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Properties

Label 693.2.g.a
Level $693$
Weight $2$
Character orbit 693.g
Analytic conductor $5.534$
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] = [693,2,Mod(197,693)]
 
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, 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("693.197");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.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: \(5.53363286007\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{4} + 64 q^{16} - 12 q^{22} - 24 q^{25} + 16 q^{31} - 32 q^{34} - 32 q^{37} - 24 q^{49} + 56 q^{55} - 112 q^{58} + 128 q^{64} + 80 q^{67} + 40 q^{70} + 16 q^{82} - 68 q^{88} - 48 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1 −2.74000 0 5.50760 − 2.49928i 0 − 1.00000i −9.61081 0 6.84802i
197.2 −2.74000 0 5.50760 2.49928i 0 1.00000i −9.61081 0 − 6.84802i
197.3 −2.42725 0 3.89157 − 1.44013i 0 − 1.00000i −4.59131 0 3.49556i
197.4 −2.42725 0 3.89157 1.44013i 0 1.00000i −4.59131 0 − 3.49556i
197.5 −1.62230 0 0.631871 − 3.18330i 0 1.00000i 2.21952 0 5.16427i
197.6 −1.62230 0 0.631871 3.18330i 0 − 1.00000i 2.21952 0 − 5.16427i
197.7 −1.08535 0 −0.822011 − 4.12615i 0 − 1.00000i 3.06288 0 4.47832i
197.8 −1.08535 0 −0.822011 4.12615i 0 1.00000i 3.06288 0 − 4.47832i
197.9 −0.884110 0 −1.21835 − 0.449010i 0 − 1.00000i 2.84537 0 0.396974i
197.10 −0.884110 0 −1.21835 0.449010i 0 1.00000i 2.84537 0 − 0.396974i
197.11 −0.0965884 0 −1.99067 − 0.565312i 0 1.00000i 0.385453 0 0.0546026i
197.12 −0.0965884 0 −1.99067 0.565312i 0 − 1.00000i 0.385453 0 − 0.0546026i
197.13 0.0965884 0 −1.99067 − 0.565312i 0 − 1.00000i −0.385453 0 − 0.0546026i
197.14 0.0965884 0 −1.99067 0.565312i 0 1.00000i −0.385453 0 0.0546026i
197.15 0.884110 0 −1.21835 − 0.449010i 0 1.00000i −2.84537 0 − 0.396974i
197.16 0.884110 0 −1.21835 0.449010i 0 − 1.00000i −2.84537 0 0.396974i
197.17 1.08535 0 −0.822011 − 4.12615i 0 1.00000i −3.06288 0 − 4.47832i
197.18 1.08535 0 −0.822011 4.12615i 0 − 1.00000i −3.06288 0 4.47832i
197.19 1.62230 0 0.631871 − 3.18330i 0 − 1.00000i −2.21952 0 − 5.16427i
197.20 1.62230 0 0.631871 3.18330i 0 1.00000i −2.21952 0 5.16427i
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 693.2.g.a ✓ 24
3.b odd 2 1 inner 693.2.g.a ✓ 24
11.b odd 2 1 inner 693.2.g.a ✓ 24
33.d even 2 1 inner 693.2.g.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.g.a ✓ 24 1.a even 1 1 trivial
693.2.g.a ✓ 24 3.b odd 2 1 inner
693.2.g.a ✓ 24 11.b odd 2 1 inner
693.2.g.a ✓ 24 33.d even 2 1 inner

Hecke kernels

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