Properties

Label 1024.2.a.b.1.1
Level $1024$
Weight $2$
Character 1024.1
Self dual yes
Analytic conductor $8.177$
Analytic rank $1$
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] = [1024,2,Mod(1,1024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1024, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1024.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.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: \(8.17668116698\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 16)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1024.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{3} +1.41421 q^{5} -2.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.41421 q^{3} +1.41421 q^{5} -2.00000 q^{7} -1.00000 q^{9} +1.41421 q^{11} +1.41421 q^{13} -2.00000 q^{15} +2.00000 q^{17} -4.24264 q^{19} +2.82843 q^{21} -6.00000 q^{23} -3.00000 q^{25} +5.65685 q^{27} +4.24264 q^{29} -8.00000 q^{31} -2.00000 q^{33} -2.82843 q^{35} -4.24264 q^{37} -2.00000 q^{39} +7.07107 q^{43} -1.41421 q^{45} -8.00000 q^{47} -3.00000 q^{49} -2.82843 q^{51} -7.07107 q^{53} +2.00000 q^{55} +6.00000 q^{57} +4.24264 q^{59} -12.7279 q^{61} +2.00000 q^{63} +2.00000 q^{65} -7.07107 q^{67} +8.48528 q^{69} -10.0000 q^{71} +4.00000 q^{73} +4.24264 q^{75} -2.82843 q^{77} -5.00000 q^{81} +1.41421 q^{83} +2.82843 q^{85} -6.00000 q^{87} +4.00000 q^{89} -2.82843 q^{91} +11.3137 q^{93} -6.00000 q^{95} -2.00000 q^{97} -1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{7} - 2 q^{9} - 4 q^{15} + 4 q^{17} - 12 q^{23} - 6 q^{25} - 16 q^{31} - 4 q^{33} - 4 q^{39} - 16 q^{47} - 6 q^{49} + 4 q^{55} + 12 q^{57} + 4 q^{63} + 4 q^{65} - 20 q^{71} + 8 q^{73} - 10 q^{81} - 12 q^{87} + 8 q^{89} - 12 q^{95} - 4 q^{97}+O(q^{100}) \) 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
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 0 0
\(5\) 1.41421 0.632456 0.316228 0.948683i \(-0.397584\pi\)
0.316228 + 0.948683i \(0.397584\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.41421 0.426401 0.213201 0.977008i \(-0.431611\pi\)
0.213201 + 0.977008i \(0.431611\pi\)
\(12\) 0 0
\(13\) 1.41421 0.392232 0.196116 0.980581i \(-0.437167\pi\)
0.196116 + 0.980581i \(0.437167\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −4.24264 −0.973329 −0.486664 0.873589i \(-0.661786\pi\)
−0.486664 + 0.873589i \(0.661786\pi\)
\(20\) 0 0
\(21\) 2.82843 0.617213
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 4.24264 0.787839 0.393919 0.919145i \(-0.371119\pi\)
0.393919 + 0.919145i \(0.371119\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) −2.82843 −0.478091
\(36\) 0 0
\(37\) −4.24264 −0.697486 −0.348743 0.937218i \(-0.613391\pi\)
−0.348743 + 0.937218i \(0.613391\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 7.07107 1.07833 0.539164 0.842201i \(-0.318740\pi\)
0.539164 + 0.842201i \(0.318740\pi\)
\(44\) 0 0
\(45\) −1.41421 −0.210819
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −2.82843 −0.396059
\(52\) 0 0
\(53\) −7.07107 −0.971286 −0.485643 0.874157i \(-0.661414\pi\)
−0.485643 + 0.874157i \(0.661414\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 0 0
\(59\) 4.24264 0.552345 0.276172 0.961108i \(-0.410934\pi\)
0.276172 + 0.961108i \(0.410934\pi\)
\(60\) 0 0
\(61\) −12.7279 −1.62964 −0.814822 0.579712i \(-0.803165\pi\)
−0.814822 + 0.579712i \(0.803165\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −7.07107 −0.863868 −0.431934 0.901905i \(-0.642169\pi\)
−0.431934 + 0.901905i \(0.642169\pi\)
\(68\) 0 0
\(69\) 8.48528 1.02151
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) 4.24264 0.489898
\(76\) 0 0
\(77\) −2.82843 −0.322329
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 1.41421 0.155230 0.0776151 0.996983i \(-0.475269\pi\)
0.0776151 + 0.996983i \(0.475269\pi\)
\(84\) 0 0
\(85\) 2.82843 0.306786
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) −2.82843 −0.296500
\(92\) 0 0
\(93\) 11.3137 1.17318
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −1.41421 −0.142134
\(100\) 0 0
\(101\) −15.5563 −1.54791 −0.773957 0.633238i \(-0.781726\pi\)
−0.773957 + 0.633238i \(0.781726\pi\)
\(102\) 0 0
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 0 0
\(105\) 4.00000 0.390360
\(106\) 0 0
\(107\) −9.89949 −0.957020 −0.478510 0.878082i \(-0.658823\pi\)
−0.478510 + 0.878082i \(0.658823\pi\)
\(108\) 0 0
\(109\) −4.24264 −0.406371 −0.203186 0.979140i \(-0.565129\pi\)
−0.203186 + 0.979140i \(0.565129\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −8.48528 −0.791257
\(116\) 0 0
\(117\) −1.41421 −0.130744
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.3137 −1.01193
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) 15.5563 1.35916 0.679582 0.733599i \(-0.262161\pi\)
0.679582 + 0.733599i \(0.262161\pi\)
\(132\) 0 0
\(133\) 8.48528 0.735767
\(134\) 0 0
\(135\) 8.00000 0.688530
\(136\) 0 0
\(137\) −8.00000 −0.683486 −0.341743 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) 0 0
\(139\) −4.24264 −0.359856 −0.179928 0.983680i \(-0.557586\pi\)
−0.179928 + 0.983680i \(0.557586\pi\)
\(140\) 0 0
\(141\) 11.3137 0.952786
\(142\) 0 0
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 4.24264 0.349927
\(148\) 0 0
\(149\) 9.89949 0.810998 0.405499 0.914095i \(-0.367098\pi\)
0.405499 + 0.914095i \(0.367098\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) −11.3137 −0.908739
\(156\) 0 0
\(157\) 21.2132 1.69300 0.846499 0.532390i \(-0.178706\pi\)
0.846499 + 0.532390i \(0.178706\pi\)
\(158\) 0 0
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) −1.41421 −0.110770 −0.0553849 0.998465i \(-0.517639\pi\)
−0.0553849 + 0.998465i \(0.517639\pi\)
\(164\) 0 0
\(165\) −2.82843 −0.220193
\(166\) 0 0
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) 0 0
\(171\) 4.24264 0.324443
\(172\) 0 0
\(173\) 1.41421 0.107521 0.0537603 0.998554i \(-0.482879\pi\)
0.0537603 + 0.998554i \(0.482879\pi\)
\(174\) 0 0
\(175\) 6.00000 0.453557
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) 0 0
\(179\) 24.0416 1.79696 0.898478 0.439019i \(-0.144674\pi\)
0.898478 + 0.439019i \(0.144674\pi\)
\(180\) 0 0
\(181\) −12.7279 −0.946059 −0.473029 0.881047i \(-0.656840\pi\)
−0.473029 + 0.881047i \(0.656840\pi\)
\(182\) 0 0
\(183\) 18.0000 1.33060
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) 2.82843 0.206835
\(188\) 0 0
\(189\) −11.3137 −0.822951
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) −2.82843 −0.202548
\(196\) 0 0
\(197\) 24.0416 1.71290 0.856448 0.516234i \(-0.172666\pi\)
0.856448 + 0.516234i \(0.172666\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) 10.0000 0.705346
\(202\) 0 0
\(203\) −8.48528 −0.595550
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 12.7279 0.876226 0.438113 0.898920i \(-0.355647\pi\)
0.438113 + 0.898920i \(0.355647\pi\)
\(212\) 0 0
\(213\) 14.1421 0.969003
\(214\) 0 0
\(215\) 10.0000 0.681994
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) 0 0
\(219\) −5.65685 −0.382255
\(220\) 0 0
\(221\) 2.82843 0.190261
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) 0 0
\(227\) 21.2132 1.40797 0.703985 0.710215i \(-0.251402\pi\)
0.703985 + 0.710215i \(0.251402\pi\)
\(228\) 0 0
\(229\) −9.89949 −0.654177 −0.327089 0.944994i \(-0.606068\pi\)
−0.327089 + 0.944994i \(0.606068\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) 4.00000 0.262049 0.131024 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 0 0
\(235\) −11.3137 −0.738025
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 0 0
\(243\) −9.89949 −0.635053
\(244\) 0 0
\(245\) −4.24264 −0.271052
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 0 0
\(249\) −2.00000 −0.126745
\(250\) 0 0
\(251\) −29.6985 −1.87455 −0.937276 0.348589i \(-0.886661\pi\)
−0.937276 + 0.348589i \(0.886661\pi\)
\(252\) 0 0
\(253\) −8.48528 −0.533465
\(254\) 0 0
\(255\) −4.00000 −0.250490
\(256\) 0 0
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 0 0
\(259\) 8.48528 0.527250
\(260\) 0 0
\(261\) −4.24264 −0.262613
\(262\) 0 0
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) −10.0000 −0.614295
\(266\) 0 0
\(267\) −5.65685 −0.346194
\(268\) 0 0
\(269\) −4.24264 −0.258678 −0.129339 0.991600i \(-0.541286\pi\)
−0.129339 + 0.991600i \(0.541286\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) 0 0
\(275\) −4.24264 −0.255841
\(276\) 0 0
\(277\) 4.24264 0.254916 0.127458 0.991844i \(-0.459318\pi\)
0.127458 + 0.991844i \(0.459318\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −20.0000 −1.19310 −0.596550 0.802576i \(-0.703462\pi\)
−0.596550 + 0.802576i \(0.703462\pi\)
\(282\) 0 0
\(283\) 21.2132 1.26099 0.630497 0.776192i \(-0.282851\pi\)
0.630497 + 0.776192i \(0.282851\pi\)
\(284\) 0 0
\(285\) 8.48528 0.502625
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 2.82843 0.165805
\(292\) 0 0
\(293\) −21.2132 −1.23929 −0.619644 0.784883i \(-0.712723\pi\)
−0.619644 + 0.784883i \(0.712723\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 0 0
\(297\) 8.00000 0.464207
\(298\) 0 0
\(299\) −8.48528 −0.490716
\(300\) 0 0
\(301\) −14.1421 −0.815139
\(302\) 0 0
\(303\) 22.0000 1.26387
\(304\) 0 0
\(305\) −18.0000 −1.03068
\(306\) 0 0
\(307\) 7.07107 0.403567 0.201784 0.979430i \(-0.435326\pi\)
0.201784 + 0.979430i \(0.435326\pi\)
\(308\) 0 0
\(309\) −8.48528 −0.482711
\(310\) 0 0
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) 16.0000 0.904373 0.452187 0.891923i \(-0.350644\pi\)
0.452187 + 0.891923i \(0.350644\pi\)
\(314\) 0 0
\(315\) 2.82843 0.159364
\(316\) 0 0
\(317\) −7.07107 −0.397151 −0.198575 0.980086i \(-0.563631\pi\)
−0.198575 + 0.980086i \(0.563631\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 14.0000 0.781404
\(322\) 0 0
\(323\) −8.48528 −0.472134
\(324\) 0 0
\(325\) −4.24264 −0.235339
\(326\) 0 0
\(327\) 6.00000 0.331801
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 1.41421 0.0777322 0.0388661 0.999244i \(-0.487625\pi\)
0.0388661 + 0.999244i \(0.487625\pi\)
\(332\) 0 0
\(333\) 4.24264 0.232495
\(334\) 0 0
\(335\) −10.0000 −0.546358
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 0 0
\(339\) −8.48528 −0.460857
\(340\) 0 0
\(341\) −11.3137 −0.612672
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 12.0000 0.646058
\(346\) 0 0
\(347\) −18.3848 −0.986947 −0.493473 0.869761i \(-0.664273\pi\)
−0.493473 + 0.869761i \(0.664273\pi\)
\(348\) 0 0
\(349\) 4.24264 0.227103 0.113552 0.993532i \(-0.463777\pi\)
0.113552 + 0.993532i \(0.463777\pi\)
\(350\) 0 0
\(351\) 8.00000 0.427008
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −14.1421 −0.750587
\(356\) 0 0
\(357\) 5.65685 0.299392
\(358\) 0 0
\(359\) −26.0000 −1.37223 −0.686114 0.727494i \(-0.740685\pi\)
−0.686114 + 0.727494i \(0.740685\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 12.7279 0.668043
\(364\) 0 0
\(365\) 5.65685 0.296093
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 14.1421 0.734223
\(372\) 0 0
\(373\) −7.07107 −0.366126 −0.183063 0.983101i \(-0.558601\pi\)
−0.183063 + 0.983101i \(0.558601\pi\)
\(374\) 0 0
\(375\) 16.0000 0.826236
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) 4.24264 0.217930 0.108965 0.994046i \(-0.465246\pi\)
0.108965 + 0.994046i \(0.465246\pi\)
\(380\) 0 0
\(381\) −11.3137 −0.579619
\(382\) 0 0
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) 0 0
\(387\) −7.07107 −0.359443
\(388\) 0 0
\(389\) 18.3848 0.932145 0.466073 0.884746i \(-0.345669\pi\)
0.466073 + 0.884746i \(0.345669\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) −22.0000 −1.10975
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 7.07107 0.354887 0.177443 0.984131i \(-0.443217\pi\)
0.177443 + 0.984131i \(0.443217\pi\)
\(398\) 0 0
\(399\) −12.0000 −0.600751
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) −11.3137 −0.563576
\(404\) 0 0
\(405\) −7.07107 −0.351364
\(406\) 0 0
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) −16.0000 −0.791149 −0.395575 0.918434i \(-0.629455\pi\)
−0.395575 + 0.918434i \(0.629455\pi\)
\(410\) 0 0
\(411\) 11.3137 0.558064
\(412\) 0 0
\(413\) −8.48528 −0.417533
\(414\) 0 0
\(415\) 2.00000 0.0981761
\(416\) 0 0
\(417\) 6.00000 0.293821
\(418\) 0 0
\(419\) 4.24264 0.207267 0.103633 0.994616i \(-0.466953\pi\)
0.103633 + 0.994616i \(0.466953\pi\)
\(420\) 0 0
\(421\) 12.7279 0.620321 0.310160 0.950684i \(-0.399617\pi\)
0.310160 + 0.950684i \(0.399617\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 25.4558 1.23189
\(428\) 0 0
\(429\) −2.82843 −0.136558
\(430\) 0 0
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) −8.48528 −0.406838
\(436\) 0 0
\(437\) 25.4558 1.21772
\(438\) 0 0
\(439\) −14.0000 −0.668184 −0.334092 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 21.2132 1.00787 0.503935 0.863742i \(-0.331885\pi\)
0.503935 + 0.863742i \(0.331885\pi\)
\(444\) 0 0
\(445\) 5.65685 0.268161
\(446\) 0 0
\(447\) −14.0000 −0.662177
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −14.1421 −0.664455
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) 32.0000 1.49690 0.748448 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) 0 0
\(459\) 11.3137 0.528079
\(460\) 0 0
\(461\) −15.5563 −0.724531 −0.362266 0.932075i \(-0.617997\pi\)
−0.362266 + 0.932075i \(0.617997\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 0 0
\(465\) 16.0000 0.741982
\(466\) 0 0
\(467\) 7.07107 0.327210 0.163605 0.986526i \(-0.447688\pi\)
0.163605 + 0.986526i \(0.447688\pi\)
\(468\) 0 0
\(469\) 14.1421 0.653023
\(470\) 0 0
\(471\) −30.0000 −1.38233
\(472\) 0 0
\(473\) 10.0000 0.459800
\(474\) 0 0
\(475\) 12.7279 0.583997
\(476\) 0 0
\(477\) 7.07107 0.323762
\(478\) 0 0
\(479\) −40.0000 −1.82765 −0.913823 0.406112i \(-0.866884\pi\)
−0.913823 + 0.406112i \(0.866884\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 0 0
\(483\) −16.9706 −0.772187
\(484\) 0 0
\(485\) −2.82843 −0.128432
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) 2.00000 0.0904431
\(490\) 0 0
\(491\) −26.8701 −1.21263 −0.606314 0.795225i \(-0.707353\pi\)
−0.606314 + 0.795225i \(0.707353\pi\)
\(492\) 0 0
\(493\) 8.48528 0.382158
\(494\) 0 0
\(495\) −2.00000 −0.0898933
\(496\) 0 0
\(497\) 20.0000 0.897123
\(498\) 0 0
\(499\) −32.5269 −1.45610 −0.728052 0.685522i \(-0.759574\pi\)
−0.728052 + 0.685522i \(0.759574\pi\)
\(500\) 0 0
\(501\) 2.82843 0.126365
\(502\) 0 0
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) −22.0000 −0.978987
\(506\) 0 0
\(507\) 15.5563 0.690882
\(508\) 0 0
\(509\) 32.5269 1.44173 0.720865 0.693075i \(-0.243745\pi\)
0.720865 + 0.693075i \(0.243745\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 0 0
\(513\) −24.0000 −1.05963
\(514\) 0 0
\(515\) 8.48528 0.373906
\(516\) 0 0
\(517\) −11.3137 −0.497576
\(518\) 0 0
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) −40.0000 −1.75243 −0.876216 0.481919i \(-0.839940\pi\)
−0.876216 + 0.481919i \(0.839940\pi\)
\(522\) 0 0
\(523\) 35.3553 1.54598 0.772991 0.634418i \(-0.218760\pi\)
0.772991 + 0.634418i \(0.218760\pi\)
\(524\) 0 0
\(525\) −8.48528 −0.370328
\(526\) 0 0
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −4.24264 −0.184115
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −14.0000 −0.605273
\(536\) 0 0
\(537\) −34.0000 −1.46721
\(538\) 0 0
\(539\) −4.24264 −0.182743
\(540\) 0 0
\(541\) −12.7279 −0.547216 −0.273608 0.961841i \(-0.588217\pi\)
−0.273608 + 0.961841i \(0.588217\pi\)
\(542\) 0 0
\(543\) 18.0000 0.772454
\(544\) 0 0
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) −7.07107 −0.302337 −0.151169 0.988508i \(-0.548304\pi\)
−0.151169 + 0.988508i \(0.548304\pi\)
\(548\) 0 0
\(549\) 12.7279 0.543214
\(550\) 0 0
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 8.48528 0.360180
\(556\) 0 0
\(557\) 35.3553 1.49805 0.749027 0.662540i \(-0.230521\pi\)
0.749027 + 0.662540i \(0.230521\pi\)
\(558\) 0 0
\(559\) 10.0000 0.422955
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) −26.8701 −1.13244 −0.566219 0.824255i \(-0.691594\pi\)
−0.566219 + 0.824255i \(0.691594\pi\)
\(564\) 0 0
\(565\) 8.48528 0.356978
\(566\) 0 0
\(567\) 10.0000 0.419961
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) −1.41421 −0.0591830 −0.0295915 0.999562i \(-0.509421\pi\)
−0.0295915 + 0.999562i \(0.509421\pi\)
\(572\) 0 0
\(573\) 11.3137 0.472637
\(574\) 0 0
\(575\) 18.0000 0.750652
\(576\) 0 0
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 0 0
\(579\) −19.7990 −0.822818
\(580\) 0 0
\(581\) −2.82843 −0.117343
\(582\) 0 0
\(583\) −10.0000 −0.414158
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 0 0
\(587\) −9.89949 −0.408596 −0.204298 0.978909i \(-0.565491\pi\)
−0.204298 + 0.978909i \(0.565491\pi\)
\(588\) 0 0
\(589\) 33.9411 1.39852
\(590\) 0 0
\(591\) −34.0000 −1.39857
\(592\) 0 0
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) −5.65685 −0.231908
\(596\) 0 0
\(597\) −19.7990 −0.810319
\(598\) 0 0
\(599\) −14.0000 −0.572024 −0.286012 0.958226i \(-0.592330\pi\)
−0.286012 + 0.958226i \(0.592330\pi\)
\(600\) 0 0
\(601\) 20.0000 0.815817 0.407909 0.913023i \(-0.366258\pi\)
0.407909 + 0.913023i \(0.366258\pi\)
\(602\) 0 0
\(603\) 7.07107 0.287956
\(604\) 0 0
\(605\) −12.7279 −0.517464
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) −11.3137 −0.457704
\(612\) 0 0
\(613\) 35.3553 1.42799 0.713994 0.700151i \(-0.246884\pi\)
0.713994 + 0.700151i \(0.246884\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.0000 0.483102 0.241551 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 0 0
\(619\) 24.0416 0.966315 0.483157 0.875534i \(-0.339490\pi\)
0.483157 + 0.875534i \(0.339490\pi\)
\(620\) 0 0
\(621\) −33.9411 −1.36201
\(622\) 0 0
\(623\) −8.00000 −0.320513
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 8.48528 0.338869
\(628\) 0 0
\(629\) −8.48528 −0.338330
\(630\) 0 0
\(631\) 10.0000 0.398094 0.199047 0.979990i \(-0.436215\pi\)
0.199047 + 0.979990i \(0.436215\pi\)
\(632\) 0 0
\(633\) −18.0000 −0.715436
\(634\) 0 0
\(635\) 11.3137 0.448971
\(636\) 0 0
\(637\) −4.24264 −0.168100
\(638\) 0 0
\(639\) 10.0000 0.395594
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −29.6985 −1.17119 −0.585597 0.810602i \(-0.699140\pi\)
−0.585597 + 0.810602i \(0.699140\pi\)
\(644\) 0 0
\(645\) −14.1421 −0.556846
\(646\) 0 0
\(647\) −42.0000 −1.65119 −0.825595 0.564263i \(-0.809160\pi\)
−0.825595 + 0.564263i \(0.809160\pi\)
\(648\) 0 0
\(649\) 6.00000 0.235521
\(650\) 0 0
\(651\) −22.6274 −0.886838
\(652\) 0 0
\(653\) −26.8701 −1.05151 −0.525753 0.850637i \(-0.676216\pi\)
−0.525753 + 0.850637i \(0.676216\pi\)
\(654\) 0 0
\(655\) 22.0000 0.859611
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) 24.0416 0.936529 0.468264 0.883588i \(-0.344879\pi\)
0.468264 + 0.883588i \(0.344879\pi\)
\(660\) 0 0
\(661\) −12.7279 −0.495059 −0.247529 0.968880i \(-0.579619\pi\)
−0.247529 + 0.968880i \(0.579619\pi\)
\(662\) 0 0
\(663\) −4.00000 −0.155347
\(664\) 0 0
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) −25.4558 −0.985654
\(668\) 0 0
\(669\) −33.9411 −1.31224
\(670\) 0 0
\(671\) −18.0000 −0.694882
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 0 0
\(675\) −16.9706 −0.653197
\(676\) 0 0
\(677\) −4.24264 −0.163058 −0.0815290 0.996671i \(-0.525980\pi\)
−0.0815290 + 0.996671i \(0.525980\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) −30.0000 −1.14960
\(682\) 0 0
\(683\) 7.07107 0.270567 0.135283 0.990807i \(-0.456805\pi\)
0.135283 + 0.990807i \(0.456805\pi\)
\(684\) 0 0
\(685\) −11.3137 −0.432275
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) 0 0
\(689\) −10.0000 −0.380970
\(690\) 0 0
\(691\) 12.7279 0.484193 0.242096 0.970252i \(-0.422165\pi\)
0.242096 + 0.970252i \(0.422165\pi\)
\(692\) 0 0
\(693\) 2.82843 0.107443
\(694\) 0 0
\(695\) −6.00000 −0.227593
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −5.65685 −0.213962
\(700\) 0 0
\(701\) 43.8406 1.65584 0.827919 0.560848i \(-0.189525\pi\)
0.827919 + 0.560848i \(0.189525\pi\)
\(702\) 0 0
\(703\) 18.0000 0.678883
\(704\) 0 0
\(705\) 16.0000 0.602595
\(706\) 0 0
\(707\) 31.1127 1.17011
\(708\) 0 0
\(709\) −38.1838 −1.43402 −0.717011 0.697062i \(-0.754490\pi\)
−0.717011 + 0.697062i \(0.754490\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 48.0000 1.79761
\(714\) 0 0
\(715\) 2.82843 0.105777
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −12.0000 −0.446903
\(722\) 0 0
\(723\) −25.4558 −0.946713
\(724\) 0 0
\(725\) −12.7279 −0.472703
\(726\) 0 0
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) 14.1421 0.523066
\(732\) 0 0
\(733\) −29.6985 −1.09694 −0.548469 0.836171i \(-0.684789\pi\)
−0.548469 + 0.836171i \(0.684789\pi\)
\(734\) 0 0
\(735\) 6.00000 0.221313
\(736\) 0 0
\(737\) −10.0000 −0.368355
\(738\) 0 0
\(739\) 32.5269 1.19652 0.598261 0.801301i \(-0.295859\pi\)
0.598261 + 0.801301i \(0.295859\pi\)
\(740\) 0 0
\(741\) 8.48528 0.311715
\(742\) 0 0
\(743\) 46.0000 1.68758 0.843788 0.536676i \(-0.180320\pi\)
0.843788 + 0.536676i \(0.180320\pi\)
\(744\) 0 0
\(745\) 14.0000 0.512920
\(746\) 0 0
\(747\) −1.41421 −0.0517434
\(748\) 0 0
\(749\) 19.7990 0.723439
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) 42.0000 1.53057
\(754\) 0 0
\(755\) 14.1421 0.514685
\(756\) 0 0
\(757\) 32.5269 1.18221 0.591105 0.806594i \(-0.298692\pi\)
0.591105 + 0.806594i \(0.298692\pi\)
\(758\) 0 0
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 8.48528 0.307188
\(764\) 0 0
\(765\) −2.82843 −0.102262
\(766\) 0 0
\(767\) 6.00000 0.216647
\(768\) 0 0
\(769\) −50.0000 −1.80305 −0.901523 0.432731i \(-0.857550\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) 31.1127 1.12050
\(772\) 0 0
\(773\) 7.07107 0.254329 0.127164 0.991882i \(-0.459412\pi\)
0.127164 + 0.991882i \(0.459412\pi\)
\(774\) 0 0
\(775\) 24.0000 0.862105
\(776\) 0 0
\(777\) −12.0000 −0.430498
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −14.1421 −0.506045
\(782\) 0 0
\(783\) 24.0000 0.857690
\(784\) 0 0
\(785\) 30.0000 1.07075
\(786\) 0 0
\(787\) −21.2132 −0.756169 −0.378085 0.925771i \(-0.623417\pi\)
−0.378085 + 0.925771i \(0.623417\pi\)
\(788\) 0 0
\(789\) −8.48528 −0.302084
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) −18.0000 −0.639199
\(794\) 0 0
\(795\) 14.1421 0.501570
\(796\) 0 0
\(797\) −35.3553 −1.25235 −0.626175 0.779682i \(-0.715381\pi\)
−0.626175 + 0.779682i \(0.715381\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −4.00000 −0.141333
\(802\) 0 0
\(803\) 5.65685 0.199626
\(804\) 0 0
\(805\) 16.9706 0.598134
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) 16.0000 0.562530 0.281265 0.959630i \(-0.409246\pi\)
0.281265 + 0.959630i \(0.409246\pi\)
\(810\) 0 0
\(811\) −55.1543 −1.93673 −0.968365 0.249537i \(-0.919722\pi\)
−0.968365 + 0.249537i \(0.919722\pi\)
\(812\) 0 0
\(813\) −11.3137 −0.396789
\(814\) 0 0
\(815\) −2.00000 −0.0700569
\(816\) 0 0
\(817\) −30.0000 −1.04957
\(818\) 0 0
\(819\) 2.82843 0.0988332
\(820\) 0 0
\(821\) 15.5563 0.542920 0.271460 0.962450i \(-0.412493\pi\)
0.271460 + 0.962450i \(0.412493\pi\)
\(822\) 0 0
\(823\) 34.0000 1.18517 0.592583 0.805510i \(-0.298108\pi\)
0.592583 + 0.805510i \(0.298108\pi\)
\(824\) 0 0
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) −46.6690 −1.62284 −0.811421 0.584462i \(-0.801305\pi\)
−0.811421 + 0.584462i \(0.801305\pi\)
\(828\) 0 0
\(829\) 32.5269 1.12971 0.564853 0.825191i \(-0.308933\pi\)
0.564853 + 0.825191i \(0.308933\pi\)
\(830\) 0 0
\(831\) −6.00000 −0.208138
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) −2.82843 −0.0978818
\(836\) 0 0
\(837\) −45.2548 −1.56424
\(838\) 0 0
\(839\) 14.0000 0.483334 0.241667 0.970359i \(-0.422306\pi\)
0.241667 + 0.970359i \(0.422306\pi\)
\(840\) 0 0
\(841\) −11.0000 −0.379310
\(842\) 0 0
\(843\) 28.2843 0.974162
\(844\) 0 0
\(845\) −15.5563 −0.535155
\(846\) 0 0
\(847\) 18.0000 0.618487
\(848\) 0 0
\(849\) −30.0000 −1.02960
\(850\) 0 0
\(851\) 25.4558 0.872615
\(852\) 0 0
\(853\) −7.07107 −0.242109 −0.121054 0.992646i \(-0.538628\pi\)
−0.121054 + 0.992646i \(0.538628\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) 0 0
\(857\) 8.00000 0.273275 0.136637 0.990621i \(-0.456370\pi\)
0.136637 + 0.990621i \(0.456370\pi\)
\(858\) 0 0
\(859\) 4.24264 0.144757 0.0723785 0.997377i \(-0.476941\pi\)
0.0723785 + 0.997377i \(0.476941\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 2.00000 0.0680020
\(866\) 0 0
\(867\) 18.3848 0.624380
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −10.0000 −0.338837
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 22.6274 0.764946
\(876\) 0 0
\(877\) 7.07107 0.238773 0.119386 0.992848i \(-0.461907\pi\)
0.119386 + 0.992848i \(0.461907\pi\)
\(878\) 0 0
\(879\) 30.0000 1.01187
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) 29.6985 0.999434 0.499717 0.866189i \(-0.333437\pi\)
0.499717 + 0.866189i \(0.333437\pi\)
\(884\) 0 0
\(885\) −8.48528 −0.285230
\(886\) 0 0
\(887\) 2.00000 0.0671534 0.0335767 0.999436i \(-0.489310\pi\)
0.0335767 + 0.999436i \(0.489310\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) −7.07107 −0.236890
\(892\) 0 0
\(893\) 33.9411 1.13580
\(894\) 0 0
\(895\) 34.0000 1.13649
\(896\) 0 0
\(897\) 12.0000 0.400668
\(898\) 0 0
\(899\) −33.9411 −1.13200
\(900\) 0 0
\(901\) −14.1421 −0.471143
\(902\) 0 0
\(903\) 20.0000 0.665558
\(904\) 0 0
\(905\) −18.0000 −0.598340
\(906\) 0 0
\(907\) −38.1838 −1.26787 −0.633936 0.773386i \(-0.718562\pi\)
−0.633936 + 0.773386i \(0.718562\pi\)
\(908\) 0 0
\(909\) 15.5563 0.515972
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) 2.00000 0.0661903
\(914\) 0 0
\(915\) 25.4558 0.841544
\(916\) 0 0
\(917\) −31.1127 −1.02743
\(918\) 0 0
\(919\) 26.0000 0.857661 0.428830 0.903385i \(-0.358926\pi\)
0.428830 + 0.903385i \(0.358926\pi\)
\(920\) 0 0
\(921\) −10.0000 −0.329511
\(922\) 0 0
\(923\) −14.1421 −0.465494
\(924\) 0 0
\(925\) 12.7279 0.418491
\(926\) 0 0
\(927\) −6.00000 −0.197066
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 12.7279 0.417141
\(932\) 0 0
\(933\) 42.4264 1.38898
\(934\) 0 0
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) −28.0000 −0.914720 −0.457360 0.889282i \(-0.651205\pi\)
−0.457360 + 0.889282i \(0.651205\pi\)
\(938\) 0 0
\(939\) −22.6274 −0.738418
\(940\) 0 0
\(941\) 41.0122 1.33696 0.668480 0.743730i \(-0.266945\pi\)
0.668480 + 0.743730i \(0.266945\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −16.0000 −0.520480
\(946\) 0 0
\(947\) 7.07107 0.229779 0.114889 0.993378i \(-0.463349\pi\)
0.114889 + 0.993378i \(0.463349\pi\)
\(948\) 0 0
\(949\) 5.65685 0.183629
\(950\) 0 0
\(951\) 10.0000 0.324272
\(952\) 0 0
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) 0 0
\(955\) −11.3137 −0.366103
\(956\) 0 0
\(957\) −8.48528 −0.274290
\(958\) 0 0
\(959\) 16.0000 0.516667
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 9.89949 0.319007
\(964\) 0 0
\(965\) 19.7990 0.637352
\(966\) 0 0
\(967\) −2.00000 −0.0643157 −0.0321578 0.999483i \(-0.510238\pi\)
−0.0321578 + 0.999483i \(0.510238\pi\)
\(968\) 0 0
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) −26.8701 −0.862301 −0.431151 0.902280i \(-0.641892\pi\)
−0.431151 + 0.902280i \(0.641892\pi\)
\(972\) 0 0
\(973\) 8.48528 0.272026
\(974\) 0 0
\(975\) 6.00000 0.192154
\(976\) 0 0
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 0 0
\(979\) 5.65685 0.180794
\(980\) 0 0
\(981\) 4.24264 0.135457
\(982\) 0 0
\(983\) 34.0000 1.08443 0.542216 0.840239i \(-0.317586\pi\)
0.542216 + 0.840239i \(0.317586\pi\)
\(984\) 0 0
\(985\) 34.0000 1.08333
\(986\) 0 0
\(987\) −22.6274 −0.720239
\(988\) 0 0
\(989\) −42.4264 −1.34908
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 0 0
\(993\) −2.00000 −0.0634681
\(994\) 0 0
\(995\) 19.7990 0.627670
\(996\) 0 0
\(997\) 52.3259 1.65718 0.828589 0.559857i \(-0.189144\pi\)
0.828589 + 0.559857i \(0.189144\pi\)
\(998\) 0 0
\(999\) −24.0000 −0.759326
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 1024.2.a.b.1.1 2
3.2 odd 2 9216.2.a.d.1.1 2
4.3 odd 2 1024.2.a.e.1.2 2
8.3 odd 2 1024.2.a.e.1.1 2
8.5 even 2 inner 1024.2.a.b.1.2 2
12.11 even 2 9216.2.a.s.1.1 2
16.3 odd 4 1024.2.b.b.513.2 2
16.5 even 4 1024.2.b.e.513.2 2
16.11 odd 4 1024.2.b.b.513.1 2
16.13 even 4 1024.2.b.e.513.1 2
24.5 odd 2 9216.2.a.d.1.2 2
24.11 even 2 9216.2.a.s.1.2 2
32.3 odd 8 64.2.e.a.17.1 2
32.5 even 8 128.2.e.b.97.1 2
32.11 odd 8 64.2.e.a.49.1 2
32.13 even 8 128.2.e.b.33.1 2
32.19 odd 8 128.2.e.a.33.1 2
32.21 even 8 16.2.e.a.5.1 2
32.27 odd 8 128.2.e.a.97.1 2
32.29 even 8 16.2.e.a.13.1 yes 2
96.5 odd 8 1152.2.k.b.865.1 2
96.11 even 8 576.2.k.a.433.1 2
96.29 odd 8 144.2.k.a.109.1 2
96.35 even 8 576.2.k.a.145.1 2
96.53 odd 8 144.2.k.a.37.1 2
96.59 even 8 1152.2.k.a.865.1 2
96.77 odd 8 1152.2.k.b.289.1 2
96.83 even 8 1152.2.k.a.289.1 2
160.3 even 8 1600.2.q.a.849.1 2
160.29 even 8 400.2.l.c.301.1 2
160.43 even 8 1600.2.q.b.49.1 2
160.53 odd 8 400.2.q.a.149.1 2
160.67 even 8 1600.2.q.b.849.1 2
160.93 odd 8 400.2.q.b.349.1 2
160.99 odd 8 1600.2.l.a.401.1 2
160.107 even 8 1600.2.q.a.49.1 2
160.117 odd 8 400.2.q.b.149.1 2
160.139 odd 8 1600.2.l.a.1201.1 2
160.149 even 8 400.2.l.c.101.1 2
160.157 odd 8 400.2.q.a.349.1 2
224.53 even 24 784.2.x.f.373.1 4
224.61 odd 24 784.2.x.c.557.1 4
224.93 even 24 784.2.x.f.557.1 4
224.117 odd 24 784.2.x.c.165.1 4
224.125 odd 8 784.2.m.b.589.1 2
224.149 even 24 784.2.x.f.165.1 4
224.157 odd 24 784.2.x.c.765.1 4
224.181 odd 8 784.2.m.b.197.1 2
224.213 odd 24 784.2.x.c.373.1 4
224.221 even 24 784.2.x.f.765.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.2.e.a.5.1 2 32.21 even 8
16.2.e.a.13.1 yes 2 32.29 even 8
64.2.e.a.17.1 2 32.3 odd 8
64.2.e.a.49.1 2 32.11 odd 8
128.2.e.a.33.1 2 32.19 odd 8
128.2.e.a.97.1 2 32.27 odd 8
128.2.e.b.33.1 2 32.13 even 8
128.2.e.b.97.1 2 32.5 even 8
144.2.k.a.37.1 2 96.53 odd 8
144.2.k.a.109.1 2 96.29 odd 8
400.2.l.c.101.1 2 160.149 even 8
400.2.l.c.301.1 2 160.29 even 8
400.2.q.a.149.1 2 160.53 odd 8
400.2.q.a.349.1 2 160.157 odd 8
400.2.q.b.149.1 2 160.117 odd 8
400.2.q.b.349.1 2 160.93 odd 8
576.2.k.a.145.1 2 96.35 even 8
576.2.k.a.433.1 2 96.11 even 8
784.2.m.b.197.1 2 224.181 odd 8
784.2.m.b.589.1 2 224.125 odd 8
784.2.x.c.165.1 4 224.117 odd 24
784.2.x.c.373.1 4 224.213 odd 24
784.2.x.c.557.1 4 224.61 odd 24
784.2.x.c.765.1 4 224.157 odd 24
784.2.x.f.165.1 4 224.149 even 24
784.2.x.f.373.1 4 224.53 even 24
784.2.x.f.557.1 4 224.93 even 24
784.2.x.f.765.1 4 224.221 even 24
1024.2.a.b.1.1 2 1.1 even 1 trivial
1024.2.a.b.1.2 2 8.5 even 2 inner
1024.2.a.e.1.1 2 8.3 odd 2
1024.2.a.e.1.2 2 4.3 odd 2
1024.2.b.b.513.1 2 16.11 odd 4
1024.2.b.b.513.2 2 16.3 odd 4
1024.2.b.e.513.1 2 16.13 even 4
1024.2.b.e.513.2 2 16.5 even 4
1152.2.k.a.289.1 2 96.83 even 8
1152.2.k.a.865.1 2 96.59 even 8
1152.2.k.b.289.1 2 96.77 odd 8
1152.2.k.b.865.1 2 96.5 odd 8
1600.2.l.a.401.1 2 160.99 odd 8
1600.2.l.a.1201.1 2 160.139 odd 8
1600.2.q.a.49.1 2 160.107 even 8
1600.2.q.a.849.1 2 160.3 even 8
1600.2.q.b.49.1 2 160.43 even 8
1600.2.q.b.849.1 2 160.67 even 8
9216.2.a.d.1.1 2 3.2 odd 2
9216.2.a.d.1.2 2 24.5 odd 2
9216.2.a.s.1.1 2 12.11 even 2
9216.2.a.s.1.2 2 24.11 even 2