Properties

Label 2200.2.a.o.1.2
Level $2200$
Weight $2$
Character 2200.1
Self dual yes
Analytic conductor $17.567$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2200,2,Mod(1,2200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2200.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: \(17.5670884447\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) 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 88)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 2200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.56155 q^{3} -3.12311 q^{7} -0.561553 q^{9} +O(q^{10})\) \(q+1.56155 q^{3} -3.12311 q^{7} -0.561553 q^{9} -1.00000 q^{11} +5.12311 q^{13} -2.00000 q^{17} -4.00000 q^{19} -4.87689 q^{21} -2.43845 q^{23} -5.56155 q^{27} -5.12311 q^{29} -5.56155 q^{31} -1.56155 q^{33} +7.56155 q^{37} +8.00000 q^{39} -1.12311 q^{41} +7.12311 q^{43} -8.00000 q^{47} +2.75379 q^{49} -3.12311 q^{51} -12.2462 q^{53} -6.24621 q^{57} +7.80776 q^{59} +1.12311 q^{61} +1.75379 q^{63} -9.56155 q^{67} -3.80776 q^{69} -8.68466 q^{71} -5.12311 q^{73} +3.12311 q^{77} -11.1231 q^{79} -7.00000 q^{81} -0.876894 q^{83} -8.00000 q^{87} +2.68466 q^{89} -16.0000 q^{91} -8.68466 q^{93} -15.5616 q^{97} +0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 4 q^{17} - 8 q^{19} - 18 q^{21} - 9 q^{23} - 7 q^{27} - 2 q^{29} - 7 q^{31} + q^{33} + 11 q^{37} + 16 q^{39} + 6 q^{41} + 6 q^{43} - 16 q^{47} + 22 q^{49} + 2 q^{51} - 8 q^{53} + 4 q^{57} - 5 q^{59} - 6 q^{61} + 20 q^{63} - 15 q^{67} + 13 q^{69} - 5 q^{71} - 2 q^{73} - 2 q^{77} - 14 q^{79} - 14 q^{81} - 10 q^{83} - 16 q^{87} - 7 q^{89} - 32 q^{91} - 5 q^{93} - 27 q^{97} - 3 q^{99}+O(q^{100}) \) 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
\(3\) 1.56155 0.901563 0.450781 0.892634i \(-0.351145\pi\)
0.450781 + 0.892634i \(0.351145\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.12311 −1.18042 −0.590211 0.807249i \(-0.700956\pi\)
−0.590211 + 0.807249i \(0.700956\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 5.12311 1.42089 0.710447 0.703751i \(-0.248493\pi\)
0.710447 + 0.703751i \(0.248493\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −4.87689 −1.06423
\(22\) 0 0
\(23\) −2.43845 −0.508451 −0.254226 0.967145i \(-0.581821\pi\)
−0.254226 + 0.967145i \(0.581821\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.56155 −1.07032
\(28\) 0 0
\(29\) −5.12311 −0.951337 −0.475668 0.879625i \(-0.657794\pi\)
−0.475668 + 0.879625i \(0.657794\pi\)
\(30\) 0 0
\(31\) −5.56155 −0.998884 −0.499442 0.866347i \(-0.666462\pi\)
−0.499442 + 0.866347i \(0.666462\pi\)
\(32\) 0 0
\(33\) −1.56155 −0.271831
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.56155 1.24311 0.621556 0.783370i \(-0.286501\pi\)
0.621556 + 0.783370i \(0.286501\pi\)
\(38\) 0 0
\(39\) 8.00000 1.28103
\(40\) 0 0
\(41\) −1.12311 −0.175400 −0.0876998 0.996147i \(-0.527952\pi\)
−0.0876998 + 0.996147i \(0.527952\pi\)
\(42\) 0 0
\(43\) 7.12311 1.08626 0.543132 0.839648i \(-0.317238\pi\)
0.543132 + 0.839648i \(0.317238\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 2.75379 0.393398
\(50\) 0 0
\(51\) −3.12311 −0.437322
\(52\) 0 0
\(53\) −12.2462 −1.68215 −0.841073 0.540921i \(-0.818076\pi\)
−0.841073 + 0.540921i \(0.818076\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.24621 −0.827331
\(58\) 0 0
\(59\) 7.80776 1.01648 0.508242 0.861214i \(-0.330295\pi\)
0.508242 + 0.861214i \(0.330295\pi\)
\(60\) 0 0
\(61\) 1.12311 0.143799 0.0718995 0.997412i \(-0.477094\pi\)
0.0718995 + 0.997412i \(0.477094\pi\)
\(62\) 0 0
\(63\) 1.75379 0.220957
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −9.56155 −1.16813 −0.584065 0.811707i \(-0.698539\pi\)
−0.584065 + 0.811707i \(0.698539\pi\)
\(68\) 0 0
\(69\) −3.80776 −0.458401
\(70\) 0 0
\(71\) −8.68466 −1.03068 −0.515340 0.856986i \(-0.672334\pi\)
−0.515340 + 0.856986i \(0.672334\pi\)
\(72\) 0 0
\(73\) −5.12311 −0.599614 −0.299807 0.954000i \(-0.596922\pi\)
−0.299807 + 0.954000i \(0.596922\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.12311 0.355911
\(78\) 0 0
\(79\) −11.1231 −1.25145 −0.625724 0.780045i \(-0.715196\pi\)
−0.625724 + 0.780045i \(0.715196\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −0.876894 −0.0962517 −0.0481258 0.998841i \(-0.515325\pi\)
−0.0481258 + 0.998841i \(0.515325\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.00000 −0.857690
\(88\) 0 0
\(89\) 2.68466 0.284573 0.142287 0.989825i \(-0.454555\pi\)
0.142287 + 0.989825i \(0.454555\pi\)
\(90\) 0 0
\(91\) −16.0000 −1.67726
\(92\) 0 0
\(93\) −8.68466 −0.900557
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −15.5616 −1.58004 −0.790018 0.613083i \(-0.789929\pi\)
−0.790018 + 0.613083i \(0.789929\pi\)
\(98\) 0 0
\(99\) 0.561553 0.0564382
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.3693 1.29246 0.646230 0.763142i \(-0.276345\pi\)
0.646230 + 0.763142i \(0.276345\pi\)
\(108\) 0 0
\(109\) 12.2462 1.17297 0.586487 0.809959i \(-0.300510\pi\)
0.586487 + 0.809959i \(0.300510\pi\)
\(110\) 0 0
\(111\) 11.8078 1.12074
\(112\) 0 0
\(113\) 0.438447 0.0412456 0.0206228 0.999787i \(-0.493435\pi\)
0.0206228 + 0.999787i \(0.493435\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.87689 −0.265969
\(118\) 0 0
\(119\) 6.24621 0.572589
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −1.75379 −0.158134
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −6.24621 −0.554262 −0.277131 0.960832i \(-0.589384\pi\)
−0.277131 + 0.960832i \(0.589384\pi\)
\(128\) 0 0
\(129\) 11.1231 0.979335
\(130\) 0 0
\(131\) 13.3693 1.16808 0.584041 0.811724i \(-0.301471\pi\)
0.584041 + 0.811724i \(0.301471\pi\)
\(132\) 0 0
\(133\) 12.4924 1.08323
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.43845 0.720945 0.360473 0.932770i \(-0.382615\pi\)
0.360473 + 0.932770i \(0.382615\pi\)
\(138\) 0 0
\(139\) 15.1231 1.28273 0.641363 0.767238i \(-0.278369\pi\)
0.641363 + 0.767238i \(0.278369\pi\)
\(140\) 0 0
\(141\) −12.4924 −1.05205
\(142\) 0 0
\(143\) −5.12311 −0.428416
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.30019 0.354673
\(148\) 0 0
\(149\) 4.24621 0.347863 0.173932 0.984758i \(-0.444353\pi\)
0.173932 + 0.984758i \(0.444353\pi\)
\(150\) 0 0
\(151\) −9.36932 −0.762464 −0.381232 0.924479i \(-0.624500\pi\)
−0.381232 + 0.924479i \(0.624500\pi\)
\(152\) 0 0
\(153\) 1.12311 0.0907977
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.43845 0.354227 0.177113 0.984190i \(-0.443324\pi\)
0.177113 + 0.984190i \(0.443324\pi\)
\(158\) 0 0
\(159\) −19.1231 −1.51656
\(160\) 0 0
\(161\) 7.61553 0.600188
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 13.2462 1.01894
\(170\) 0 0
\(171\) 2.24621 0.171772
\(172\) 0 0
\(173\) −12.2462 −0.931062 −0.465531 0.885032i \(-0.654137\pi\)
−0.465531 + 0.885032i \(0.654137\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 12.1922 0.916425
\(178\) 0 0
\(179\) −6.43845 −0.481232 −0.240616 0.970620i \(-0.577349\pi\)
−0.240616 + 0.970620i \(0.577349\pi\)
\(180\) 0 0
\(181\) −1.31534 −0.0977686 −0.0488843 0.998804i \(-0.515567\pi\)
−0.0488843 + 0.998804i \(0.515567\pi\)
\(182\) 0 0
\(183\) 1.75379 0.129644
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) 0 0
\(189\) 17.3693 1.26343
\(190\) 0 0
\(191\) −10.4384 −0.755300 −0.377650 0.925949i \(-0.623268\pi\)
−0.377650 + 0.925949i \(0.623268\pi\)
\(192\) 0 0
\(193\) 9.12311 0.656696 0.328348 0.944557i \(-0.393508\pi\)
0.328348 + 0.944557i \(0.393508\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.4924 1.03254 0.516271 0.856425i \(-0.327320\pi\)
0.516271 + 0.856425i \(0.327320\pi\)
\(198\) 0 0
\(199\) −12.4924 −0.885564 −0.442782 0.896629i \(-0.646009\pi\)
−0.442782 + 0.896629i \(0.646009\pi\)
\(200\) 0 0
\(201\) −14.9309 −1.05314
\(202\) 0 0
\(203\) 16.0000 1.12298
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.36932 0.0951741
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 8.49242 0.584642 0.292321 0.956320i \(-0.405572\pi\)
0.292321 + 0.956320i \(0.405572\pi\)
\(212\) 0 0
\(213\) −13.5616 −0.929222
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 17.3693 1.17911
\(218\) 0 0
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) −10.2462 −0.689235
\(222\) 0 0
\(223\) 11.8078 0.790706 0.395353 0.918529i \(-0.370622\pi\)
0.395353 + 0.918529i \(0.370622\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −23.1231 −1.53473 −0.767367 0.641208i \(-0.778434\pi\)
−0.767367 + 0.641208i \(0.778434\pi\)
\(228\) 0 0
\(229\) 14.6847 0.970390 0.485195 0.874406i \(-0.338749\pi\)
0.485195 + 0.874406i \(0.338749\pi\)
\(230\) 0 0
\(231\) 4.87689 0.320876
\(232\) 0 0
\(233\) 7.36932 0.482780 0.241390 0.970428i \(-0.422397\pi\)
0.241390 + 0.970428i \(0.422397\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −17.3693 −1.12826
\(238\) 0 0
\(239\) −4.87689 −0.315460 −0.157730 0.987482i \(-0.550418\pi\)
−0.157730 + 0.987482i \(0.550418\pi\)
\(240\) 0 0
\(241\) 29.1231 1.87598 0.937992 0.346657i \(-0.112683\pi\)
0.937992 + 0.346657i \(0.112683\pi\)
\(242\) 0 0
\(243\) 5.75379 0.369106
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −20.4924 −1.30390
\(248\) 0 0
\(249\) −1.36932 −0.0867769
\(250\) 0 0
\(251\) 1.56155 0.0985643 0.0492822 0.998785i \(-0.484307\pi\)
0.0492822 + 0.998785i \(0.484307\pi\)
\(252\) 0 0
\(253\) 2.43845 0.153304
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −11.7538 −0.733181 −0.366591 0.930382i \(-0.619475\pi\)
−0.366591 + 0.930382i \(0.619475\pi\)
\(258\) 0 0
\(259\) −23.6155 −1.46740
\(260\) 0 0
\(261\) 2.87689 0.178075
\(262\) 0 0
\(263\) −19.1231 −1.17918 −0.589591 0.807702i \(-0.700711\pi\)
−0.589591 + 0.807702i \(0.700711\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 4.19224 0.256561
\(268\) 0 0
\(269\) −20.7386 −1.26446 −0.632228 0.774782i \(-0.717860\pi\)
−0.632228 + 0.774782i \(0.717860\pi\)
\(270\) 0 0
\(271\) −28.4924 −1.73079 −0.865396 0.501089i \(-0.832933\pi\)
−0.865396 + 0.501089i \(0.832933\pi\)
\(272\) 0 0
\(273\) −24.9848 −1.51215
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 18.0000 1.08152 0.540758 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) 0 0
\(279\) 3.12311 0.186975
\(280\) 0 0
\(281\) 16.2462 0.969168 0.484584 0.874745i \(-0.338971\pi\)
0.484584 + 0.874745i \(0.338971\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.50758 0.207046
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −24.3002 −1.42450
\(292\) 0 0
\(293\) 3.36932 0.196838 0.0984188 0.995145i \(-0.468622\pi\)
0.0984188 + 0.995145i \(0.468622\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.56155 0.322714
\(298\) 0 0
\(299\) −12.4924 −0.722455
\(300\) 0 0
\(301\) −22.2462 −1.28225
\(302\) 0 0
\(303\) −3.12311 −0.179418
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 32.4924 1.85444 0.927220 0.374516i \(-0.122191\pi\)
0.927220 + 0.374516i \(0.122191\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.75379 0.553087 0.276543 0.961001i \(-0.410811\pi\)
0.276543 + 0.961001i \(0.410811\pi\)
\(312\) 0 0
\(313\) 9.80776 0.554368 0.277184 0.960817i \(-0.410599\pi\)
0.277184 + 0.960817i \(0.410599\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.1922 0.797115 0.398558 0.917143i \(-0.369511\pi\)
0.398558 + 0.917143i \(0.369511\pi\)
\(318\) 0 0
\(319\) 5.12311 0.286839
\(320\) 0 0
\(321\) 20.8769 1.16523
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 19.1231 1.05751
\(328\) 0 0
\(329\) 24.9848 1.37746
\(330\) 0 0
\(331\) 34.9309 1.91997 0.959987 0.280044i \(-0.0903491\pi\)
0.959987 + 0.280044i \(0.0903491\pi\)
\(332\) 0 0
\(333\) −4.24621 −0.232691
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 16.7386 0.911811 0.455906 0.890028i \(-0.349315\pi\)
0.455906 + 0.890028i \(0.349315\pi\)
\(338\) 0 0
\(339\) 0.684658 0.0371855
\(340\) 0 0
\(341\) 5.56155 0.301175
\(342\) 0 0
\(343\) 13.2614 0.716046
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −22.7386 −1.22067 −0.610337 0.792142i \(-0.708966\pi\)
−0.610337 + 0.792142i \(0.708966\pi\)
\(348\) 0 0
\(349\) −32.2462 −1.72610 −0.863050 0.505118i \(-0.831449\pi\)
−0.863050 + 0.505118i \(0.831449\pi\)
\(350\) 0 0
\(351\) −28.4924 −1.52081
\(352\) 0 0
\(353\) 24.0540 1.28026 0.640132 0.768265i \(-0.278880\pi\)
0.640132 + 0.768265i \(0.278880\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 9.75379 0.516225
\(358\) 0 0
\(359\) 4.49242 0.237101 0.118550 0.992948i \(-0.462175\pi\)
0.118550 + 0.992948i \(0.462175\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 1.56155 0.0819603
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −22.9309 −1.19698 −0.598491 0.801130i \(-0.704233\pi\)
−0.598491 + 0.801130i \(0.704233\pi\)
\(368\) 0 0
\(369\) 0.630683 0.0328321
\(370\) 0 0
\(371\) 38.2462 1.98564
\(372\) 0 0
\(373\) 8.24621 0.426973 0.213486 0.976946i \(-0.431518\pi\)
0.213486 + 0.976946i \(0.431518\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −26.2462 −1.35175
\(378\) 0 0
\(379\) 0.192236 0.00987450 0.00493725 0.999988i \(-0.498428\pi\)
0.00493725 + 0.999988i \(0.498428\pi\)
\(380\) 0 0
\(381\) −9.75379 −0.499702
\(382\) 0 0
\(383\) 2.05398 0.104953 0.0524766 0.998622i \(-0.483288\pi\)
0.0524766 + 0.998622i \(0.483288\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) 3.56155 0.180578 0.0902889 0.995916i \(-0.471221\pi\)
0.0902889 + 0.995916i \(0.471221\pi\)
\(390\) 0 0
\(391\) 4.87689 0.246635
\(392\) 0 0
\(393\) 20.8769 1.05310
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −10.4924 −0.526600 −0.263300 0.964714i \(-0.584811\pi\)
−0.263300 + 0.964714i \(0.584811\pi\)
\(398\) 0 0
\(399\) 19.5076 0.976600
\(400\) 0 0
\(401\) 30.4924 1.52272 0.761359 0.648330i \(-0.224532\pi\)
0.761359 + 0.648330i \(0.224532\pi\)
\(402\) 0 0
\(403\) −28.4924 −1.41931
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −7.56155 −0.374812
\(408\) 0 0
\(409\) 22.4924 1.11218 0.556089 0.831123i \(-0.312301\pi\)
0.556089 + 0.831123i \(0.312301\pi\)
\(410\) 0 0
\(411\) 13.1771 0.649977
\(412\) 0 0
\(413\) −24.3845 −1.19988
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 23.6155 1.15646
\(418\) 0 0
\(419\) −32.4924 −1.58736 −0.793679 0.608336i \(-0.791837\pi\)
−0.793679 + 0.608336i \(0.791837\pi\)
\(420\) 0 0
\(421\) 2.49242 0.121473 0.0607366 0.998154i \(-0.480655\pi\)
0.0607366 + 0.998154i \(0.480655\pi\)
\(422\) 0 0
\(423\) 4.49242 0.218429
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −3.50758 −0.169744
\(428\) 0 0
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) 27.1231 1.30647 0.653237 0.757153i \(-0.273411\pi\)
0.653237 + 0.757153i \(0.273411\pi\)
\(432\) 0 0
\(433\) 22.6847 1.09016 0.545078 0.838386i \(-0.316500\pi\)
0.545078 + 0.838386i \(0.316500\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 9.75379 0.466587
\(438\) 0 0
\(439\) 4.49242 0.214412 0.107206 0.994237i \(-0.465810\pi\)
0.107206 + 0.994237i \(0.465810\pi\)
\(440\) 0 0
\(441\) −1.54640 −0.0736380
\(442\) 0 0
\(443\) −11.3153 −0.537608 −0.268804 0.963195i \(-0.586628\pi\)
−0.268804 + 0.963195i \(0.586628\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.63068 0.313621
\(448\) 0 0
\(449\) −36.5464 −1.72473 −0.862366 0.506286i \(-0.831018\pi\)
−0.862366 + 0.506286i \(0.831018\pi\)
\(450\) 0 0
\(451\) 1.12311 0.0528850
\(452\) 0 0
\(453\) −14.6307 −0.687409
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −23.8617 −1.11621 −0.558103 0.829772i \(-0.688470\pi\)
−0.558103 + 0.829772i \(0.688470\pi\)
\(458\) 0 0
\(459\) 11.1231 0.519182
\(460\) 0 0
\(461\) 1.12311 0.0523082 0.0261541 0.999658i \(-0.491674\pi\)
0.0261541 + 0.999658i \(0.491674\pi\)
\(462\) 0 0
\(463\) −15.3153 −0.711764 −0.355882 0.934531i \(-0.615820\pi\)
−0.355882 + 0.934531i \(0.615820\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −28.3002 −1.30958 −0.654788 0.755812i \(-0.727242\pi\)
−0.654788 + 0.755812i \(0.727242\pi\)
\(468\) 0 0
\(469\) 29.8617 1.37889
\(470\) 0 0
\(471\) 6.93087 0.319358
\(472\) 0 0
\(473\) −7.12311 −0.327521
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.87689 0.314871
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) 38.7386 1.76633
\(482\) 0 0
\(483\) 11.8920 0.541107
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −14.9309 −0.676582 −0.338291 0.941041i \(-0.609849\pi\)
−0.338291 + 0.941041i \(0.609849\pi\)
\(488\) 0 0
\(489\) 6.24621 0.282463
\(490\) 0 0
\(491\) 13.7538 0.620700 0.310350 0.950622i \(-0.399554\pi\)
0.310350 + 0.950622i \(0.399554\pi\)
\(492\) 0 0
\(493\) 10.2462 0.461466
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 27.1231 1.21664
\(498\) 0 0
\(499\) −28.9848 −1.29754 −0.648770 0.760985i \(-0.724716\pi\)
−0.648770 + 0.760985i \(0.724716\pi\)
\(500\) 0 0
\(501\) −12.4924 −0.558120
\(502\) 0 0
\(503\) 31.6155 1.40967 0.704833 0.709373i \(-0.251022\pi\)
0.704833 + 0.709373i \(0.251022\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 20.6847 0.918638
\(508\) 0 0
\(509\) −18.3002 −0.811142 −0.405571 0.914064i \(-0.632927\pi\)
−0.405571 + 0.914064i \(0.632927\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 0 0
\(513\) 22.2462 0.982194
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) −19.1231 −0.839411
\(520\) 0 0
\(521\) 1.31534 0.0576262 0.0288131 0.999585i \(-0.490827\pi\)
0.0288131 + 0.999585i \(0.490827\pi\)
\(522\) 0 0
\(523\) 12.0000 0.524723 0.262362 0.964970i \(-0.415499\pi\)
0.262362 + 0.964970i \(0.415499\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.1231 0.484530
\(528\) 0 0
\(529\) −17.0540 −0.741477
\(530\) 0 0
\(531\) −4.38447 −0.190270
\(532\) 0 0
\(533\) −5.75379 −0.249224
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −10.0540 −0.433861
\(538\) 0 0
\(539\) −2.75379 −0.118614
\(540\) 0 0
\(541\) −23.8617 −1.02590 −0.512948 0.858420i \(-0.671447\pi\)
−0.512948 + 0.858420i \(0.671447\pi\)
\(542\) 0 0
\(543\) −2.05398 −0.0881445
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 42.2462 1.80632 0.903159 0.429307i \(-0.141242\pi\)
0.903159 + 0.429307i \(0.141242\pi\)
\(548\) 0 0
\(549\) −0.630683 −0.0269169
\(550\) 0 0
\(551\) 20.4924 0.873007
\(552\) 0 0
\(553\) 34.7386 1.47724
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.75379 0.159053 0.0795266 0.996833i \(-0.474659\pi\)
0.0795266 + 0.996833i \(0.474659\pi\)
\(558\) 0 0
\(559\) 36.4924 1.54347
\(560\) 0 0
\(561\) 3.12311 0.131858
\(562\) 0 0
\(563\) −24.4924 −1.03223 −0.516116 0.856519i \(-0.672623\pi\)
−0.516116 + 0.856519i \(0.672623\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 21.8617 0.918107
\(568\) 0 0
\(569\) −26.8769 −1.12674 −0.563369 0.826205i \(-0.690495\pi\)
−0.563369 + 0.826205i \(0.690495\pi\)
\(570\) 0 0
\(571\) 16.4924 0.690186 0.345093 0.938568i \(-0.387847\pi\)
0.345093 + 0.938568i \(0.387847\pi\)
\(572\) 0 0
\(573\) −16.3002 −0.680950
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −15.5616 −0.647836 −0.323918 0.946085i \(-0.605000\pi\)
−0.323918 + 0.946085i \(0.605000\pi\)
\(578\) 0 0
\(579\) 14.2462 0.592052
\(580\) 0 0
\(581\) 2.73863 0.113618
\(582\) 0 0
\(583\) 12.2462 0.507186
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 24.4924 1.01091 0.505455 0.862853i \(-0.331325\pi\)
0.505455 + 0.862853i \(0.331325\pi\)
\(588\) 0 0
\(589\) 22.2462 0.916639
\(590\) 0 0
\(591\) 22.6307 0.930902
\(592\) 0 0
\(593\) −3.36932 −0.138361 −0.0691806 0.997604i \(-0.522038\pi\)
−0.0691806 + 0.997604i \(0.522038\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −19.5076 −0.798392
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) 3.75379 0.153120 0.0765601 0.997065i \(-0.475606\pi\)
0.0765601 + 0.997065i \(0.475606\pi\)
\(602\) 0 0
\(603\) 5.36932 0.218655
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 45.8617 1.86147 0.930735 0.365694i \(-0.119168\pi\)
0.930735 + 0.365694i \(0.119168\pi\)
\(608\) 0 0
\(609\) 24.9848 1.01244
\(610\) 0 0
\(611\) −40.9848 −1.65807
\(612\) 0 0
\(613\) −11.8617 −0.479091 −0.239546 0.970885i \(-0.576998\pi\)
−0.239546 + 0.970885i \(0.576998\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.49242 0.100341 0.0501706 0.998741i \(-0.484024\pi\)
0.0501706 + 0.998741i \(0.484024\pi\)
\(618\) 0 0
\(619\) 18.9309 0.760896 0.380448 0.924802i \(-0.375770\pi\)
0.380448 + 0.924802i \(0.375770\pi\)
\(620\) 0 0
\(621\) 13.5616 0.544206
\(622\) 0 0
\(623\) −8.38447 −0.335917
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 6.24621 0.249450
\(628\) 0 0
\(629\) −15.1231 −0.602998
\(630\) 0 0
\(631\) −42.0540 −1.67414 −0.837071 0.547094i \(-0.815734\pi\)
−0.837071 + 0.547094i \(0.815734\pi\)
\(632\) 0 0
\(633\) 13.2614 0.527092
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 14.1080 0.558977
\(638\) 0 0
\(639\) 4.87689 0.192927
\(640\) 0 0
\(641\) −46.3002 −1.82875 −0.914374 0.404871i \(-0.867316\pi\)
−0.914374 + 0.404871i \(0.867316\pi\)
\(642\) 0 0
\(643\) 9.17708 0.361909 0.180954 0.983491i \(-0.442081\pi\)
0.180954 + 0.983491i \(0.442081\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −13.5616 −0.533160 −0.266580 0.963813i \(-0.585894\pi\)
−0.266580 + 0.963813i \(0.585894\pi\)
\(648\) 0 0
\(649\) −7.80776 −0.306482
\(650\) 0 0
\(651\) 27.1231 1.06304
\(652\) 0 0
\(653\) −35.1771 −1.37659 −0.688293 0.725433i \(-0.741640\pi\)
−0.688293 + 0.725433i \(0.741640\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.87689 0.112238
\(658\) 0 0
\(659\) −11.6155 −0.452477 −0.226238 0.974072i \(-0.572643\pi\)
−0.226238 + 0.974072i \(0.572643\pi\)
\(660\) 0 0
\(661\) 41.8078 1.62613 0.813067 0.582170i \(-0.197796\pi\)
0.813067 + 0.582170i \(0.197796\pi\)
\(662\) 0 0
\(663\) −16.0000 −0.621389
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12.4924 0.483709
\(668\) 0 0
\(669\) 18.4384 0.712872
\(670\) 0 0
\(671\) −1.12311 −0.0433570
\(672\) 0 0
\(673\) −33.2311 −1.28096 −0.640482 0.767974i \(-0.721265\pi\)
−0.640482 + 0.767974i \(0.721265\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 20.7386 0.797050 0.398525 0.917157i \(-0.369522\pi\)
0.398525 + 0.917157i \(0.369522\pi\)
\(678\) 0 0
\(679\) 48.6004 1.86511
\(680\) 0 0
\(681\) −36.1080 −1.38366
\(682\) 0 0
\(683\) −6.73863 −0.257847 −0.128923 0.991655i \(-0.541152\pi\)
−0.128923 + 0.991655i \(0.541152\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 22.9309 0.874867
\(688\) 0 0
\(689\) −62.7386 −2.39015
\(690\) 0 0
\(691\) −9.94602 −0.378365 −0.189182 0.981942i \(-0.560584\pi\)
−0.189182 + 0.981942i \(0.560584\pi\)
\(692\) 0 0
\(693\) −1.75379 −0.0666209
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.24621 0.0850813
\(698\) 0 0
\(699\) 11.5076 0.435257
\(700\) 0 0
\(701\) 50.4924 1.90707 0.953536 0.301278i \(-0.0974133\pi\)
0.953536 + 0.301278i \(0.0974133\pi\)
\(702\) 0 0
\(703\) −30.2462 −1.14076
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.24621 0.234913
\(708\) 0 0
\(709\) 2.19224 0.0823311 0.0411656 0.999152i \(-0.486893\pi\)
0.0411656 + 0.999152i \(0.486893\pi\)
\(710\) 0 0
\(711\) 6.24621 0.234251
\(712\) 0 0
\(713\) 13.5616 0.507884
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −7.61553 −0.284407
\(718\) 0 0
\(719\) 35.4233 1.32107 0.660533 0.750797i \(-0.270330\pi\)
0.660533 + 0.750797i \(0.270330\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 45.4773 1.69132
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 23.3153 0.864718 0.432359 0.901702i \(-0.357681\pi\)
0.432359 + 0.901702i \(0.357681\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) −14.2462 −0.526915
\(732\) 0 0
\(733\) −1.12311 −0.0414829 −0.0207414 0.999785i \(-0.506603\pi\)
−0.0207414 + 0.999785i \(0.506603\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.56155 0.352204
\(738\) 0 0
\(739\) −2.63068 −0.0967712 −0.0483856 0.998829i \(-0.515408\pi\)
−0.0483856 + 0.998829i \(0.515408\pi\)
\(740\) 0 0
\(741\) −32.0000 −1.17555
\(742\) 0 0
\(743\) −10.7386 −0.393962 −0.196981 0.980407i \(-0.563114\pi\)
−0.196981 + 0.980407i \(0.563114\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0.492423 0.0180168
\(748\) 0 0
\(749\) −41.7538 −1.52565
\(750\) 0 0
\(751\) 5.56155 0.202944 0.101472 0.994838i \(-0.467645\pi\)
0.101472 + 0.994838i \(0.467645\pi\)
\(752\) 0 0
\(753\) 2.43845 0.0888620
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −15.7538 −0.572581 −0.286291 0.958143i \(-0.592422\pi\)
−0.286291 + 0.958143i \(0.592422\pi\)
\(758\) 0 0
\(759\) 3.80776 0.138213
\(760\) 0 0
\(761\) 5.12311 0.185712 0.0928562 0.995680i \(-0.470400\pi\)
0.0928562 + 0.995680i \(0.470400\pi\)
\(762\) 0 0
\(763\) −38.2462 −1.38461
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 40.0000 1.44432
\(768\) 0 0
\(769\) 25.6155 0.923720 0.461860 0.886953i \(-0.347182\pi\)
0.461860 + 0.886953i \(0.347182\pi\)
\(770\) 0 0
\(771\) −18.3542 −0.661009
\(772\) 0 0
\(773\) −40.7386 −1.46527 −0.732633 0.680623i \(-0.761709\pi\)
−0.732633 + 0.680623i \(0.761709\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −36.8769 −1.32295
\(778\) 0 0
\(779\) 4.49242 0.160958
\(780\) 0 0
\(781\) 8.68466 0.310762
\(782\) 0 0
\(783\) 28.4924 1.01824
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 29.7538 1.06061 0.530304 0.847808i \(-0.322078\pi\)
0.530304 + 0.847808i \(0.322078\pi\)
\(788\) 0 0
\(789\) −29.8617 −1.06311
\(790\) 0 0
\(791\) −1.36932 −0.0486873
\(792\) 0 0
\(793\) 5.75379 0.204323
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 14.1922 0.502715 0.251357 0.967894i \(-0.419123\pi\)
0.251357 + 0.967894i \(0.419123\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −1.50758 −0.0532676
\(802\) 0 0
\(803\) 5.12311 0.180790
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −32.3845 −1.13999
\(808\) 0 0
\(809\) −45.6155 −1.60376 −0.801878 0.597487i \(-0.796166\pi\)
−0.801878 + 0.597487i \(0.796166\pi\)
\(810\) 0 0
\(811\) −7.12311 −0.250126 −0.125063 0.992149i \(-0.539913\pi\)
−0.125063 + 0.992149i \(0.539913\pi\)
\(812\) 0 0
\(813\) −44.4924 −1.56042
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −28.4924 −0.996824
\(818\) 0 0
\(819\) 8.98485 0.313956
\(820\) 0 0
\(821\) −42.9848 −1.50018 −0.750091 0.661335i \(-0.769990\pi\)
−0.750091 + 0.661335i \(0.769990\pi\)
\(822\) 0 0
\(823\) 54.5464 1.90137 0.950684 0.310161i \(-0.100383\pi\)
0.950684 + 0.310161i \(0.100383\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −38.7386 −1.34707 −0.673537 0.739153i \(-0.735226\pi\)
−0.673537 + 0.739153i \(0.735226\pi\)
\(828\) 0 0
\(829\) 15.0691 0.523373 0.261686 0.965153i \(-0.415721\pi\)
0.261686 + 0.965153i \(0.415721\pi\)
\(830\) 0 0
\(831\) 28.1080 0.975054
\(832\) 0 0
\(833\) −5.50758 −0.190826
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 30.9309 1.06913
\(838\) 0 0
\(839\) −19.8078 −0.683840 −0.341920 0.939729i \(-0.611077\pi\)
−0.341920 + 0.939729i \(0.611077\pi\)
\(840\) 0 0
\(841\) −2.75379 −0.0949582
\(842\) 0 0
\(843\) 25.3693 0.873766
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −3.12311 −0.107311
\(848\) 0 0
\(849\) −31.2311 −1.07185
\(850\) 0 0
\(851\) −18.4384 −0.632062
\(852\) 0 0
\(853\) 46.4924 1.59187 0.795935 0.605382i \(-0.206980\pi\)
0.795935 + 0.605382i \(0.206980\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −30.1080 −1.02847 −0.514234 0.857650i \(-0.671924\pi\)
−0.514234 + 0.857650i \(0.671924\pi\)
\(858\) 0 0
\(859\) 30.0540 1.02543 0.512714 0.858559i \(-0.328640\pi\)
0.512714 + 0.858559i \(0.328640\pi\)
\(860\) 0 0
\(861\) 5.47727 0.186665
\(862\) 0 0
\(863\) 36.4924 1.24222 0.621108 0.783725i \(-0.286683\pi\)
0.621108 + 0.783725i \(0.286683\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −20.3002 −0.689430
\(868\) 0 0
\(869\) 11.1231 0.377326
\(870\) 0 0
\(871\) −48.9848 −1.65979
\(872\) 0 0
\(873\) 8.73863 0.295758
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −55.3693 −1.86969 −0.934844 0.355057i \(-0.884461\pi\)
−0.934844 + 0.355057i \(0.884461\pi\)
\(878\) 0 0
\(879\) 5.26137 0.177461
\(880\) 0 0
\(881\) 34.3002 1.15560 0.577801 0.816177i \(-0.303911\pi\)
0.577801 + 0.816177i \(0.303911\pi\)
\(882\) 0 0
\(883\) −8.49242 −0.285793 −0.142896 0.989738i \(-0.545642\pi\)
−0.142896 + 0.989738i \(0.545642\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −31.6155 −1.06155 −0.530773 0.847514i \(-0.678098\pi\)
−0.530773 + 0.847514i \(0.678098\pi\)
\(888\) 0 0
\(889\) 19.5076 0.654263
\(890\) 0 0
\(891\) 7.00000 0.234509
\(892\) 0 0
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −19.5076 −0.651339
\(898\) 0 0
\(899\) 28.4924 0.950275
\(900\) 0 0
\(901\) 24.4924 0.815961
\(902\) 0 0
\(903\) −34.7386 −1.15603
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −16.4924 −0.547622 −0.273811 0.961784i \(-0.588284\pi\)
−0.273811 + 0.961784i \(0.588284\pi\)
\(908\) 0 0
\(909\) 1.12311 0.0372511
\(910\) 0 0
\(911\) −26.7386 −0.885890 −0.442945 0.896549i \(-0.646066\pi\)
−0.442945 + 0.896549i \(0.646066\pi\)
\(912\) 0 0
\(913\) 0.876894 0.0290210
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −41.7538 −1.37883
\(918\) 0 0
\(919\) 6.63068 0.218726 0.109363 0.994002i \(-0.465119\pi\)
0.109363 + 0.994002i \(0.465119\pi\)
\(920\) 0 0
\(921\) 50.7386 1.67189
\(922\) 0 0
\(923\) −44.4924 −1.46449
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 46.4924 1.52537 0.762683 0.646772i \(-0.223881\pi\)
0.762683 + 0.646772i \(0.223881\pi\)
\(930\) 0 0
\(931\) −11.0152 −0.361007
\(932\) 0 0
\(933\) 15.2311 0.498642
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 42.1080 1.37561 0.687803 0.725897i \(-0.258575\pi\)
0.687803 + 0.725897i \(0.258575\pi\)
\(938\) 0 0
\(939\) 15.3153 0.499797
\(940\) 0 0
\(941\) −32.2462 −1.05120 −0.525598 0.850733i \(-0.676158\pi\)
−0.525598 + 0.850733i \(0.676158\pi\)
\(942\) 0 0
\(943\) 2.73863 0.0891822
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.6847 0.412196 0.206098 0.978531i \(-0.433923\pi\)
0.206098 + 0.978531i \(0.433923\pi\)
\(948\) 0 0
\(949\) −26.2462 −0.851988
\(950\) 0 0
\(951\) 22.1619 0.718650
\(952\) 0 0
\(953\) −0.246211 −0.00797556 −0.00398778 0.999992i \(-0.501269\pi\)
−0.00398778 + 0.999992i \(0.501269\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 8.00000 0.258603
\(958\) 0 0
\(959\) −26.3542 −0.851020
\(960\) 0 0
\(961\) −0.0691303 −0.00223001
\(962\) 0 0
\(963\) −7.50758 −0.241928
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) 12.4924 0.401314
\(970\) 0 0
\(971\) −34.5464 −1.10865 −0.554323 0.832301i \(-0.687023\pi\)
−0.554323 + 0.832301i \(0.687023\pi\)
\(972\) 0 0
\(973\) −47.2311 −1.51416
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −53.8078 −1.72146 −0.860731 0.509059i \(-0.829993\pi\)
−0.860731 + 0.509059i \(0.829993\pi\)
\(978\) 0 0
\(979\) −2.68466 −0.0858021
\(980\) 0 0
\(981\) −6.87689 −0.219562
\(982\) 0 0
\(983\) −30.9309 −0.986542 −0.493271 0.869876i \(-0.664199\pi\)
−0.493271 + 0.869876i \(0.664199\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 39.0152 1.24187
\(988\) 0 0
\(989\) −17.3693 −0.552312
\(990\) 0 0
\(991\) −4.49242 −0.142707 −0.0713533 0.997451i \(-0.522732\pi\)
−0.0713533 + 0.997451i \(0.522732\pi\)
\(992\) 0 0
\(993\) 54.5464 1.73098
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −52.2462 −1.65465 −0.827327 0.561721i \(-0.810140\pi\)
−0.827327 + 0.561721i \(0.810140\pi\)
\(998\) 0 0
\(999\) −42.0540 −1.33053
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 2200.2.a.o.1.2 2
4.3 odd 2 4400.2.a.bp.1.1 2
5.2 odd 4 2200.2.b.g.1849.2 4
5.3 odd 4 2200.2.b.g.1849.3 4
5.4 even 2 88.2.a.b.1.1 2
15.14 odd 2 792.2.a.h.1.1 2
20.3 even 4 4400.2.b.v.4049.2 4
20.7 even 4 4400.2.b.v.4049.3 4
20.19 odd 2 176.2.a.d.1.2 2
35.34 odd 2 4312.2.a.n.1.2 2
40.19 odd 2 704.2.a.p.1.1 2
40.29 even 2 704.2.a.m.1.2 2
55.4 even 10 968.2.i.r.753.2 8
55.9 even 10 968.2.i.r.81.1 8
55.14 even 10 968.2.i.r.9.2 8
55.19 odd 10 968.2.i.q.9.2 8
55.24 odd 10 968.2.i.q.81.1 8
55.29 odd 10 968.2.i.q.753.2 8
55.39 odd 10 968.2.i.q.729.1 8
55.49 even 10 968.2.i.r.729.1 8
55.54 odd 2 968.2.a.j.1.1 2
60.59 even 2 1584.2.a.t.1.1 2
80.19 odd 4 2816.2.c.p.1409.3 4
80.29 even 4 2816.2.c.w.1409.2 4
80.59 odd 4 2816.2.c.p.1409.2 4
80.69 even 4 2816.2.c.w.1409.3 4
120.29 odd 2 6336.2.a.cu.1.2 2
120.59 even 2 6336.2.a.cx.1.2 2
140.139 even 2 8624.2.a.cb.1.1 2
165.164 even 2 8712.2.a.bb.1.1 2
220.219 even 2 1936.2.a.r.1.2 2
440.109 odd 2 7744.2.a.by.1.2 2
440.219 even 2 7744.2.a.cl.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
88.2.a.b.1.1 2 5.4 even 2
176.2.a.d.1.2 2 20.19 odd 2
704.2.a.m.1.2 2 40.29 even 2
704.2.a.p.1.1 2 40.19 odd 2
792.2.a.h.1.1 2 15.14 odd 2
968.2.a.j.1.1 2 55.54 odd 2
968.2.i.q.9.2 8 55.19 odd 10
968.2.i.q.81.1 8 55.24 odd 10
968.2.i.q.729.1 8 55.39 odd 10
968.2.i.q.753.2 8 55.29 odd 10
968.2.i.r.9.2 8 55.14 even 10
968.2.i.r.81.1 8 55.9 even 10
968.2.i.r.729.1 8 55.49 even 10
968.2.i.r.753.2 8 55.4 even 10
1584.2.a.t.1.1 2 60.59 even 2
1936.2.a.r.1.2 2 220.219 even 2
2200.2.a.o.1.2 2 1.1 even 1 trivial
2200.2.b.g.1849.2 4 5.2 odd 4
2200.2.b.g.1849.3 4 5.3 odd 4
2816.2.c.p.1409.2 4 80.59 odd 4
2816.2.c.p.1409.3 4 80.19 odd 4
2816.2.c.w.1409.2 4 80.29 even 4
2816.2.c.w.1409.3 4 80.69 even 4
4312.2.a.n.1.2 2 35.34 odd 2
4400.2.a.bp.1.1 2 4.3 odd 2
4400.2.b.v.4049.2 4 20.3 even 4
4400.2.b.v.4049.3 4 20.7 even 4
6336.2.a.cu.1.2 2 120.29 odd 2
6336.2.a.cx.1.2 2 120.59 even 2
7744.2.a.by.1.2 2 440.109 odd 2
7744.2.a.cl.1.1 2 440.219 even 2
8624.2.a.cb.1.1 2 140.139 even 2
8712.2.a.bb.1.1 2 165.164 even 2