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Properties

Label 231.2.g.a
Level $231$
Weight $2$
Character orbit 231.g
Analytic conductor $1.845$
Analytic rank $0$
Dimension $24$
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] = [231,2,Mod(197,231)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(231, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("231.197");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.g (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: \(1.84454428669\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{3} + 24 q^{4} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 6 q^{3} + 24 q^{4} + 10 q^{9} - 20 q^{12} - 10 q^{15} - 8 q^{16} - 12 q^{25} - 20 q^{31} + 14 q^{33} - 8 q^{34} - 12 q^{36} + 4 q^{37} + 6 q^{45} - 48 q^{48} - 24 q^{49} - 28 q^{55} + 44 q^{58} + 32 q^{60} - 52 q^{64} + 12 q^{66} - 4 q^{67} + 54 q^{69} - 20 q^{70} + 68 q^{75} - 20 q^{78} + 2 q^{81} + 16 q^{82} - 44 q^{88} + 24 q^{91} + 26 q^{93} - 12 q^{97} - 6 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1 −2.45155 −1.72491 − 0.157123i 4.01010 − 3.63499i 4.22870 + 0.385196i 1.00000i −4.92788 2.95062 + 0.542048i 8.91135i
197.2 −2.45155 −1.72491 + 0.157123i 4.01010 3.63499i 4.22870 − 0.385196i − 1.00000i −4.92788 2.95062 − 0.542048i − 8.91135i
197.3 −2.30127 −0.220342 − 1.71798i 3.29586 1.52521i 0.507068 + 3.95354i 1.00000i −2.98213 −2.90290 + 0.757087i − 3.50993i
197.4 −2.30127 −0.220342 + 1.71798i 3.29586 − 1.52521i 0.507068 − 3.95354i − 1.00000i −2.98213 −2.90290 − 0.757087i 3.50993i
197.5 −1.92194 1.23500 − 1.21441i 1.69384 0.413489i −2.37358 + 2.33402i − 1.00000i 0.588415 0.0504282 − 2.99958i − 0.794699i
197.6 −1.92194 1.23500 + 1.21441i 1.69384 − 0.413489i −2.37358 − 2.33402i 1.00000i 0.588415 0.0504282 + 2.99958i 0.794699i
197.7 −1.36666 −0.859522 − 1.50374i −0.132249 − 3.05646i 1.17467 + 2.05509i − 1.00000i 2.91405 −1.52244 + 2.58499i 4.17714i
197.8 −1.36666 −0.859522 + 1.50374i −0.132249 3.05646i 1.17467 − 2.05509i 1.00000i 2.91405 −1.52244 − 2.58499i − 4.17714i
197.9 −0.858468 1.60964 − 0.639565i −1.26303 − 2.44714i −1.38183 + 0.549047i 1.00000i 2.80121 2.18191 − 2.05895i 2.10079i
197.10 −0.858468 1.60964 + 0.639565i −1.26303 2.44714i −1.38183 − 0.549047i − 1.00000i 2.80121 2.18191 + 2.05895i − 2.10079i
197.11 −0.628865 −1.53987 − 0.792976i −1.60453 1.39971i 0.968369 + 0.498675i − 1.00000i 2.26676 1.74238 + 2.44216i − 0.880226i
197.12 −0.628865 −1.53987 + 0.792976i −1.60453 − 1.39971i 0.968369 − 0.498675i 1.00000i 2.26676 1.74238 − 2.44216i 0.880226i
197.13 0.628865 −1.53987 − 0.792976i −1.60453 1.39971i −0.968369 − 0.498675i 1.00000i −2.26676 1.74238 + 2.44216i 0.880226i
197.14 0.628865 −1.53987 + 0.792976i −1.60453 − 1.39971i −0.968369 + 0.498675i − 1.00000i −2.26676 1.74238 − 2.44216i − 0.880226i
197.15 0.858468 1.60964 − 0.639565i −1.26303 − 2.44714i 1.38183 − 0.549047i − 1.00000i −2.80121 2.18191 − 2.05895i − 2.10079i
197.16 0.858468 1.60964 + 0.639565i −1.26303 2.44714i 1.38183 + 0.549047i 1.00000i −2.80121 2.18191 + 2.05895i 2.10079i
197.17 1.36666 −0.859522 − 1.50374i −0.132249 − 3.05646i −1.17467 − 2.05509i 1.00000i −2.91405 −1.52244 + 2.58499i − 4.17714i
197.18 1.36666 −0.859522 + 1.50374i −0.132249 3.05646i −1.17467 + 2.05509i − 1.00000i −2.91405 −1.52244 − 2.58499i 4.17714i
197.19 1.92194 1.23500 − 1.21441i 1.69384 0.413489i 2.37358 − 2.33402i 1.00000i −0.588415 0.0504282 − 2.99958i 0.794699i
197.20 1.92194 1.23500 + 1.21441i 1.69384 − 0.413489i 2.37358 + 2.33402i − 1.00000i −0.588415 0.0504282 + 2.99958i − 0.794699i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.g.a ✓ 24
3.b odd 2 1 inner 231.2.g.a ✓ 24
11.b odd 2 1 inner 231.2.g.a ✓ 24
33.d even 2 1 inner 231.2.g.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.g.a ✓ 24 1.a even 1 1 trivial
231.2.g.a ✓ 24 3.b odd 2 1 inner
231.2.g.a ✓ 24 11.b odd 2 1 inner
231.2.g.a ✓ 24 33.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(231, [\chi])\).