↑

Properties

Label 222.3.d.a
Level $222$
Weight $3$
Character orbit 222.d
Analytic conductor $6.049$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [222,3,Mod(221,222)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(222, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("222.221");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 222 = 2 \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 222.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: \(6.04906186880\)
Analytic rank: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{3} + 48 q^{4} + 8 q^{7} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{3} + 48 q^{4} + 8 q^{7} - 20 q^{9} - 16 q^{10} - 8 q^{12} + 96 q^{16} + 12 q^{21} + 24 q^{25} - 16 q^{27} + 16 q^{28} + 48 q^{30} + 20 q^{33} + 64 q^{34} - 40 q^{36} - 120 q^{37} - 32 q^{40} - 144 q^{46} - 16 q^{48} + 48 q^{49} + 32 q^{58} + 308 q^{63} + 192 q^{64} - 672 q^{67} + 112 q^{70} - 8 q^{73} - 252 q^{75} - 256 q^{78} - 116 q^{81} + 24 q^{84} - 272 q^{85} - 128 q^{90} - 236 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1 −1.41421 −2.76313 − 1.16837i 2.00000 −1.31297 3.90766 + 1.65232i 3.89381 −2.82843 6.26982 + 6.45673i 1.85683
221.2 −1.41421 −2.76313 + 1.16837i 2.00000 −1.31297 3.90766 − 1.65232i 3.89381 −2.82843 6.26982 − 6.45673i 1.85683
221.3 −1.41421 −2.37052 − 1.83866i 2.00000 7.01121 3.35241 + 2.60025i −0.574284 −2.82843 2.23868 + 8.71713i −9.91535
221.4 −1.41421 −2.37052 + 1.83866i 2.00000 7.01121 3.35241 − 2.60025i −0.574284 −2.82843 2.23868 − 8.71713i −9.91535
221.5 −1.41421 −0.462568 − 2.96412i 2.00000 −7.97168 0.654171 + 4.19190i 3.16343 −2.82843 −8.57206 + 2.74222i 11.2737
221.6 −1.41421 −0.462568 + 2.96412i 2.00000 −7.97168 0.654171 − 4.19190i 3.16343 −2.82843 −8.57206 − 2.74222i 11.2737
221.7 −1.41421 −0.0470154 − 2.99963i 2.00000 3.54197 0.0664898 + 4.24212i −12.3309 −2.82843 −8.99558 + 0.282058i −5.00910
221.8 −1.41421 −0.0470154 + 2.99963i 2.00000 3.54197 0.0664898 − 4.24212i −12.3309 −2.82843 −8.99558 − 0.282058i −5.00910
221.9 −1.41421 1.96824 − 2.26408i 2.00000 4.50884 −2.78351 + 3.20189i 10.9116 −2.82843 −1.25210 − 8.91248i −6.37646
221.10 −1.41421 1.96824 + 2.26408i 2.00000 4.50884 −2.78351 − 3.20189i 10.9116 −2.82843 −1.25210 + 8.91248i −6.37646
221.11 −1.41421 2.67500 − 1.35808i 2.00000 −2.94894 −3.78302 + 1.92062i −3.06368 −2.82843 5.31123 − 7.26573i 4.17043
221.12 −1.41421 2.67500 + 1.35808i 2.00000 −2.94894 −3.78302 − 1.92062i −3.06368 −2.82843 5.31123 + 7.26573i 4.17043
221.13 1.41421 −2.76313 − 1.16837i 2.00000 1.31297 −3.90766 − 1.65232i 3.89381 2.82843 6.26982 + 6.45673i 1.85683
221.14 1.41421 −2.76313 + 1.16837i 2.00000 1.31297 −3.90766 + 1.65232i 3.89381 2.82843 6.26982 − 6.45673i 1.85683
221.15 1.41421 −2.37052 − 1.83866i 2.00000 −7.01121 −3.35241 − 2.60025i −0.574284 2.82843 2.23868 + 8.71713i −9.91535
221.16 1.41421 −2.37052 + 1.83866i 2.00000 −7.01121 −3.35241 + 2.60025i −0.574284 2.82843 2.23868 − 8.71713i −9.91535
221.17 1.41421 −0.462568 − 2.96412i 2.00000 7.97168 −0.654171 − 4.19190i 3.16343 2.82843 −8.57206 + 2.74222i 11.2737
221.18 1.41421 −0.462568 + 2.96412i 2.00000 7.97168 −0.654171 + 4.19190i 3.16343 2.82843 −8.57206 − 2.74222i 11.2737
221.19 1.41421 −0.0470154 − 2.99963i 2.00000 −3.54197 −0.0664898 − 4.24212i −12.3309 2.82843 −8.99558 + 0.282058i −5.00910
221.20 1.41421 −0.0470154 + 2.99963i 2.00000 −3.54197 −0.0664898 + 4.24212i −12.3309 2.82843 −8.99558 − 0.282058i −5.00910
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 221.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 222.3.d.a ✓ 24
3.b odd 2 1 inner 222.3.d.a ✓ 24
37.b even 2 1 inner 222.3.d.a ✓ 24
111.d odd 2 1 inner 222.3.d.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.3.d.a ✓ 24 1.a even 1 1 trivial
222.3.d.a ✓ 24 3.b odd 2 1 inner
222.3.d.a ✓ 24 37.b even 2 1 inner
222.3.d.a ✓ 24 111.d odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(222, [\chi])\).