Properties

Label 8092.2.a.l.1.2
Level $8092$
Weight $2$
Character 8092.1
Self dual yes
Analytic conductor $64.615$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8092,2,Mod(1,8092)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8092, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8092.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8092 = 2^{2} \cdot 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8092.a (trivial)

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: yes
Analytic conductor: \(64.6149453156\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 476)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 8092.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.30278 q^{3} +1.30278 q^{5} +1.00000 q^{7} +2.30278 q^{9} +O(q^{10})\) \(q+2.30278 q^{3} +1.30278 q^{5} +1.00000 q^{7} +2.30278 q^{9} -4.00000 q^{11} -4.60555 q^{13} +3.00000 q^{15} +8.60555 q^{19} +2.30278 q^{21} -4.00000 q^{23} -3.30278 q^{25} -1.60555 q^{27} -9.21110 q^{29} -7.30278 q^{31} -9.21110 q^{33} +1.30278 q^{35} -9.81665 q^{37} -10.6056 q^{39} +11.5139 q^{41} -4.30278 q^{43} +3.00000 q^{45} -2.60555 q^{47} +1.00000 q^{49} +0.697224 q^{53} -5.21110 q^{55} +19.8167 q^{57} -8.00000 q^{59} -15.5139 q^{61} +2.30278 q^{63} -6.00000 q^{65} +2.69722 q^{67} -9.21110 q^{69} +3.39445 q^{71} -7.51388 q^{73} -7.60555 q^{75} -4.00000 q^{77} +2.60555 q^{79} -10.6056 q^{81} +3.21110 q^{83} -21.2111 q^{87} +7.81665 q^{89} -4.60555 q^{91} -16.8167 q^{93} +11.2111 q^{95} +13.3028 q^{97} -9.21110 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - q^{5} + 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - q^{5} + 2 q^{7} + q^{9} - 8 q^{11} - 2 q^{13} + 6 q^{15} + 10 q^{19} + q^{21} - 8 q^{23} - 3 q^{25} + 4 q^{27} - 4 q^{29} - 11 q^{31} - 4 q^{33} - q^{35} + 2 q^{37} - 14 q^{39} + 5 q^{41} - 5 q^{43} + 6 q^{45} + 2 q^{47} + 2 q^{49} + 5 q^{53} + 4 q^{55} + 18 q^{57} - 16 q^{59} - 13 q^{61} + q^{63} - 12 q^{65} + 9 q^{67} - 4 q^{69} + 14 q^{71} + 3 q^{73} - 8 q^{75} - 8 q^{77} - 2 q^{79} - 14 q^{81} - 8 q^{83} - 28 q^{87} - 6 q^{89} - 2 q^{91} - 12 q^{93} + 8 q^{95} + 23 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 2.30278 1.32951 0.664754 0.747062i \(-0.268536\pi\)
0.664754 + 0.747062i \(0.268536\pi\)
\(4\) 0 0
\(5\) 1.30278 0.582619 0.291309 0.956629i \(-0.405909\pi\)
0.291309 + 0.956629i \(0.405909\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 2.30278 0.767592
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −4.60555 −1.27735 −0.638675 0.769477i \(-0.720517\pi\)
−0.638675 + 0.769477i \(0.720517\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 0 0
\(17\) 0 0
\(18\) 0 0
\(19\) 8.60555 1.97425 0.987124 0.159954i \(-0.0511347\pi\)
0.987124 + 0.159954i \(0.0511347\pi\)
\(20\) 0 0
\(21\) 2.30278 0.502507
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −3.30278 −0.660555
\(26\) 0 0
\(27\) −1.60555 −0.308988
\(28\) 0 0
\(29\) −9.21110 −1.71046 −0.855229 0.518250i \(-0.826584\pi\)
−0.855229 + 0.518250i \(0.826584\pi\)
\(30\) 0 0
\(31\) −7.30278 −1.31162 −0.655809 0.754927i \(-0.727672\pi\)
−0.655809 + 0.754927i \(0.727672\pi\)
\(32\) 0 0
\(33\) −9.21110 −1.60345
\(34\) 0 0
\(35\) 1.30278 0.220209
\(36\) 0 0
\(37\) −9.81665 −1.61385 −0.806924 0.590655i \(-0.798869\pi\)
−0.806924 + 0.590655i \(0.798869\pi\)
\(38\) 0 0
\(39\) −10.6056 −1.69825
\(40\) 0 0
\(41\) 11.5139 1.79817 0.899083 0.437779i \(-0.144235\pi\)
0.899083 + 0.437779i \(0.144235\pi\)
\(42\) 0 0
\(43\) −4.30278 −0.656167 −0.328084 0.944649i \(-0.606403\pi\)
−0.328084 + 0.944649i \(0.606403\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) 0 0
\(47\) −2.60555 −0.380059 −0.190029 0.981778i \(-0.560858\pi\)
−0.190029 + 0.981778i \(0.560858\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.697224 0.0957711 0.0478856 0.998853i \(-0.484752\pi\)
0.0478856 + 0.998853i \(0.484752\pi\)
\(54\) 0 0
\(55\) −5.21110 −0.702665
\(56\) 0 0
\(57\) 19.8167 2.62478
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −15.5139 −1.98635 −0.993174 0.116640i \(-0.962788\pi\)
−0.993174 + 0.116640i \(0.962788\pi\)
\(62\) 0 0
\(63\) 2.30278 0.290122
\(64\) 0 0
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) 2.69722 0.329518 0.164759 0.986334i \(-0.447315\pi\)
0.164759 + 0.986334i \(0.447315\pi\)
\(68\) 0 0
\(69\) −9.21110 −1.10889
\(70\) 0 0
\(71\) 3.39445 0.402847 0.201423 0.979504i \(-0.435443\pi\)
0.201423 + 0.979504i \(0.435443\pi\)
\(72\) 0 0
\(73\) −7.51388 −0.879433 −0.439716 0.898137i \(-0.644921\pi\)
−0.439716 + 0.898137i \(0.644921\pi\)
\(74\) 0 0
\(75\) −7.60555 −0.878213
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) 2.60555 0.293147 0.146574 0.989200i \(-0.453175\pi\)
0.146574 + 0.989200i \(0.453175\pi\)
\(80\) 0 0
\(81\) −10.6056 −1.17839
\(82\) 0 0
\(83\) 3.21110 0.352464 0.176232 0.984349i \(-0.443609\pi\)
0.176232 + 0.984349i \(0.443609\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −21.2111 −2.27407
\(88\) 0 0
\(89\) 7.81665 0.828564 0.414282 0.910149i \(-0.364033\pi\)
0.414282 + 0.910149i \(0.364033\pi\)
\(90\) 0 0
\(91\) −4.60555 −0.482793
\(92\) 0 0
\(93\) −16.8167 −1.74381
\(94\) 0 0
\(95\) 11.2111 1.15023
\(96\) 0 0
\(97\) 13.3028 1.35069 0.675346 0.737501i \(-0.263994\pi\)
0.675346 + 0.737501i \(0.263994\pi\)
\(98\) 0 0
\(99\) −9.21110 −0.925751
\(100\) 0 0
\(101\) −12.6056 −1.25430 −0.627150 0.778899i \(-0.715779\pi\)
−0.627150 + 0.778899i \(0.715779\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 0 0
\(107\) 4.60555 0.445235 0.222618 0.974906i \(-0.428540\pi\)
0.222618 + 0.974906i \(0.428540\pi\)
\(108\) 0 0
\(109\) −4.78890 −0.458693 −0.229347 0.973345i \(-0.573659\pi\)
−0.229347 + 0.973345i \(0.573659\pi\)
\(110\) 0 0
\(111\) −22.6056 −2.14562
\(112\) 0 0
\(113\) 9.39445 0.883755 0.441878 0.897075i \(-0.354313\pi\)
0.441878 + 0.897075i \(0.354313\pi\)
\(114\) 0 0
\(115\) −5.21110 −0.485938
\(116\) 0 0
\(117\) −10.6056 −0.980484
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 26.5139 2.39068
\(124\) 0 0
\(125\) −10.8167 −0.967471
\(126\) 0 0
\(127\) 14.5139 1.28790 0.643949 0.765068i \(-0.277295\pi\)
0.643949 + 0.765068i \(0.277295\pi\)
\(128\) 0 0
\(129\) −9.90833 −0.872380
\(130\) 0 0
\(131\) 9.21110 0.804778 0.402389 0.915469i \(-0.368180\pi\)
0.402389 + 0.915469i \(0.368180\pi\)
\(132\) 0 0
\(133\) 8.60555 0.746196
\(134\) 0 0
\(135\) −2.09167 −0.180023
\(136\) 0 0
\(137\) 18.1194 1.54805 0.774024 0.633157i \(-0.218241\pi\)
0.774024 + 0.633157i \(0.218241\pi\)
\(138\) 0 0
\(139\) −8.51388 −0.722138 −0.361069 0.932539i \(-0.617588\pi\)
−0.361069 + 0.932539i \(0.617588\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 18.4222 1.54054
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 0 0
\(147\) 2.30278 0.189930
\(148\) 0 0
\(149\) −4.30278 −0.352497 −0.176249 0.984346i \(-0.556396\pi\)
−0.176249 + 0.984346i \(0.556396\pi\)
\(150\) 0 0
\(151\) 11.6972 0.951907 0.475953 0.879471i \(-0.342103\pi\)
0.475953 + 0.879471i \(0.342103\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −9.51388 −0.764173
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 1.60555 0.127328
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) −1.39445 −0.109222 −0.0546108 0.998508i \(-0.517392\pi\)
−0.0546108 + 0.998508i \(0.517392\pi\)
\(164\) 0 0
\(165\) −12.0000 −0.934199
\(166\) 0 0
\(167\) −12.1194 −0.937830 −0.468915 0.883243i \(-0.655355\pi\)
−0.468915 + 0.883243i \(0.655355\pi\)
\(168\) 0 0
\(169\) 8.21110 0.631623
\(170\) 0 0
\(171\) 19.8167 1.51542
\(172\) 0 0
\(173\) −20.7250 −1.57569 −0.787846 0.615873i \(-0.788803\pi\)
−0.787846 + 0.615873i \(0.788803\pi\)
\(174\) 0 0
\(175\) −3.30278 −0.249666
\(176\) 0 0
\(177\) −18.4222 −1.38470
\(178\) 0 0
\(179\) 9.51388 0.711101 0.355550 0.934657i \(-0.384293\pi\)
0.355550 + 0.934657i \(0.384293\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) −35.7250 −2.64087
\(184\) 0 0
\(185\) −12.7889 −0.940258
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.60555 −0.116787
\(190\) 0 0
\(191\) −11.7250 −0.848390 −0.424195 0.905571i \(-0.639443\pi\)
−0.424195 + 0.905571i \(0.639443\pi\)
\(192\) 0 0
\(193\) 15.0278 1.08172 0.540861 0.841112i \(-0.318099\pi\)
0.540861 + 0.841112i \(0.318099\pi\)
\(194\) 0 0
\(195\) −13.8167 −0.989431
\(196\) 0 0
\(197\) 6.60555 0.470626 0.235313 0.971920i \(-0.424388\pi\)
0.235313 + 0.971920i \(0.424388\pi\)
\(198\) 0 0
\(199\) −9.69722 −0.687418 −0.343709 0.939076i \(-0.611683\pi\)
−0.343709 + 0.939076i \(0.611683\pi\)
\(200\) 0 0
\(201\) 6.21110 0.438097
\(202\) 0 0
\(203\) −9.21110 −0.646493
\(204\) 0 0
\(205\) 15.0000 1.04765
\(206\) 0 0
\(207\) −9.21110 −0.640216
\(208\) 0 0
\(209\) −34.4222 −2.38103
\(210\) 0 0
\(211\) −6.18335 −0.425679 −0.212840 0.977087i \(-0.568271\pi\)
−0.212840 + 0.977087i \(0.568271\pi\)
\(212\) 0 0
\(213\) 7.81665 0.535588
\(214\) 0 0
\(215\) −5.60555 −0.382295
\(216\) 0 0
\(217\) −7.30278 −0.495745
\(218\) 0 0
\(219\) −17.3028 −1.16921
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 13.8167 0.925232 0.462616 0.886559i \(-0.346911\pi\)
0.462616 + 0.886559i \(0.346911\pi\)
\(224\) 0 0
\(225\) −7.60555 −0.507037
\(226\) 0 0
\(227\) −2.09167 −0.138829 −0.0694146 0.997588i \(-0.522113\pi\)
−0.0694146 + 0.997588i \(0.522113\pi\)
\(228\) 0 0
\(229\) −1.21110 −0.0800319 −0.0400160 0.999199i \(-0.512741\pi\)
−0.0400160 + 0.999199i \(0.512741\pi\)
\(230\) 0 0
\(231\) −9.21110 −0.606046
\(232\) 0 0
\(233\) −26.6056 −1.74299 −0.871494 0.490407i \(-0.836848\pi\)
−0.871494 + 0.490407i \(0.836848\pi\)
\(234\) 0 0
\(235\) −3.39445 −0.221429
\(236\) 0 0
\(237\) 6.00000 0.389742
\(238\) 0 0
\(239\) 18.1194 1.17205 0.586024 0.810294i \(-0.300692\pi\)
0.586024 + 0.810294i \(0.300692\pi\)
\(240\) 0 0
\(241\) 26.5139 1.70791 0.853955 0.520348i \(-0.174198\pi\)
0.853955 + 0.520348i \(0.174198\pi\)
\(242\) 0 0
\(243\) −19.6056 −1.25770
\(244\) 0 0
\(245\) 1.30278 0.0832313
\(246\) 0 0
\(247\) −39.6333 −2.52181
\(248\) 0 0
\(249\) 7.39445 0.468604
\(250\) 0 0
\(251\) −2.18335 −0.137812 −0.0689058 0.997623i \(-0.521951\pi\)
−0.0689058 + 0.997623i \(0.521951\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4.42221 −0.275850 −0.137925 0.990443i \(-0.544043\pi\)
−0.137925 + 0.990443i \(0.544043\pi\)
\(258\) 0 0
\(259\) −9.81665 −0.609977
\(260\) 0 0
\(261\) −21.2111 −1.31293
\(262\) 0 0
\(263\) −18.4222 −1.13596 −0.567981 0.823042i \(-0.692275\pi\)
−0.567981 + 0.823042i \(0.692275\pi\)
\(264\) 0 0
\(265\) 0.908327 0.0557981
\(266\) 0 0
\(267\) 18.0000 1.10158
\(268\) 0 0
\(269\) −12.7889 −0.779753 −0.389876 0.920867i \(-0.627482\pi\)
−0.389876 + 0.920867i \(0.627482\pi\)
\(270\) 0 0
\(271\) −10.4222 −0.633104 −0.316552 0.948575i \(-0.602525\pi\)
−0.316552 + 0.948575i \(0.602525\pi\)
\(272\) 0 0
\(273\) −10.6056 −0.641877
\(274\) 0 0
\(275\) 13.2111 0.796659
\(276\) 0 0
\(277\) −1.57779 −0.0948005 −0.0474003 0.998876i \(-0.515094\pi\)
−0.0474003 + 0.998876i \(0.515094\pi\)
\(278\) 0 0
\(279\) −16.8167 −1.00679
\(280\) 0 0
\(281\) −19.1194 −1.14057 −0.570285 0.821447i \(-0.693167\pi\)
−0.570285 + 0.821447i \(0.693167\pi\)
\(282\) 0 0
\(283\) −20.3305 −1.20852 −0.604262 0.796785i \(-0.706532\pi\)
−0.604262 + 0.796785i \(0.706532\pi\)
\(284\) 0 0
\(285\) 25.8167 1.52925
\(286\) 0 0
\(287\) 11.5139 0.679643
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) 30.6333 1.79576
\(292\) 0 0
\(293\) −11.2111 −0.654960 −0.327480 0.944858i \(-0.606199\pi\)
−0.327480 + 0.944858i \(0.606199\pi\)
\(294\) 0 0
\(295\) −10.4222 −0.606804
\(296\) 0 0
\(297\) 6.42221 0.372654
\(298\) 0 0
\(299\) 18.4222 1.06538
\(300\) 0 0
\(301\) −4.30278 −0.248008
\(302\) 0 0
\(303\) −29.0278 −1.66760
\(304\) 0 0
\(305\) −20.2111 −1.15728
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) 32.2389 1.83400
\(310\) 0 0
\(311\) 17.7250 1.00509 0.502546 0.864551i \(-0.332397\pi\)
0.502546 + 0.864551i \(0.332397\pi\)
\(312\) 0 0
\(313\) −3.90833 −0.220912 −0.110456 0.993881i \(-0.535231\pi\)
−0.110456 + 0.993881i \(0.535231\pi\)
\(314\) 0 0
\(315\) 3.00000 0.169031
\(316\) 0 0
\(317\) 33.6333 1.88903 0.944517 0.328461i \(-0.106530\pi\)
0.944517 + 0.328461i \(0.106530\pi\)
\(318\) 0 0
\(319\) 36.8444 2.06289
\(320\) 0 0
\(321\) 10.6056 0.591944
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 15.2111 0.843760
\(326\) 0 0
\(327\) −11.0278 −0.609836
\(328\) 0 0
\(329\) −2.60555 −0.143649
\(330\) 0 0
\(331\) 3.72498 0.204743 0.102372 0.994746i \(-0.467357\pi\)
0.102372 + 0.994746i \(0.467357\pi\)
\(332\) 0 0
\(333\) −22.6056 −1.23878
\(334\) 0 0
\(335\) 3.51388 0.191984
\(336\) 0 0
\(337\) 14.4222 0.785628 0.392814 0.919618i \(-0.371502\pi\)
0.392814 + 0.919618i \(0.371502\pi\)
\(338\) 0 0
\(339\) 21.6333 1.17496
\(340\) 0 0
\(341\) 29.2111 1.58187
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −12.0000 −0.646058
\(346\) 0 0
\(347\) −26.8444 −1.44108 −0.720542 0.693412i \(-0.756107\pi\)
−0.720542 + 0.693412i \(0.756107\pi\)
\(348\) 0 0
\(349\) 18.4222 0.986118 0.493059 0.869996i \(-0.335879\pi\)
0.493059 + 0.869996i \(0.335879\pi\)
\(350\) 0 0
\(351\) 7.39445 0.394686
\(352\) 0 0
\(353\) −16.0000 −0.851594 −0.425797 0.904819i \(-0.640006\pi\)
−0.425797 + 0.904819i \(0.640006\pi\)
\(354\) 0 0
\(355\) 4.42221 0.234706
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.48612 −0.0784345 −0.0392173 0.999231i \(-0.512486\pi\)
−0.0392173 + 0.999231i \(0.512486\pi\)
\(360\) 0 0
\(361\) 55.0555 2.89766
\(362\) 0 0
\(363\) 11.5139 0.604322
\(364\) 0 0
\(365\) −9.78890 −0.512374
\(366\) 0 0
\(367\) −6.90833 −0.360612 −0.180306 0.983611i \(-0.557709\pi\)
−0.180306 + 0.983611i \(0.557709\pi\)
\(368\) 0 0
\(369\) 26.5139 1.38026
\(370\) 0 0
\(371\) 0.697224 0.0361981
\(372\) 0 0
\(373\) −15.3305 −0.793785 −0.396892 0.917865i \(-0.629911\pi\)
−0.396892 + 0.917865i \(0.629911\pi\)
\(374\) 0 0
\(375\) −24.9083 −1.28626
\(376\) 0 0
\(377\) 42.4222 2.18485
\(378\) 0 0
\(379\) −9.57779 −0.491978 −0.245989 0.969273i \(-0.579113\pi\)
−0.245989 + 0.969273i \(0.579113\pi\)
\(380\) 0 0
\(381\) 33.4222 1.71227
\(382\) 0 0
\(383\) −1.57779 −0.0806216 −0.0403108 0.999187i \(-0.512835\pi\)
−0.0403108 + 0.999187i \(0.512835\pi\)
\(384\) 0 0
\(385\) −5.21110 −0.265582
\(386\) 0 0
\(387\) −9.90833 −0.503669
\(388\) 0 0
\(389\) −29.7250 −1.50712 −0.753558 0.657381i \(-0.771664\pi\)
−0.753558 + 0.657381i \(0.771664\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 21.2111 1.06996
\(394\) 0 0
\(395\) 3.39445 0.170793
\(396\) 0 0
\(397\) 12.0917 0.606864 0.303432 0.952853i \(-0.401868\pi\)
0.303432 + 0.952853i \(0.401868\pi\)
\(398\) 0 0
\(399\) 19.8167 0.992074
\(400\) 0 0
\(401\) −2.60555 −0.130115 −0.0650575 0.997882i \(-0.520723\pi\)
−0.0650575 + 0.997882i \(0.520723\pi\)
\(402\) 0 0
\(403\) 33.6333 1.67539
\(404\) 0 0
\(405\) −13.8167 −0.686555
\(406\) 0 0
\(407\) 39.2666 1.94637
\(408\) 0 0
\(409\) −5.81665 −0.287615 −0.143808 0.989606i \(-0.545935\pi\)
−0.143808 + 0.989606i \(0.545935\pi\)
\(410\) 0 0
\(411\) 41.7250 2.05814
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 4.18335 0.205352
\(416\) 0 0
\(417\) −19.6056 −0.960088
\(418\) 0 0
\(419\) 8.51388 0.415930 0.207965 0.978136i \(-0.433316\pi\)
0.207965 + 0.978136i \(0.433316\pi\)
\(420\) 0 0
\(421\) 11.0917 0.540575 0.270288 0.962780i \(-0.412881\pi\)
0.270288 + 0.962780i \(0.412881\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −15.5139 −0.750769
\(428\) 0 0
\(429\) 42.4222 2.04816
\(430\) 0 0
\(431\) 28.8444 1.38939 0.694693 0.719306i \(-0.255540\pi\)
0.694693 + 0.719306i \(0.255540\pi\)
\(432\) 0 0
\(433\) −7.39445 −0.355355 −0.177677 0.984089i \(-0.556858\pi\)
−0.177677 + 0.984089i \(0.556858\pi\)
\(434\) 0 0
\(435\) −27.6333 −1.32492
\(436\) 0 0
\(437\) −34.4222 −1.64664
\(438\) 0 0
\(439\) −21.0917 −1.00665 −0.503325 0.864097i \(-0.667890\pi\)
−0.503325 + 0.864097i \(0.667890\pi\)
\(440\) 0 0
\(441\) 2.30278 0.109656
\(442\) 0 0
\(443\) −33.2111 −1.57791 −0.788954 0.614453i \(-0.789377\pi\)
−0.788954 + 0.614453i \(0.789377\pi\)
\(444\) 0 0
\(445\) 10.1833 0.482737
\(446\) 0 0
\(447\) −9.90833 −0.468648
\(448\) 0 0
\(449\) 5.02776 0.237274 0.118637 0.992938i \(-0.462147\pi\)
0.118637 + 0.992938i \(0.462147\pi\)
\(450\) 0 0
\(451\) −46.0555 −2.16867
\(452\) 0 0
\(453\) 26.9361 1.26557
\(454\) 0 0
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) −13.9083 −0.650604 −0.325302 0.945610i \(-0.605466\pi\)
−0.325302 + 0.945610i \(0.605466\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 23.3944 1.08959 0.544794 0.838570i \(-0.316608\pi\)
0.544794 + 0.838570i \(0.316608\pi\)
\(462\) 0 0
\(463\) −6.72498 −0.312536 −0.156268 0.987715i \(-0.549946\pi\)
−0.156268 + 0.987715i \(0.549946\pi\)
\(464\) 0 0
\(465\) −21.9083 −1.01597
\(466\) 0 0
\(467\) −16.6056 −0.768413 −0.384207 0.923247i \(-0.625525\pi\)
−0.384207 + 0.923247i \(0.625525\pi\)
\(468\) 0 0
\(469\) 2.69722 0.124546
\(470\) 0 0
\(471\) 4.60555 0.212213
\(472\) 0 0
\(473\) 17.2111 0.791367
\(474\) 0 0
\(475\) −28.4222 −1.30410
\(476\) 0 0
\(477\) 1.60555 0.0735131
\(478\) 0 0
\(479\) −17.0917 −0.780938 −0.390469 0.920616i \(-0.627687\pi\)
−0.390469 + 0.920616i \(0.627687\pi\)
\(480\) 0 0
\(481\) 45.2111 2.06145
\(482\) 0 0
\(483\) −9.21110 −0.419120
\(484\) 0 0
\(485\) 17.3305 0.786939
\(486\) 0 0
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) 0 0
\(489\) −3.21110 −0.145211
\(490\) 0 0
\(491\) −22.5139 −1.01604 −0.508019 0.861346i \(-0.669622\pi\)
−0.508019 + 0.861346i \(0.669622\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −12.0000 −0.539360
\(496\) 0 0
\(497\) 3.39445 0.152262
\(498\) 0 0
\(499\) 29.2111 1.30767 0.653834 0.756638i \(-0.273159\pi\)
0.653834 + 0.756638i \(0.273159\pi\)
\(500\) 0 0
\(501\) −27.9083 −1.24685
\(502\) 0 0
\(503\) 31.1194 1.38755 0.693773 0.720193i \(-0.255947\pi\)
0.693773 + 0.720193i \(0.255947\pi\)
\(504\) 0 0
\(505\) −16.4222 −0.730779
\(506\) 0 0
\(507\) 18.9083 0.839748
\(508\) 0 0
\(509\) 6.60555 0.292786 0.146393 0.989227i \(-0.453234\pi\)
0.146393 + 0.989227i \(0.453234\pi\)
\(510\) 0 0
\(511\) −7.51388 −0.332394
\(512\) 0 0
\(513\) −13.8167 −0.610020
\(514\) 0 0
\(515\) 18.2389 0.803700
\(516\) 0 0
\(517\) 10.4222 0.458368
\(518\) 0 0
\(519\) −47.7250 −2.09489
\(520\) 0 0
\(521\) 0.486122 0.0212974 0.0106487 0.999943i \(-0.496610\pi\)
0.0106487 + 0.999943i \(0.496610\pi\)
\(522\) 0 0
\(523\) 38.0555 1.66405 0.832026 0.554737i \(-0.187181\pi\)
0.832026 + 0.554737i \(0.187181\pi\)
\(524\) 0 0
\(525\) −7.60555 −0.331933
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −18.4222 −0.799456
\(532\) 0 0
\(533\) −53.0278 −2.29689
\(534\) 0 0
\(535\) 6.00000 0.259403
\(536\) 0 0
\(537\) 21.9083 0.945414
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) −4.00000 −0.171973 −0.0859867 0.996296i \(-0.527404\pi\)
−0.0859867 + 0.996296i \(0.527404\pi\)
\(542\) 0 0
\(543\) 13.8167 0.592929
\(544\) 0 0
\(545\) −6.23886 −0.267243
\(546\) 0 0
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) 0 0
\(549\) −35.7250 −1.52471
\(550\) 0 0
\(551\) −79.2666 −3.37687
\(552\) 0 0
\(553\) 2.60555 0.110799
\(554\) 0 0
\(555\) −29.4500 −1.25008
\(556\) 0 0
\(557\) −20.7889 −0.880854 −0.440427 0.897788i \(-0.645173\pi\)
−0.440427 + 0.897788i \(0.645173\pi\)
\(558\) 0 0
\(559\) 19.8167 0.838155
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) 12.2389 0.514893
\(566\) 0 0
\(567\) −10.6056 −0.445391
\(568\) 0 0
\(569\) 22.1194 0.927295 0.463647 0.886020i \(-0.346540\pi\)
0.463647 + 0.886020i \(0.346540\pi\)
\(570\) 0 0
\(571\) 45.4500 1.90202 0.951011 0.309158i \(-0.100047\pi\)
0.951011 + 0.309158i \(0.100047\pi\)
\(572\) 0 0
\(573\) −27.0000 −1.12794
\(574\) 0 0
\(575\) 13.2111 0.550941
\(576\) 0 0
\(577\) −8.18335 −0.340677 −0.170339 0.985386i \(-0.554486\pi\)
−0.170339 + 0.985386i \(0.554486\pi\)
\(578\) 0 0
\(579\) 34.6056 1.43816
\(580\) 0 0
\(581\) 3.21110 0.133219
\(582\) 0 0
\(583\) −2.78890 −0.115504
\(584\) 0 0
\(585\) −13.8167 −0.571248
\(586\) 0 0
\(587\) −11.3944 −0.470299 −0.235150 0.971959i \(-0.575558\pi\)
−0.235150 + 0.971959i \(0.575558\pi\)
\(588\) 0 0
\(589\) −62.8444 −2.58946
\(590\) 0 0
\(591\) 15.2111 0.625701
\(592\) 0 0
\(593\) 22.1833 0.910961 0.455480 0.890246i \(-0.349468\pi\)
0.455480 + 0.890246i \(0.349468\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −22.3305 −0.913928
\(598\) 0 0
\(599\) 26.7250 1.09195 0.545977 0.837800i \(-0.316159\pi\)
0.545977 + 0.837800i \(0.316159\pi\)
\(600\) 0 0
\(601\) −45.2666 −1.84646 −0.923232 0.384243i \(-0.874462\pi\)
−0.923232 + 0.384243i \(0.874462\pi\)
\(602\) 0 0
\(603\) 6.21110 0.252936
\(604\) 0 0
\(605\) 6.51388 0.264827
\(606\) 0 0
\(607\) −0.330532 −0.0134159 −0.00670794 0.999978i \(-0.502135\pi\)
−0.00670794 + 0.999978i \(0.502135\pi\)
\(608\) 0 0
\(609\) −21.2111 −0.859517
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) −21.7250 −0.877464 −0.438732 0.898618i \(-0.644572\pi\)
−0.438732 + 0.898618i \(0.644572\pi\)
\(614\) 0 0
\(615\) 34.5416 1.39285
\(616\) 0 0
\(617\) −7.63331 −0.307305 −0.153653 0.988125i \(-0.549104\pi\)
−0.153653 + 0.988125i \(0.549104\pi\)
\(618\) 0 0
\(619\) −37.2111 −1.49564 −0.747820 0.663901i \(-0.768900\pi\)
−0.747820 + 0.663901i \(0.768900\pi\)
\(620\) 0 0
\(621\) 6.42221 0.257714
\(622\) 0 0
\(623\) 7.81665 0.313168
\(624\) 0 0
\(625\) 2.42221 0.0968882
\(626\) 0 0
\(627\) −79.2666 −3.16560
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 9.11943 0.363039 0.181519 0.983387i \(-0.441898\pi\)
0.181519 + 0.983387i \(0.441898\pi\)
\(632\) 0 0
\(633\) −14.2389 −0.565944
\(634\) 0 0
\(635\) 18.9083 0.750354
\(636\) 0 0
\(637\) −4.60555 −0.182479
\(638\) 0 0
\(639\) 7.81665 0.309222
\(640\) 0 0
\(641\) −43.6333 −1.72341 −0.861706 0.507408i \(-0.830604\pi\)
−0.861706 + 0.507408i \(0.830604\pi\)
\(642\) 0 0
\(643\) −7.88057 −0.310779 −0.155390 0.987853i \(-0.549663\pi\)
−0.155390 + 0.987853i \(0.549663\pi\)
\(644\) 0 0
\(645\) −12.9083 −0.508265
\(646\) 0 0
\(647\) −35.6333 −1.40089 −0.700445 0.713706i \(-0.747015\pi\)
−0.700445 + 0.713706i \(0.747015\pi\)
\(648\) 0 0
\(649\) 32.0000 1.25611
\(650\) 0 0
\(651\) −16.8167 −0.659097
\(652\) 0 0
\(653\) −40.4222 −1.58184 −0.790922 0.611918i \(-0.790398\pi\)
−0.790922 + 0.611918i \(0.790398\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) −17.3028 −0.675046
\(658\) 0 0
\(659\) 21.1194 0.822696 0.411348 0.911478i \(-0.365058\pi\)
0.411348 + 0.911478i \(0.365058\pi\)
\(660\) 0 0
\(661\) −5.02776 −0.195557 −0.0977785 0.995208i \(-0.531174\pi\)
−0.0977785 + 0.995208i \(0.531174\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 11.2111 0.434748
\(666\) 0 0
\(667\) 36.8444 1.42662
\(668\) 0 0
\(669\) 31.8167 1.23010
\(670\) 0 0
\(671\) 62.0555 2.39563
\(672\) 0 0
\(673\) 7.02776 0.270900 0.135450 0.990784i \(-0.456752\pi\)
0.135450 + 0.990784i \(0.456752\pi\)
\(674\) 0 0
\(675\) 5.30278 0.204104
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 13.3028 0.510514
\(680\) 0 0
\(681\) −4.81665 −0.184575
\(682\) 0 0
\(683\) −13.3944 −0.512524 −0.256262 0.966607i \(-0.582491\pi\)
−0.256262 + 0.966607i \(0.582491\pi\)
\(684\) 0 0
\(685\) 23.6056 0.901922
\(686\) 0 0
\(687\) −2.78890 −0.106403
\(688\) 0 0
\(689\) −3.21110 −0.122333
\(690\) 0 0
\(691\) 1.93608 0.0736521 0.0368260 0.999322i \(-0.488275\pi\)
0.0368260 + 0.999322i \(0.488275\pi\)
\(692\) 0 0
\(693\) −9.21110 −0.349901
\(694\) 0 0
\(695\) −11.0917 −0.420731
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −61.2666 −2.31732
\(700\) 0 0
\(701\) −11.5778 −0.437287 −0.218644 0.975805i \(-0.570163\pi\)
−0.218644 + 0.975805i \(0.570163\pi\)
\(702\) 0 0
\(703\) −84.4777 −3.18614
\(704\) 0 0
\(705\) −7.81665 −0.294392
\(706\) 0 0
\(707\) −12.6056 −0.474081
\(708\) 0 0
\(709\) 4.23886 0.159194 0.0795968 0.996827i \(-0.474637\pi\)
0.0795968 + 0.996827i \(0.474637\pi\)
\(710\) 0 0
\(711\) 6.00000 0.225018
\(712\) 0 0
\(713\) 29.2111 1.09396
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) 0 0
\(717\) 41.7250 1.55825
\(718\) 0 0
\(719\) −9.69722 −0.361645 −0.180823 0.983516i \(-0.557876\pi\)
−0.180823 + 0.983516i \(0.557876\pi\)
\(720\) 0 0
\(721\) 14.0000 0.521387
\(722\) 0 0
\(723\) 61.0555 2.27068
\(724\) 0 0
\(725\) 30.4222 1.12985
\(726\) 0 0
\(727\) −14.4222 −0.534890 −0.267445 0.963573i \(-0.586179\pi\)
−0.267445 + 0.963573i \(0.586179\pi\)
\(728\) 0 0
\(729\) −13.3305 −0.493723
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 16.2389 0.599796 0.299898 0.953971i \(-0.403047\pi\)
0.299898 + 0.953971i \(0.403047\pi\)
\(734\) 0 0
\(735\) 3.00000 0.110657
\(736\) 0 0
\(737\) −10.7889 −0.397414
\(738\) 0 0
\(739\) −23.3305 −0.858227 −0.429114 0.903250i \(-0.641174\pi\)
−0.429114 + 0.903250i \(0.641174\pi\)
\(740\) 0 0
\(741\) −91.2666 −3.35276
\(742\) 0 0
\(743\) 34.6056 1.26955 0.634777 0.772695i \(-0.281092\pi\)
0.634777 + 0.772695i \(0.281092\pi\)
\(744\) 0 0
\(745\) −5.60555 −0.205372
\(746\) 0 0
\(747\) 7.39445 0.270549
\(748\) 0 0
\(749\) 4.60555 0.168283
\(750\) 0 0
\(751\) −3.39445 −0.123865 −0.0619326 0.998080i \(-0.519726\pi\)
−0.0619326 + 0.998080i \(0.519726\pi\)
\(752\) 0 0
\(753\) −5.02776 −0.183222
\(754\) 0 0
\(755\) 15.2389 0.554599
\(756\) 0 0
\(757\) −7.69722 −0.279760 −0.139880 0.990168i \(-0.544672\pi\)
−0.139880 + 0.990168i \(0.544672\pi\)
\(758\) 0 0
\(759\) 36.8444 1.33737
\(760\) 0 0
\(761\) −50.2389 −1.82116 −0.910579 0.413336i \(-0.864364\pi\)
−0.910579 + 0.413336i \(0.864364\pi\)
\(762\) 0 0
\(763\) −4.78890 −0.173370
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 36.8444 1.33037
\(768\) 0 0
\(769\) −36.0555 −1.30020 −0.650098 0.759851i \(-0.725272\pi\)
−0.650098 + 0.759851i \(0.725272\pi\)
\(770\) 0 0
\(771\) −10.1833 −0.366744
\(772\) 0 0
\(773\) 17.0278 0.612446 0.306223 0.951960i \(-0.400935\pi\)
0.306223 + 0.951960i \(0.400935\pi\)
\(774\) 0 0
\(775\) 24.1194 0.866395
\(776\) 0 0
\(777\) −22.6056 −0.810970
\(778\) 0 0
\(779\) 99.0833 3.55003
\(780\) 0 0
\(781\) −13.5778 −0.485852
\(782\) 0 0
\(783\) 14.7889 0.528512
\(784\) 0 0
\(785\) 2.60555 0.0929961
\(786\) 0 0
\(787\) −50.4222 −1.79736 −0.898679 0.438607i \(-0.855472\pi\)
−0.898679 + 0.438607i \(0.855472\pi\)
\(788\) 0 0
\(789\) −42.4222 −1.51027
\(790\) 0 0
\(791\) 9.39445 0.334028
\(792\) 0 0
\(793\) 71.4500 2.53726
\(794\) 0 0
\(795\) 2.09167 0.0741840
\(796\) 0 0
\(797\) 48.0555 1.70221 0.851107 0.524993i \(-0.175932\pi\)
0.851107 + 0.524993i \(0.175932\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 0 0
\(803\) 30.0555 1.06064
\(804\) 0 0
\(805\) −5.21110 −0.183667
\(806\) 0 0
\(807\) −29.4500 −1.03669
\(808\) 0 0
\(809\) 19.8167 0.696716 0.348358 0.937361i \(-0.386739\pi\)
0.348358 + 0.937361i \(0.386739\pi\)
\(810\) 0 0
\(811\) 4.48612 0.157529 0.0787645 0.996893i \(-0.474902\pi\)
0.0787645 + 0.996893i \(0.474902\pi\)
\(812\) 0 0
\(813\) −24.0000 −0.841717
\(814\) 0 0
\(815\) −1.81665 −0.0636346
\(816\) 0 0
\(817\) −37.0278 −1.29544
\(818\) 0 0
\(819\) −10.6056 −0.370588
\(820\) 0 0
\(821\) 8.23886 0.287538 0.143769 0.989611i \(-0.454078\pi\)
0.143769 + 0.989611i \(0.454078\pi\)
\(822\) 0 0
\(823\) 38.6056 1.34570 0.672852 0.739777i \(-0.265069\pi\)
0.672852 + 0.739777i \(0.265069\pi\)
\(824\) 0 0
\(825\) 30.4222 1.05917
\(826\) 0 0
\(827\) −45.2111 −1.57214 −0.786072 0.618135i \(-0.787889\pi\)
−0.786072 + 0.618135i \(0.787889\pi\)
\(828\) 0 0
\(829\) −16.2389 −0.563999 −0.281999 0.959415i \(-0.590998\pi\)
−0.281999 + 0.959415i \(0.590998\pi\)
\(830\) 0 0
\(831\) −3.63331 −0.126038
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −15.7889 −0.546397
\(836\) 0 0
\(837\) 11.7250 0.405275
\(838\) 0 0
\(839\) −18.4222 −0.636005 −0.318003 0.948090i \(-0.603012\pi\)
−0.318003 + 0.948090i \(0.603012\pi\)
\(840\) 0 0
\(841\) 55.8444 1.92567
\(842\) 0 0
\(843\) −44.0278 −1.51640
\(844\) 0 0
\(845\) 10.6972 0.367996
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) 0 0
\(849\) −46.8167 −1.60674
\(850\) 0 0
\(851\) 39.2666 1.34604
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) 25.8167 0.882911
\(856\) 0 0
\(857\) 45.9638 1.57009 0.785047 0.619436i \(-0.212639\pi\)
0.785047 + 0.619436i \(0.212639\pi\)
\(858\) 0 0
\(859\) −54.6056 −1.86312 −0.931559 0.363591i \(-0.881551\pi\)
−0.931559 + 0.363591i \(0.881551\pi\)
\(860\) 0 0
\(861\) 26.5139 0.903591
\(862\) 0 0
\(863\) 23.3583 0.795125 0.397563 0.917575i \(-0.369856\pi\)
0.397563 + 0.917575i \(0.369856\pi\)
\(864\) 0 0
\(865\) −27.0000 −0.918028
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −10.4222 −0.353549
\(870\) 0 0
\(871\) −12.4222 −0.420910
\(872\) 0 0
\(873\) 30.6333 1.03678
\(874\) 0 0
\(875\) −10.8167 −0.365670
\(876\) 0 0
\(877\) 1.39445 0.0470872 0.0235436 0.999723i \(-0.492505\pi\)
0.0235436 + 0.999723i \(0.492505\pi\)
\(878\) 0 0
\(879\) −25.8167 −0.870774
\(880\) 0 0
\(881\) 13.1194 0.442005 0.221002 0.975273i \(-0.429067\pi\)
0.221002 + 0.975273i \(0.429067\pi\)
\(882\) 0 0
\(883\) 12.0917 0.406917 0.203459 0.979084i \(-0.434782\pi\)
0.203459 + 0.979084i \(0.434782\pi\)
\(884\) 0 0
\(885\) −24.0000 −0.806751
\(886\) 0 0
\(887\) −55.7805 −1.87293 −0.936463 0.350767i \(-0.885921\pi\)
−0.936463 + 0.350767i \(0.885921\pi\)
\(888\) 0 0
\(889\) 14.5139 0.486780
\(890\) 0 0
\(891\) 42.4222 1.42120
\(892\) 0 0
\(893\) −22.4222 −0.750330
\(894\) 0 0
\(895\) 12.3944 0.414301
\(896\) 0 0
\(897\) 42.4222 1.41644
\(898\) 0 0
\(899\) 67.2666 2.24347
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −9.90833 −0.329728
\(904\) 0 0
\(905\) 7.81665 0.259834
\(906\) 0 0
\(907\) −27.2111 −0.903530 −0.451765 0.892137i \(-0.649205\pi\)
−0.451765 + 0.892137i \(0.649205\pi\)
\(908\) 0 0
\(909\) −29.0278 −0.962790
\(910\) 0 0
\(911\) 15.8167 0.524029 0.262015 0.965064i \(-0.415613\pi\)
0.262015 + 0.965064i \(0.415613\pi\)
\(912\) 0 0
\(913\) −12.8444 −0.425088
\(914\) 0 0
\(915\) −46.5416 −1.53862
\(916\) 0 0
\(917\) 9.21110 0.304177
\(918\) 0 0
\(919\) 26.3583 0.869480 0.434740 0.900556i \(-0.356840\pi\)
0.434740 + 0.900556i \(0.356840\pi\)
\(920\) 0 0
\(921\) 9.21110 0.303516
\(922\) 0 0
\(923\) −15.6333 −0.514577
\(924\) 0 0
\(925\) 32.4222 1.06604
\(926\) 0 0
\(927\) 32.2389 1.05886
\(928\) 0 0
\(929\) −36.0917 −1.18413 −0.592065 0.805890i \(-0.701687\pi\)
−0.592065 + 0.805890i \(0.701687\pi\)
\(930\) 0 0
\(931\) 8.60555 0.282036
\(932\) 0 0
\(933\) 40.8167 1.33628
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −37.2111 −1.21563 −0.607817 0.794077i \(-0.707955\pi\)
−0.607817 + 0.794077i \(0.707955\pi\)
\(938\) 0 0
\(939\) −9.00000 −0.293704
\(940\) 0 0
\(941\) 33.6972 1.09850 0.549249 0.835659i \(-0.314914\pi\)
0.549249 + 0.835659i \(0.314914\pi\)
\(942\) 0 0
\(943\) −46.0555 −1.49977
\(944\) 0 0
\(945\) −2.09167 −0.0680421
\(946\) 0 0
\(947\) −24.8444 −0.807335 −0.403667 0.914906i \(-0.632265\pi\)
−0.403667 + 0.914906i \(0.632265\pi\)
\(948\) 0 0
\(949\) 34.6056 1.12334
\(950\) 0 0
\(951\) 77.4500 2.51149
\(952\) 0 0
\(953\) −21.3583 −0.691863 −0.345931 0.938260i \(-0.612437\pi\)
−0.345931 + 0.938260i \(0.612437\pi\)
\(954\) 0 0
\(955\) −15.2750 −0.494288
\(956\) 0 0
\(957\) 84.8444 2.74263
\(958\) 0 0
\(959\) 18.1194 0.585107
\(960\) 0 0
\(961\) 22.3305 0.720340
\(962\) 0 0
\(963\) 10.6056 0.341759
\(964\) 0 0
\(965\) 19.5778 0.630232
\(966\) 0 0
\(967\) −9.51388 −0.305946 −0.152973 0.988230i \(-0.548885\pi\)
−0.152973 + 0.988230i \(0.548885\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2.78890 0.0895000 0.0447500 0.998998i \(-0.485751\pi\)
0.0447500 + 0.998998i \(0.485751\pi\)
\(972\) 0 0
\(973\) −8.51388 −0.272942
\(974\) 0 0
\(975\) 35.0278 1.12179
\(976\) 0 0
\(977\) −8.69722 −0.278249 −0.139124 0.990275i \(-0.544429\pi\)
−0.139124 + 0.990275i \(0.544429\pi\)
\(978\) 0 0
\(979\) −31.2666 −0.999285
\(980\) 0 0
\(981\) −11.0278 −0.352089
\(982\) 0 0
\(983\) −47.7805 −1.52396 −0.761981 0.647600i \(-0.775773\pi\)
−0.761981 + 0.647600i \(0.775773\pi\)
\(984\) 0 0
\(985\) 8.60555 0.274196
\(986\) 0 0
\(987\) −6.00000 −0.190982
\(988\) 0 0
\(989\) 17.2111 0.547281
\(990\) 0 0
\(991\) 11.2111 0.356132 0.178066 0.984019i \(-0.443016\pi\)
0.178066 + 0.984019i \(0.443016\pi\)
\(992\) 0 0
\(993\) 8.57779 0.272208
\(994\) 0 0
\(995\) −12.6333 −0.400503
\(996\) 0 0
\(997\) 45.6972 1.44725 0.723623 0.690196i \(-0.242476\pi\)
0.723623 + 0.690196i \(0.242476\pi\)
\(998\) 0 0
\(999\) 15.7611 0.498660
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 8092.2.a.l.1.2 2
17.16 even 2 476.2.a.c.1.1 2
51.50 odd 2 4284.2.a.l.1.2 2
68.67 odd 2 1904.2.a.k.1.2 2
119.118 odd 2 3332.2.a.k.1.2 2
136.67 odd 2 7616.2.a.o.1.1 2
136.101 even 2 7616.2.a.t.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
476.2.a.c.1.1 2 17.16 even 2
1904.2.a.k.1.2 2 68.67 odd 2
3332.2.a.k.1.2 2 119.118 odd 2
4284.2.a.l.1.2 2 51.50 odd 2
7616.2.a.o.1.1 2 136.67 odd 2
7616.2.a.t.1.2 2 136.101 even 2
8092.2.a.l.1.2 2 1.1 even 1 trivial