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Properties

Label 2169.2.d.f
Level $2169$
Weight $2$
Character orbit 2169.d
Analytic conductor $17.320$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2169,2,Mod(1927,2169)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2169.1927");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2169 = 3^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2169.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(17.3195521984\)
Analytic rank: \(0\)
Dimension: \(36\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 52 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 52 q^{4} - 12 q^{10} + 36 q^{16} + 68 q^{25} - 64 q^{40} - 48 q^{49} + 24 q^{58} + 16 q^{61} + 112 q^{64} - 24 q^{79} - 52 q^{82} - 24 q^{91} + 52 q^{94} - 36 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1927.1 −2.71730 0 5.38370 −1.25803 0 − 2.48123i −9.19452 0 3.41843
1927.2 −2.71730 0 5.38370 −1.25803 0 2.48123i −9.19452 0 3.41843
1927.3 −2.57149 0 4.61255 4.10310 0 2.43913i −6.71814 0 −10.5511
1927.4 −2.57149 0 4.61255 4.10310 0 − 2.43913i −6.71814 0 −10.5511
1927.5 −2.22529 0 2.95192 −2.90861 0 − 1.93383i −2.11829 0 6.47250
1927.6 −2.22529 0 2.95192 −2.90861 0 1.93383i −2.11829 0 6.47250
1927.7 −2.14764 0 2.61238 1.43134 0 − 2.81865i −1.31517 0 −3.07402
1927.8 −2.14764 0 2.61238 1.43134 0 2.81865i −1.31517 0 −3.07402
1927.9 −1.71428 0 0.938764 0.702701 0 3.07594i 1.81926 0 −1.20463
1927.10 −1.71428 0 0.938764 0.702701 0 − 3.07594i 1.81926 0 −1.20463
1927.11 −1.31010 0 −0.283641 −2.61155 0 − 0.280997i 2.99180 0 3.42139
1927.12 −1.31010 0 −0.283641 −2.61155 0 0.280997i 2.99180 0 3.42139
1927.13 −1.14514 0 −0.688654 −0.522560 0 4.74048i 3.07889 0 0.598405
1927.14 −1.14514 0 −0.688654 −0.522560 0 − 4.74048i 3.07889 0 0.598405
1927.15 −1.12770 0 −0.728288 3.34840 0 − 0.846010i 3.07670 0 −3.77600
1927.16 −1.12770 0 −0.728288 3.34840 0 0.846010i 3.07670 0 −3.77600
1927.17 −0.448633 0 −1.79873 −3.77814 0 − 4.29901i 1.70423 0 1.69500
1927.18 −0.448633 0 −1.79873 −3.77814 0 4.29901i 1.70423 0 1.69500
1927.19 0.448633 0 −1.79873 3.77814 0 − 4.29901i −1.70423 0 1.69500
1927.20 0.448633 0 −1.79873 3.77814 0 4.29901i −1.70423 0 1.69500
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1927.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
241.b even 2 1 inner
723.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.d.f ✓ 36
3.b odd 2 1 inner 2169.2.d.f ✓ 36
241.b even 2 1 inner 2169.2.d.f ✓ 36
723.b odd 2 1 inner 2169.2.d.f ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2169.2.d.f ✓ 36 1.a even 1 1 trivial
2169.2.d.f ✓ 36 3.b odd 2 1 inner
2169.2.d.f ✓ 36 241.b even 2 1 inner
2169.2.d.f ✓ 36 723.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 31 T_{2}^{16} + 401 T_{2}^{14} - 2816 T_{2}^{12} + 11709 T_{2}^{10} - 29528 T_{2}^{8} + \cdots - 1888 \) acting on \(S_{2}^{\mathrm{new}}(2169, [\chi])\). Copy content Toggle raw display