0153510i
\(738\) 0 0
\(739\) −958.000 −1.29635 −0.648173 0.761493i \(-0.724467\pi\)
−0.648173 + 0.761493i \(0.724467\pi\)
\(740\) 0 0
\(741\) −20.0000 56.5685i −0.0269906 0.0763408i
\(742\) 0 0
\(743\) 1006.92i 1.35521i 0.735427 + 0.677604i \(0.236982\pi\)
−0.735427 + 0.677604i \(0.763018\pi\)
\(744\) 0 0
\(745\) −928.000 −1.24564
\(746\) 0 0
\(747\) −416.000 514.774i −0.556894 0.689121i
\(748\) 0 0
\(749\) 305.470i 0.407837i
\(750\) 0 0
\(751\) 630.000 0.838881 0.419441 0.907783i \(-0.362226\pi\)
0.419441 + 0.907783i \(0.362226\pi\)
\(752\) 0 0
\(753\) −304.000 + 107.480i −0.403718 + 0.142736i
\(754\) 0 0
\(755\) 1233.19i 1.63337i
\(756\) 0 0
\(757\) −602.000 −0.795244 −0.397622 0.917549i \(-0.630165\pi\)
−0.397622 + 0.917549i \(0.630165\pi\)
\(758\) 0 0
\(759\) −64.0000 181.019i −0.0843215 0.238497i
\(760\) 0 0
\(761\) 1097.43i 1.44209i −0.692889 0.721044i \(-0.743662\pi\)
0.692889 0.721044i \(-0.256338\pi\)
\(762\) 0 0
\(763\) −60.0000 −0.0786370
\(764\) 0 0
\(765\) 896.000 724.077i 1.17124 0.946506i
\(766\) 0 0
\(767\) 735.391i 0.958789i
\(768\) 0 0
\(769\) 770.000 1.00130 0.500650 0.865650i \(-0.333094\pi\)
0.500650 + 0.865650i \(0.333094\pi\)
\(770\) 0 0
\(771\) 512.000 181.019i 0.664073 0.234785i
\(772\) 0 0
\(773\) 186.676i 0.241496i 0.992683 + 0.120748i \(0.0385293\pi\)
−0.992683 + 0.120748i \(0.961471\pi\)
\(774\) 0 0
\(775\) −154.000 −0.198710
\(776\) 0 0
\(777\) 36.0000 + 101.823i 0.0463320 + 0.131047i
\(778\) 0 0
\(779\) 67.8823i 0.0871402i
\(780\) 0 0
\(781\) −704.000 −0.901408
\(782\) 0 0
\(783\) 240.000 390.323i 0.306513 0.498497i
\(784\) 0 0
\(785\) 486.489i 0.619732i
\(786\) 0 0
\(787\) 514.000 0.653113 0.326557 0.945178i \(-0.394112\pi\)
0.326557 + 0.945178i \(0.394112\pi\)
\(788\) 0 0
\(789\) 608.000 214.960i 0.770596 0.272447i
\(790\) 0 0
\(791\) 407.294i 0.514910i
\(792\) 0 0
\(793\) −860.000 −1.08449
\(794\) 0 0
\(795\) 352.000 + 995.606i 0.442767 + 1.25234i
\(796\) 0 0
\(797\) 707.107i 0.887211i −0.896222 0.443605i \(-0.853699\pi\)
0.896222 0.443605i \(-0.146301\pi\)
\(798\) 0 0
\(799\) −1536.00 −1.92240
\(800\) 0 0
\(801\) −192.000 237.588i −0.239700 0.296614i
\(802\) 0 0
\(803\) 463.862i 0.577661i
\(804\) 0 0
\(805\) 384.000 0.477019
\(806\) 0 0
\(807\) −1136.00 + 401.637i −1.40768 + 0.497691i
\(808\) 0 0
\(809\) 758.018i 0.936982i −0.883468 0.468491i \(-0.844798\pi\)
0.883468 0.468491i \(-0.155202\pi\)
\(810\) 0 0
\(811\) −1454.00 −1.79285 −0.896424 0.443197i \(-0.853844\pi\)
−0.896424 + 0.443197i \(0.853844\pi\)
\(812\) 0 0
\(813\) −266.000 752.362i −0.327183 0.925414i
\(814\) 0 0
\(815\) 1255.82i 1.54089i
\(816\) 0 0
\(817\) 164.000 0.200734
\(818\) 0 0
\(819\) 420.000 339.411i 0.512821 0.414422i
\(820\) 0 0
\(821\) 967.322i 1.17822i −0.808051 0.589112i \(-0.799478\pi\)
0.808051 0.589112i \(-0.200522\pi\)
\(822\) 0 0
\(823\) 166.000 0.201701 0.100851 0.994902i \(-0.467844\pi\)
0.100851 + 0.994902i \(0.467844\pi\)
\(824\) 0 0
\(825\) −112.000 + 39.5980i −0.135758 + 0.0479976i
\(826\) 0 0
\(827\) 978.636i 1.18336i −0.806174 0.591678i \(-0.798466\pi\)
0.806174 0.591678i \(-0.201534\pi\)
\(828\) 0 0
\(829\) −1258.00 −1.51749 −0.758745 0.651387i \(-0.774187\pi\)
−0.758745 + 0.651387i \(0.774187\pi\)
\(830\) 0 0
\(831\) −346.000 978.636i −0.416366 1.17766i
\(832\) 0 0
\(833\) 294.156i 0.353129i
\(834\) 0 0
\(835\) 960.000 1.14970
\(836\) 0 0
\(837\) −506.000 311.127i −0.604540 0.371717i
\(838\) 0 0
\(839\) 1323.70i 1.57772i 0.614575 + 0.788858i \(0.289327\pi\)
−0.614575 + 0.788858i \(0.710673\pi\)
\(840\) 0 0
\(841\) 553.000 0.657551
\(842\) 0 0
\(843\) −352.000 + 124.451i −0.417556 + 0.147628i
\(844\) 0 0
\(845\) 390.323i 0.461921i
\(846\) 0 0
\(847\) 534.000 0.630460
\(848\) 0 0
\(849\) −46.0000 130.108i −0.0541814 0.153248i
\(850\) 0 0
\(851\) 67.8823i 0.0797676i
\(852\) 0 0
\(853\) 742.000 0.869871 0.434936 0.900462i \(-0.356771\pi\)
0.434936 + 0.900462i \(0.356771\pi\)
\(854\) 0 0
\(855\) −64.0000 79.1960i −0.0748538 0.0926269i
\(856\) 0 0
\(857\) 237.588i 0.277232i −0.990346 0.138616i \(-0.955735\pi\)
0.990346 0.138616i \(-0.0442654\pi\)
\(858\) 0 0
\(859\) −1230.00 −1.43190 −0.715949 0.698153i \(-0.754006\pi\)
−0.715949 + 0.698153i \(0.754006\pi\)
\(860\) 0 0
\(861\) 576.000 203.647i 0.668990 0.236524i
\(862\) 0 0
\(863\) 45.2548i 0.0524390i −0.999656 0.0262195i \(-0.991653\pi\)
0.999656 0.0262195i \(-0.00834688\pi\)
\(864\) 0 0
\(865\) 1056.00 1.22081
\(866\) 0 0
\(867\) −223.000 630.739i −0.257209 0.727496i
\(868\) 0 0
\(869\) 56.5685i 0.0650961i
\(870\) 0 0
\(871\) −20.0000 −0.0229621
\(872\) 0 0
\(873\) 658.000 531.744i 0.753723 0.609100i
\(874\) 0 0
\(875\) 610.940i 0.698217i
\(876\) 0 0
\(877\) 822.000 0.937286 0.468643 0.883388i \(-0.344743\pi\)
0.468643 + 0.883388i \(0.344743\pi\)
\(878\) 0 0
\(879\) 624.000 220.617i 0.709898 0.250987i
\(880\) 0 0
\(881\) 656.195i 0.744830i −0.928066 0.372415i \(-0.878530\pi\)
0.928066 0.372415i \(-0.121470\pi\)
\(882\) 0 0
\(883\) 962.000 1.08947 0.544734 0.838609i \(-0.316631\pi\)
0.544734 + 0.838609i \(0.316631\pi\)
\(884\) 0 0
\(885\) 416.000 + 1176.63i 0.470056 + 1.32952i
\(886\) 0 0
\(887\) 1142.68i 1.28826i −0.764917 0.644129i \(-0.777220\pi\)
0.764917 0.644129i \(-0.222780\pi\)
\(888\) 0 0
\(889\) −636.000 −0.715411
\(890\) 0 0
\(891\) −448.000 96.1665i −0.502806 0.107931i
\(892\) 0 0
\(893\) 135.765i 0.152032i
\(894\) 0 0
\(895\) 864.000 0.965363
\(896\) 0 0
\(897\) −320.000 + 113.137i −0.356745 + 0.126128i
\(898\) 0 0
\(899\) 373.352i 0.415297i
\(900\) 0 0
\(901\) 1408.00 1.56271
\(902\) 0 0
\(903\) 492.000 + 1391.59i 0.544850 + 1.54107i
\(904\) 0 0
\(905\) 509.117i 0.562560i
\(906\) 0 0
\(907\) 1042.00 1.14884 0.574421 0.818560i \(-0.305227\pi\)
0.574421 + 0.818560i \(0.305227\pi\)
\(908\) 0 0
\(909\) −288.000 356.382i −0.316832 0.392059i
\(910\) 0 0
\(911\) 1606.55i 1.76350i −0.471719 0.881749i \(-0.656366\pi\)
0.471719 0.881749i \(-0.343634\pi\)
\(912\) 0 0
\(913\) 416.000 0.455641
\(914\) 0 0
\(915\) −1376.00 + 486.489i −1.50383 + 0.531682i
\(916\) 0 0
\(917\) 33.9411i 0.0370132i
\(918\) 0 0
\(919\) 614.000 0.668118 0.334059 0.942552i \(-0.391582\pi\)
0.334059 + 0.942552i \(0.391582\pi\)
\(920\) 0 0
\(921\) −30.0000 84.8528i −0.0325733 0.0921312i
\(922\) 0 0
\(923\) 1244.51i 1.34833i
\(924\) 0 0
\(925\) −42.0000 −0.0454054
\(926\) 0 0
\(927\) −938.000 + 758.018i −1.01187 + 0.817711i
\(928\) 0 0
\(929\) 45.2548i 0.0487135i −0.999703 0.0243567i \(-0.992246\pi\)
0.999703 0.0243567i \(-0.00775376\pi\)
\(930\) 0 0
\(931\) −26.0000 −0.0279270
\(932\) 0 0
\(933\) −1632.00 + 576.999i −1.74920 + 0.618434i
\(934\) 0 0
\(935\) 724.077i 0.774414i
\(936\) 0 0
\(937\) −462.000 −0.493063 −0.246531 0.969135i \(-0.579291\pi\)
−0.246531 + 0.969135i \(0.579291\pi\)
\(938\) 0 0
\(939\) 210.000 + 593.970i 0.223642 + 0.632556i
\(940\) 0 0
\(941\) 356.382i 0.378727i 0.981907 + 0.189363i \(0.0606424\pi\)
−0.981907 + 0.189363i \(0.939358\pi\)
\(942\) 0 0
\(943\) −384.000 −0.407211
\(944\) 0 0
\(945\) 480.000 780.646i 0.507937 0.826080i
\(946\) 0 0
\(947\) 695.793i 0.734734i −0.930076 0.367367i \(-0.880259\pi\)
0.930076 0.367367i \(-0.119741\pi\)
\(948\) 0 0
\(949\) −820.000 −0.864067
\(950\) 0 0
\(951\) −432.000 + 152.735i −0.454259 + 0.160605i
\(952\) 0 0
\(953\) 1527.35i 1.60268i 0.598212 + 0.801338i \(0.295878\pi\)
−0.598212 + 0.801338i \(0.704122\pi\)
\(954\) 0 0
\(955\) 1536.00 1.60838
\(956\) 0 0
\(957\) 96.0000 + 271.529i 0.100313 + 0.283729i
\(958\) 0 0
\(959\) 610.940i 0.637060i
\(960\) 0 0
\(961\) −477.000 −0.496358
\(962\) 0 0
\(963\) 288.000 + 356.382i 0.299065 + 0.370075i
\(964\) 0 0
\(965\) 11.3137i 0.0117241i
\(966\) 0 0
\(967\) 70.0000 0.0723888 0.0361944 0.999345i \(-0.488476\pi\)
0.0361944 + 0.999345i \(0.488476\pi\)
\(968\) 0 0
\(969\) −128.000 + 45.2548i −0.132095 + 0.0467026i
\(970\) 0 0
\(971\) 627.911i 0.646664i 0.946286 + 0.323332i \(0.104803\pi\)
−0.946286 + 0.323332i \(0.895197\pi\)
\(972\) 0 0
\(973\) −468.000 −0.480987
\(974\) 0 0
\(975\) 70.0000 + 197.990i 0.0717949 + 0.203067i
\(976\) 0 0
\(977\) 1018.23i 1.04220i 0.853494 + 0.521102i \(0.174479\pi\)
−0.853494 + 0.521102i \(0.825521\pi\)
\(978\) 0 0
\(979\) 192.000 0.196118
\(980\) 0 0
\(981\) 70.0000 56.5685i 0.0713558 0.0576642i
\(982\) 0 0
\(983\) 644.881i 0.656034i −0.944672 0.328017i \(-0.893620\pi\)
0.944672 0.328017i \(-0.106380\pi\)
\(984\) 0 0
\(985\) 480.000 0.487310
\(986\) 0 0
\(987\) −1152.00 + 407.294i −1.16717 + 0.412658i
\(988\) 0 0
\(989\) 927.724i 0.938043i
\(990\) 0 0
\(991\) 854.000 0.861756 0.430878 0.902410i \(-0.358204\pi\)
0.430878 + 0.902410i \(0.358204\pi\)
\(992\) 0 0
\(993\) 434.000 + 1227.54i 0.437059 + 1.23619i
\(994\) 0 0
\(995\) 1414.21i 1.42132i
\(996\) 0 0
\(997\) 518.000 0.519559 0.259779 0.965668i \(-0.416350\pi\)
0.259779 + 0.965668i \(0.416350\pi\)
\(998\) 0 0
\(999\) −138.000 84.8528i −0.138138 0.0849378i
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 192.3.e.d.65.2 2
3.2 odd 2 inner 192.3.e.d.65.1 2
4.3 odd 2 192.3.e.c.65.1 2
8.3 odd 2 24.3.e.a.17.2 yes 2
8.5 even 2 48.3.e.b.17.1 2
12.11 even 2 192.3.e.c.65.2 2
16.3 odd 4 768.3.h.d.641.3 4
16.5 even 4 768.3.h.c.641.3 4
16.11 odd 4 768.3.h.d.641.2 4
16.13 even 4 768.3.h.c.641.2 4
24.5 odd 2 48.3.e.b.17.2 2
24.11 even 2 24.3.e.a.17.1 2
40.3 even 4 600.3.c.a.449.2 4
40.13 odd 4 1200.3.c.i.449.3 4
40.19 odd 2 600.3.l.b.401.1 2
40.27 even 4 600.3.c.a.449.3 4
40.29 even 2 1200.3.l.n.401.2 2
40.37 odd 4 1200.3.c.i.449.2 4
48.5 odd 4 768.3.h.c.641.1 4
48.11 even 4 768.3.h.d.641.4 4
48.29 odd 4 768.3.h.c.641.4 4
48.35 even 4 768.3.h.d.641.1 4
56.27 even 2 1176.3.d.a.785.1 2
72.5 odd 6 1296.3.q.e.593.2 4
72.11 even 6 648.3.m.d.377.1 4
72.13 even 6 1296.3.q.e.593.1 4
72.29 odd 6 1296.3.q.e.1025.1 4
72.43 odd 6 648.3.m.d.377.2 4
72.59 even 6 648.3.m.d.593.2 4
72.61 even 6 1296.3.q.e.1025.2 4
72.67 odd 6 648.3.m.d.593.1 4
120.29 odd 2 1200.3.l.n.401.1 2
120.53 even 4 1200.3.c.i.449.1 4
120.59 even 2 600.3.l.b.401.2 2
120.77 even 4 1200.3.c.i.449.4 4
120.83 odd 4 600.3.c.a.449.4 4
120.107 odd 4 600.3.c.a.449.1 4
168.83 odd 2 1176.3.d.a.785.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.3.e.a.17.1 2 24.11 even 2
24.3.e.a.17.2 yes 2 8.3 odd 2
48.3.e.b.17.1 2 8.5 even 2
48.3.e.b.17.2 2 24.5 odd 2
192.3.e.c.65.1 2 4.3 odd 2
192.3.e.c.65.2 2 12.11 even 2
192.3.e.d.65.1 2 3.2 odd 2 inner
192.3.e.d.65.2 2 1.1 even 1 trivial
600.3.c.a.449.1 4 120.107 odd 4
600.3.c.a.449.2 4 40.3 even 4
600.3.c.a.449.3 4 40.27 even 4
600.3.c.a.449.4 4 120.83 odd 4
600.3.l.b.401.1 2 40.19 odd 2
600.3.l.b.401.2 2 120.59 even 2
648.3.m.d.377.1 4 72.11 even 6
648.3.m.d.377.2 4 72.43 odd 6
648.3.m.d.593.1 4 72.67 odd 6
648.3.m.d.593.2 4 72.59 even 6
768.3.h.c.641.1 4 48.5 odd 4
768.3.h.c.641.2 4 16.13 even 4
768.3.h.c.641.3 4 16.5 even 4
768.3.h.c.641.4 4 48.29 odd 4
768.3.h.d.641.1 4 48.35 even 4
768.3.h.d.641.2 4 16.11 odd 4
768.3.h.d.641.3 4 16.3 odd 4
768.3.h.d.641.4 4 48.11 even 4
1176.3.d.a.785.1 2 56.27 even 2
1176.3.d.a.785.2 2 168.83 odd 2
1200.3.c.i.449.1 4 120.53 even 4
1200.3.c.i.449.2 4 40.37 odd 4
1200.3.c.i.449.3 4 40.13 odd 4
1200.3.c.i.449.4 4 120.77 even 4
1200.3.l.n.401.1 2 120.29 odd 2
1200.3.l.n.401.2 2 40.29 even 2
1296.3.q.e.593.1 4 72.13 even 6
1296.3.q.e.593.2 4 72.5 odd 6
1296.3.q.e.1025.1 4 72.29 odd 6
1296.3.q.e.1025.2 4 72.61 even 6