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Properties

Label 693.2.e.a
Level $693$
Weight $2$
Character orbit 693.e
Analytic conductor $5.534$
Analytic rank $0$
Dimension $24$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [693,2,Mod(188,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, 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("693.188");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.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: \(5.53363286007\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 16 q^{4} + 8 q^{7} + 8 q^{25} + 8 q^{28} + 32 q^{37} + 16 q^{43} - 64 q^{46} - 48 q^{49} + 88 q^{58} + 56 q^{64} - 16 q^{67} - 64 q^{70} - 96 q^{79} + 16 q^{85} - 24 q^{88} - 16 q^{91}+O(q^{100}) \) 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 − 2.52403i 0 −4.37071 −0.561855 0 0.401306 + 2.61514i 5.98373i 0 1.41814i
188.2 − 2.52403i 0 −4.37071 0.561855 0 0.401306 − 2.61514i 5.98373i 0 − 1.41814i
188.3 − 2.05962i 0 −2.24205 −1.75955 0 −1.33464 − 2.28446i 0.498535i 0 3.62401i
188.4 − 2.05962i 0 −2.24205 1.75955 0 −1.33464 + 2.28446i 0.498535i 0 − 3.62401i
188.5 − 1.68283i 0 −0.831927 −2.60272 0 2.07110 + 1.64638i − 1.96567i 0 4.37995i
188.6 − 1.68283i 0 −0.831927 2.60272 0 2.07110 − 1.64638i − 1.96567i 0 − 4.37995i
188.7 − 1.52101i 0 −0.313458 −3.64105 0 −0.560972 + 2.58560i − 2.56524i 0 5.53805i
188.8 − 1.52101i 0 −0.313458 3.64105 0 −0.560972 − 2.58560i − 2.56524i 0 − 5.53805i
188.9 − 0.464402i 0 1.78433 −2.02559 0 2.64044 − 0.167598i − 1.75745i 0 0.940689i
188.10 − 0.464402i 0 1.78433 2.02559 0 2.64044 + 0.167598i − 1.75745i 0 − 0.940689i
188.11 − 0.161828i 0 1.97381 −2.11043 0 −1.21723 − 2.34912i − 0.643072i 0 0.341526i
188.12 − 0.161828i 0 1.97381 2.11043 0 −1.21723 + 2.34912i − 0.643072i 0 − 0.341526i
188.13 0.161828i 0 1.97381 −2.11043 0 −1.21723 + 2.34912i 0.643072i 0 − 0.341526i
188.14 0.161828i 0 1.97381 2.11043 0 −1.21723 − 2.34912i 0.643072i 0 0.341526i
188.15 0.464402i 0 1.78433 −2.02559 0 2.64044 + 0.167598i 1.75745i 0 − 0.940689i
188.16 0.464402i 0 1.78433 2.02559 0 2.64044 − 0.167598i 1.75745i 0 0.940689i
188.17 1.52101i 0 −0.313458 −3.64105 0 −0.560972 − 2.58560i 2.56524i 0 − 5.53805i
188.18 1.52101i 0 −0.313458 3.64105 0 −0.560972 + 2.58560i 2.56524i 0 5.53805i
188.19 1.68283i 0 −0.831927 −2.60272 0 2.07110 − 1.64638i 1.96567i 0 − 4.37995i
188.20 1.68283i 0 −0.831927 2.60272 0 2.07110 + 1.64638i 1.96567i 0 4.37995i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 188.24
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 693.2.e.a ✓ 24
3.b odd 2 1 inner 693.2.e.a ✓ 24
7.b odd 2 1 inner 693.2.e.a ✓ 24
21.c even 2 1 inner 693.2.e.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.e.a ✓ 24 1.a even 1 1 trivial
693.2.e.a ✓ 24 3.b odd 2 1 inner
693.2.e.a ✓ 24 7.b odd 2 1 inner
693.2.e.a ✓ 24 21.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(693, [\chi])\).