↑

Properties

Label 231.2.e.a
Level $231$
Weight $2$
Character orbit 231.e
Analytic conductor $1.845$
Analytic rank $0$
Dimension $28$
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] = [231,2,Mod(188,231)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("231.188");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.e (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: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 32 q^{4} - 8 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 32 q^{4} - 8 q^{7} - 8 q^{9} - 20 q^{15} + 40 q^{16} - 12 q^{18} - 10 q^{21} + 36 q^{25} + 12 q^{28} - 4 q^{30} + 24 q^{36} - 24 q^{37} + 16 q^{39} - 40 q^{43} - 16 q^{46} + 4 q^{49} - 8 q^{51} - 4 q^{57} - 44 q^{58} + 52 q^{60} + 6 q^{63} - 68 q^{64} + 40 q^{67} + 20 q^{70} + 24 q^{72} - 28 q^{78} + 56 q^{79} + 32 q^{81} + 100 q^{84} - 8 q^{85} + 12 q^{88} + 8 q^{91} - 36 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
188.1 − 2.73839i −0.280799 − 1.70914i −5.49880 2.40492 −4.68029 + 0.768937i −1.28455 − 2.31299i 9.58108i −2.84230 + 0.959847i − 6.58561i
188.2 − 2.73839i 0.280799 + 1.70914i −5.49880 −2.40492 4.68029 − 0.768937i −1.28455 + 2.31299i 9.58108i −2.84230 + 0.959847i 6.58561i
188.3 − 2.33290i −1.41254 + 1.00236i −3.44244 4.22171 2.33841 + 3.29532i 2.13159 + 1.56727i 3.36508i 0.990548 − 2.83175i − 9.84884i
188.4 − 2.33290i 1.41254 − 1.00236i −3.44244 −4.22171 −2.33841 − 3.29532i 2.13159 − 1.56727i 3.36508i 0.990548 − 2.83175i 9.84884i
188.5 − 2.06094i −1.71028 − 0.273753i −2.24749 −1.49849 −0.564190 + 3.52479i −2.62644 + 0.319041i 0.510055i 2.85012 + 0.936390i 3.08830i
188.6 − 2.06094i 1.71028 + 0.273753i −2.24749 1.49849 0.564190 − 3.52479i −2.62644 − 0.319041i 0.510055i 2.85012 + 0.936390i − 3.08830i
188.7 − 1.68989i −1.02930 + 1.39303i −0.855732 −1.45819 2.35408 + 1.73940i 0.366782 − 2.62020i − 1.93369i −0.881093 − 2.86770i 2.46419i
188.8 − 1.68989i 1.02930 − 1.39303i −0.855732 1.45819 −2.35408 − 1.73940i 0.366782 + 2.62020i − 1.93369i −0.881093 − 2.86770i − 2.46419i
188.9 − 1.12088i −0.799601 − 1.53644i 0.743632 0.911250 −1.72216 + 0.896255i 2.18571 − 1.49086i − 3.07528i −1.72128 + 2.45707i − 1.02140i
188.10 − 1.12088i 0.799601 + 1.53644i 0.743632 −0.911250 1.72216 − 0.896255i 2.18571 + 1.49086i − 3.07528i −1.72128 + 2.45707i 1.02140i
188.11 − 0.776672i −0.233960 − 1.71618i 1.39678 −3.58495 −1.33291 + 0.181710i −2.64177 + 0.145081i − 2.63818i −2.89053 + 0.803035i 2.78433i
188.12 − 0.776672i 0.233960 + 1.71618i 1.39678 3.58495 1.33291 − 0.181710i −2.64177 − 0.145081i − 2.63818i −2.89053 + 0.803035i − 2.78433i
188.13 − 0.309769i −1.65749 − 0.502726i 1.90404 1.52955 −0.155729 + 0.513438i −0.131314 + 2.64249i − 1.20935i 2.49453 + 1.66652i − 0.473807i
188.14 − 0.309769i 1.65749 + 0.502726i 1.90404 −1.52955 0.155729 − 0.513438i −0.131314 − 2.64249i − 1.20935i 2.49453 + 1.66652i 0.473807i
188.15 0.309769i −1.65749 + 0.502726i 1.90404 1.52955 −0.155729 − 0.513438i −0.131314 − 2.64249i 1.20935i 2.49453 − 1.66652i 0.473807i
188.16 0.309769i 1.65749 − 0.502726i 1.90404 −1.52955 0.155729 + 0.513438i −0.131314 + 2.64249i 1.20935i 2.49453 − 1.66652i − 0.473807i
188.17 0.776672i −0.233960 + 1.71618i 1.39678 −3.58495 −1.33291 − 0.181710i −2.64177 − 0.145081i 2.63818i −2.89053 − 0.803035i − 2.78433i
188.18 0.776672i 0.233960 − 1.71618i 1.39678 3.58495 1.33291 + 0.181710i −2.64177 + 0.145081i 2.63818i −2.89053 − 0.803035i 2.78433i
188.19 1.12088i −0.799601 + 1.53644i 0.743632 0.911250 −1.72216 − 0.896255i 2.18571 + 1.49086i 3.07528i −1.72128 − 2.45707i 1.02140i
188.20 1.12088i 0.799601 − 1.53644i 0.743632 −0.911250 1.72216 + 0.896255i 2.18571 − 1.49086i 3.07528i −1.72128 − 2.45707i − 1.02140i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 188.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c 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.e.a ✓ 28
3.b odd 2 1 inner 231.2.e.a ✓ 28
7.b odd 2 1 inner 231.2.e.a ✓ 28
21.c even 2 1 inner 231.2.e.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.e.a ✓ 28 1.a even 1 1 trivial
231.2.e.a ✓ 28 3.b odd 2 1 inner
231.2.e.a ✓ 28 7.b odd 2 1 inner
231.2.e.a ✓ 28 21.c even 2 1 inner

Hecke kernels

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