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Properties

Label 2169.2.a.k
Level $2169$
Weight $2$
Character orbit 2169.a
Self dual yes
Analytic conductor $17.320$
Analytic rank $0$
Dimension $26$
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] = [2169,2,Mod(1,2169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2169, 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("2169.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2169 = 3^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2169.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: \(17.3195521984\)
Analytic rank: \(0\)
Dimension: \(26\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 26 q + 40 q^{4} + 22 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q + 40 q^{4} + 22 q^{7} + 10 q^{10} + 12 q^{13} + 68 q^{16} + 22 q^{19} + 12 q^{22} + 46 q^{25} + 54 q^{28} + 74 q^{31} + 34 q^{34} + 30 q^{37} + 16 q^{40} + 30 q^{43} + 58 q^{46} + 60 q^{49} + 60 q^{52} + 42 q^{55} + 24 q^{58} + 30 q^{61} + 106 q^{64} + 22 q^{67} - 20 q^{70} + 8 q^{73} + 6 q^{76} + 70 q^{79} - 6 q^{82} + 32 q^{85} - 30 q^{88} + 76 q^{91} - 6 q^{94} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.80481 0 5.86696 2.90028 0 4.11673 −10.8461 0 −8.13474
1.2 −2.75618 0 5.59653 −0.473377 0 −1.95106 −9.91269 0 1.30471
1.3 −2.45758 0 4.03972 −2.99095 0 3.38349 −5.01279 0 7.35051
1.4 −2.42053 0 3.85896 −2.12516 0 −1.61468 −4.49968 0 5.14401
1.5 −2.28827 0 3.23617 −3.28238 0 1.28763 −2.82869 0 7.51097
1.6 −2.11054 0 2.45440 0.798150 0 3.81630 −0.959021 0 −1.68453
1.7 −1.82912 0 1.34567 4.36507 0 1.56287 1.19685 0 −7.98422
1.8 −1.50897 0 0.277004 −1.36085 0 −4.01853 2.59996 0 2.05349
1.9 −1.24417 0 −0.452035 2.73311 0 −3.56359 3.05075 0 −3.40046
1.10 −0.973004 0 −1.05326 −2.80719 0 5.15037 2.97084 0 2.73141
1.11 −0.904052 0 −1.18269 −0.599999 0 1.57515 2.87732 0 0.542431
1.12 −0.0993763 0 −1.99012 2.63352 0 −1.47981 0.396524 0 −0.261710
1.13 −0.0519595 0 −1.99730 3.30753 0 2.73513 0.207698 0 −0.171857
1.14 0.0519595 0 −1.99730 −3.30753 0 2.73513 −0.207698 0 −0.171857
1.15 0.0993763 0 −1.99012 −2.63352 0 −1.47981 −0.396524 0 −0.261710
1.16 0.904052 0 −1.18269 0.599999 0 1.57515 −2.87732 0 0.542431
1.17 0.973004 0 −1.05326 2.80719 0 5.15037 −2.97084 0 2.73141
1.18 1.24417 0 −0.452035 −2.73311 0 −3.56359 −3.05075 0 −3.40046
1.19 1.50897 0 0.277004 1.36085 0 −4.01853 −2.59996 0 2.05349
1.20 1.82912 0 1.34567 −4.36507 0 1.56287 −1.19685 0 −7.98422
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.26
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(241\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2169.2.a.k ✓ 26
3.b odd 2 1 inner 2169.2.a.k ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2169.2.a.k ✓ 26 1.a even 1 1 trivial
2169.2.a.k ✓ 26 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{26} - 46 T_{2}^{24} + 929 T_{2}^{22} - 10833 T_{2}^{20} + 80695 T_{2}^{18} - 401172 T_{2}^{16} + \cdots - 12 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2169))\). Copy content Toggle raw display