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Properties

Label 1.16.a.a.1.1
Level $1$
Weight $16$
Character 1.1
Self dual yes
Analytic conductor $1.427$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1,16,Mod(1,1)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.42693505100\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+216.000 q^{2} -3348.00 q^{3} +13888.0 q^{4} +52110.0 q^{5} -723168. q^{6} +2.82246e6 q^{7} -4.07808e6 q^{8} -3.13980e6 q^{9} +1.12558e7 q^{10} +2.05869e7 q^{11} -4.64970e7 q^{12} -1.90073e8 q^{13} +6.09650e8 q^{14} -1.74464e8 q^{15} -1.33595e9 q^{16} +1.64653e9 q^{17} -6.78197e8 q^{18} +1.56326e9 q^{19} +7.23704e8 q^{20} -9.44958e9 q^{21} +4.44676e9 q^{22} +9.45112e9 q^{23} +1.36534e10 q^{24} -2.78021e10 q^{25} -4.10558e10 q^{26} +5.85522e10 q^{27} +3.91983e10 q^{28} -3.69026e10 q^{29} -3.76843e10 q^{30} +7.15885e10 q^{31} -1.54934e11 q^{32} -6.89248e10 q^{33} +3.55650e11 q^{34} +1.47078e11 q^{35} -4.36056e10 q^{36} -1.03365e12 q^{37} +3.37664e11 q^{38} +6.36366e11 q^{39} -2.12509e11 q^{40} +1.64197e12 q^{41} -2.04111e12 q^{42} -4.92403e11 q^{43} +2.85910e11 q^{44} -1.63615e11 q^{45} +2.04144e12 q^{46} -3.41068e12 q^{47} +4.47275e12 q^{48} +3.21870e12 q^{49} -6.00526e12 q^{50} -5.51258e12 q^{51} -2.63974e12 q^{52} +6.79715e12 q^{53} +1.26473e13 q^{54} +1.07278e12 q^{55} -1.15102e13 q^{56} -5.23379e12 q^{57} -7.97095e12 q^{58} +9.85886e12 q^{59} -2.42296e12 q^{60} +4.93184e12 q^{61} +1.54631e13 q^{62} -8.86196e12 q^{63} +1.03106e13 q^{64} -9.90472e12 q^{65} -1.48878e13 q^{66} -2.88378e13 q^{67} +2.28670e13 q^{68} -3.16423e13 q^{69} +3.17689e13 q^{70} +1.25050e14 q^{71} +1.28044e13 q^{72} -8.21715e13 q^{73} -2.23269e14 q^{74} +9.30815e13 q^{75} +2.17105e13 q^{76} +5.81055e13 q^{77} +1.37455e14 q^{78} -2.54131e13 q^{79} -6.96162e13 q^{80} -1.50980e14 q^{81} +3.54666e14 q^{82} -2.81737e14 q^{83} -1.31236e14 q^{84} +8.58006e13 q^{85} -1.06359e14 q^{86} +1.23550e14 q^{87} -8.39548e13 q^{88} +7.15619e14 q^{89} -3.53409e13 q^{90} -5.36474e14 q^{91} +1.31257e14 q^{92} -2.39678e14 q^{93} -7.36708e14 q^{94} +8.14613e13 q^{95} +5.18719e14 q^{96} +6.12786e14 q^{97} +6.95238e14 q^{98} -6.46387e13 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 216.000 1.19324 0.596621 − 0.802523i \(-0.296509\pi\)
0.596621 + 0.802523i \(0.296509\pi\)
\(3\) −3348.00 −0.883845 −0.441922 − 0.897053i \(-0.645703\pi\)
−0.441922 + 0.897053i \(0.645703\pi\)
\(4\) 13888.0 0.423828
\(5\) 52110.0 0.298295 0.149148 − 0.988815i \(-0.452347\pi\)
0.149148 + 0.988815i \(0.452347\pi\)
\(6\) −723168. −1.05464
\(7\) 2.82246e6 1.29536 0.647682 − 0.761911i \(-0.275739\pi\)
0.647682 + 0.761911i \(0.275739\pi\)
\(8\) −4.07808e6 −0.687513
\(9\) −3.13980e6 −0.218818
\(10\) 1.12558e7 0.355938
\(11\) 2.05869e7 0.318526 0.159263 − 0.987236i \(-0.449088\pi\)
0.159263 + 0.987236i \(0.449088\pi\)
\(12\) −4.64970e7 −0.374598
\(13\) −1.90073e8 −0.840129 −0.420065 − 0.907494i \(-0.637993\pi\)
−0.420065 + 0.907494i \(0.637993\pi\)
\(14\) 6.09650e8 1.54568
\(15\) −1.74464e8 −0.263647
\(16\) −1.33595e9 −1.24420
\(17\) 1.64653e9 0.973200 0.486600 − 0.873625i \(-0.338237\pi\)
0.486600 + 0.873625i \(0.338237\pi\)
\(18\) −6.78197e8 −0.261103
\(19\) 1.56326e9 0.401216 0.200608 − 0.979672i \(-0.435708\pi\)
0.200608 + 0.979672i \(0.435708\pi\)
\(20\) 7.23704e8 0.126426
\(21\) −9.44958e9 −1.14490
\(22\) 4.44676e9 0.380079
\(23\) 9.45112e9 0.578794 0.289397 − 0.957209i \(-0.406545\pi\)
0.289397 + 0.957209i \(0.406545\pi\)
\(24\) 1.36534e10 0.607655
\(25\) −2.78021e10 −0.911020
\(26\) −4.10558e10 −1.00248
\(27\) 5.85522e10 1.07725
\(28\) 3.91983e10 0.549012
\(29\) −3.69026e10 −0.397257 −0.198629 − 0.980075i \(-0.563649\pi\)
−0.198629 + 0.980075i \(0.563649\pi\)
\(30\) −3.76843e10 −0.314594
\(31\) 7.15885e10 0.467337 0.233669 − 0.972316i \(-0.424927\pi\)
0.233669 + 0.972316i \(0.424927\pi\)
\(32\) −1.54934e11 −0.797117
\(33\) −6.89248e10 −0.281528
\(34\) 3.55650e11 1.16126
\(35\) 1.47078e11 0.386401
\(36\) −4.36056e10 −0.0927413
\(37\) −1.03365e12 −1.79003 −0.895017 − 0.446031i \(-0.852837\pi\)
−0.895017 + 0.446031i \(0.852837\pi\)
\(38\) 3.37664e11 0.478748
\(39\) 6.36366e11 0.742544
\(40\) −2.12509e11 −0.205082
\(41\) 1.64197e12 1.31670 0.658351 − 0.752711i \(-0.271254\pi\)
0.658351 + 0.752711i \(0.271254\pi\)
\(42\) −2.04111e12 −1.36614
\(43\) −4.92403e11 −0.276253 −0.138127 − 0.990415i \(-0.544108\pi\)
−0.138127 + 0.990415i \(0.544108\pi\)
\(44\) 2.85910e11 0.135000
\(45\) −1.63615e11 −0.0652724
\(46\) 2.04144e12 0.690642
\(47\) −3.41068e12 −0.981991 −0.490996 − 0.871162i \(-0.663367\pi\)
−0.490996 + 0.871162i \(0.663367\pi\)
\(48\) 4.47275e12 1.09968
\(49\) 3.21870e12 0.677968
\(50\) −6.00526e12 −1.08707
\(51\) −5.51258e12 −0.860158
\(52\) −2.63974e12 −0.356070
\(53\) 6.79715e12 0.794800 0.397400 − 0.917645i \(-0.369913\pi\)
0.397400 + 0.917645i \(0.369913\pi\)
\(54\) 1.26473e13 1.28542
\(55\) 1.07278e12 0.0950147
\(56\) −1.15102e13 −0.890580
\(57\) −5.23379e12 −0.354613
\(58\) −7.97095e12 −0.474024
\(59\) 9.85886e12 0.515747 0.257873 − 0.966179i \(-0.416978\pi\)
0.257873 + 0.966179i \(0.416978\pi\)
\(60\) −2.42296e12 −0.111741
\(61\) 4.93184e12 0.200926 0.100463 − 0.994941i \(-0.467968\pi\)
0.100463 + 0.994941i \(0.467968\pi\)
\(62\) 1.54631e13 0.557647
\(63\) −8.86196e12 −0.283449
\(64\) 1.03106e13 0.293044
\(65\) −9.90472e12 −0.250606
\(66\) −1.48878e13 −0.335931
\(67\) −2.88378e13 −0.581302 −0.290651 − 0.956829i \(-0.593872\pi\)
−0.290651 + 0.956829i \(0.593872\pi\)
\(68\) 2.28670e13 0.412470
\(69\) −3.16423e13 −0.511564
\(70\) 3.17689e13 0.461070
\(71\) 1.25050e14 1.63172 0.815862 − 0.578247i \(-0.196263\pi\)
0.815862 + 0.578247i \(0.196263\pi\)
\(72\) 1.28044e13 0.150440
\(73\) −8.21715e13 −0.870562 −0.435281 − 0.900295i \(-0.643351\pi\)
−0.435281 + 0.900295i \(0.643351\pi\)
\(74\) −2.23269e14 −2.13595
\(75\) 9.30815e13 0.805200
\(76\) 2.17105e13 0.170047
\(77\) 5.81055e13 0.412607
\(78\) 1.37455e14 0.886035
\(79\) −2.54131e13 −0.148886 −0.0744430 − 0.997225i \(-0.523718\pi\)
−0.0744430 + 0.997225i \(0.523718\pi\)
\(80\) −6.96162e13 −0.371138
\(81\) −1.50980e14 −0.733300
\(82\) 3.54666e14 1.57114
\(83\) −2.81737e14 −1.13961 −0.569807 − 0.821779i \(-0.692982\pi\)
−0.569807 + 0.821779i \(0.692982\pi\)
\(84\) −1.31236e14 −0.485241
\(85\) 8.58006e13 0.290301
\(86\) −1.06359e14 −0.329637
\(87\) 1.23550e14 0.351114
\(88\) −8.39548e13 −0.218991
\(89\) 7.15619e14 1.71497 0.857485 − 0.514509i \(-0.172026\pi\)
0.857485 + 0.514509i \(0.172026\pi\)
\(90\) −3.53409e13 −0.0778858
\(91\) −5.36474e14 −1.08827
\(92\) 1.31257e14 0.245309
\(93\) −2.39678e14 −0.413054
\(94\) −7.36708e14 −1.17175
\(95\) 8.14613e13 0.119681
\(96\) 5.18719e14 0.704528
\(97\) 6.12786e14 0.770054 0.385027 − 0.922905i \(-0.374192\pi\)
0.385027 + 0.922905i \(0.374192\pi\)
\(98\) 6.95238e14 0.808981
\(99\) −6.46387e13 −0.0696993
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1.16.a.a.1.1 ✓ 1
3.2 odd 2 9.16.a.a.1.1 1
4.3 odd 2 16.16.a.d.1.1 1
5.2 odd 4 25.16.b.a.24.2 2
5.3 odd 4 25.16.b.a.24.1 2
5.4 even 2 25.16.a.a.1.1 1
7.2 even 3 49.16.c.c.18.1 2
7.3 odd 6 49.16.c.b.30.1 2
7.4 even 3 49.16.c.c.30.1 2
7.5 odd 6 49.16.c.b.18.1 2
7.6 odd 2 49.16.a.a.1.1 1
8.3 odd 2 64.16.a.c.1.1 1
8.5 even 2 64.16.a.i.1.1 1
11.10 odd 2 121.16.a.a.1.1 1
12.11 even 2 144.16.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.16.a.a.1.1 ✓ 1 1.1 even 1 trivial
9.16.a.a.1.1 1 3.2 odd 2
16.16.a.d.1.1 1 4.3 odd 2
25.16.a.a.1.1 1 5.4 even 2
25.16.b.a.24.1 2 5.3 odd 4
25.16.b.a.24.2 2 5.2 odd 4
49.16.a.a.1.1 1 7.6 odd 2
49.16.c.b.18.1 2 7.5 odd 6
49.16.c.b.30.1 2 7.3 odd 6
49.16.c.c.18.1 2 7.2 even 3
49.16.c.c.30.1 2 7.4 even 3
64.16.a.c.1.1 1 8.3 odd 2
64.16.a.i.1.1 1 8.5 even 2
121.16.a.a.1.1 1 11.10 odd 2
144.16.a.f.1.1 1 12.11 even 2