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Properties

Label 693.2.bs.b
Level $693$
Weight $2$
Character orbit 693.bs
Analytic conductor $5.534$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [693,2,Mod(125,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.125");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.bs (of order \(10\), degree \(4\), 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: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 16 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 16 q^{4} - 8 q^{7} + 120 q^{22} - 48 q^{25} + 72 q^{28} - 32 q^{37} - 16 q^{43} + 64 q^{46} + 28 q^{49} - 48 q^{58} + 184 q^{64} - 144 q^{67} - 76 q^{70} - 24 q^{79} + 64 q^{85} + 264 q^{88} - 104 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1 −1.54066 + 2.12053i 0 −1.50500 − 4.63192i −2.82290 + 2.05095i 0 2.63468 + 0.241804i 7.15516 + 2.32485i 0 − 9.14586i
125.2 −1.54066 + 2.12053i 0 −1.50500 − 4.63192i 2.82290 − 2.05095i 0 −1.98937 + 1.74425i 7.15516 + 2.32485i 0 9.14586i
125.3 −1.15301 + 1.58698i 0 −0.571036 − 1.75747i −2.30863 + 1.67732i 0 −1.13082 + 2.39191i −0.283737 − 0.0921918i 0 − 5.59770i
125.4 −1.15301 + 1.58698i 0 −0.571036 − 1.75747i 2.30863 − 1.67732i 0 2.32078 + 1.27042i −0.283737 − 0.0921918i 0 5.59770i
125.5 −0.982869 + 1.35280i 0 −0.246010 − 0.757142i −3.21914 + 2.33884i 0 −0.667590 − 2.56014i −1.91457 − 0.622081i 0 − 6.65363i
125.6 −0.982869 + 1.35280i 0 −0.246010 − 0.757142i 3.21914 − 2.33884i 0 −0.964722 − 2.46360i −1.91457 − 0.622081i 0 6.65363i
125.7 −0.617328 + 0.849679i 0 0.277174 + 0.853053i −1.31636 + 0.956388i 0 1.37465 − 2.26060i −2.89364 − 0.940201i 0 − 1.70888i
125.8 −0.617328 + 0.849679i 0 0.277174 + 0.853053i 1.31636 − 0.956388i 0 −2.44087 − 1.02087i −2.89364 − 0.940201i 0 1.70888i
125.9 −0.499650 + 0.687709i 0 0.394740 + 1.21489i −0.731293 + 0.531315i 0 2.06797 − 1.65030i −2.64962 − 0.860914i 0 − 0.768389i
125.10 −0.499650 + 0.687709i 0 0.394740 + 1.21489i 0.731293 − 0.531315i 0 −2.64305 − 0.119600i −2.64962 − 0.860914i 0 0.768389i
125.11 −0.477539 + 0.657276i 0 0.414066 + 1.27436i −3.32153 + 2.41323i 0 −2.20500 + 1.46218i −2.58069 − 0.838516i 0 − 3.33557i
125.12 −0.477539 + 0.657276i 0 0.414066 + 1.27436i 3.32153 − 2.41323i 0 2.64333 − 0.113137i −2.58069 − 0.838516i 0 3.33557i
125.13 0.477539 − 0.657276i 0 0.414066 + 1.27436i −3.32153 + 2.41323i 0 2.64333 − 0.113137i 2.58069 + 0.838516i 0 3.33557i
125.14 0.477539 − 0.657276i 0 0.414066 + 1.27436i 3.32153 − 2.41323i 0 −2.20500 + 1.46218i 2.58069 + 0.838516i 0 − 3.33557i
125.15 0.499650 − 0.687709i 0 0.394740 + 1.21489i −0.731293 + 0.531315i 0 −2.64305 − 0.119600i 2.64962 + 0.860914i 0 0.768389i
125.16 0.499650 − 0.687709i 0 0.394740 + 1.21489i 0.731293 − 0.531315i 0 2.06797 − 1.65030i 2.64962 + 0.860914i 0 − 0.768389i
125.17 0.617328 − 0.849679i 0 0.277174 + 0.853053i −1.31636 + 0.956388i 0 −2.44087 − 1.02087i 2.89364 + 0.940201i 0 1.70888i
125.18 0.617328 − 0.849679i 0 0.277174 + 0.853053i 1.31636 − 0.956388i 0 1.37465 − 2.26060i 2.89364 + 0.940201i 0 − 1.70888i
125.19 0.982869 − 1.35280i 0 −0.246010 − 0.757142i −3.21914 + 2.33884i 0 −0.964722 − 2.46360i 1.91457 + 0.622081i 0 6.65363i
125.20 0.982869 − 1.35280i 0 −0.246010 − 0.757142i 3.21914 − 2.33884i 0 −0.667590 − 2.56014i 1.91457 + 0.622081i 0 − 6.65363i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.24
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
11.c even 5 1 inner
21.c even 2 1 inner
33.h odd 10 1 inner
77.j odd 10 1 inner
231.u even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.bs.b ✓ 96
3.b odd 2 1 inner 693.2.bs.b ✓ 96
7.b odd 2 1 inner 693.2.bs.b ✓ 96
11.c even 5 1 inner 693.2.bs.b ✓ 96
21.c even 2 1 inner 693.2.bs.b ✓ 96
33.h odd 10 1 inner 693.2.bs.b ✓ 96
77.j odd 10 1 inner 693.2.bs.b ✓ 96
231.u even 10 1 inner 693.2.bs.b ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.bs.b ✓ 96 1.a even 1 1 trivial
693.2.bs.b ✓ 96 3.b odd 2 1 inner
693.2.bs.b ✓ 96 7.b odd 2 1 inner
693.2.bs.b ✓ 96 11.c even 5 1 inner
693.2.bs.b ✓ 96 21.c even 2 1 inner
693.2.bs.b ✓ 96 33.h odd 10 1 inner
693.2.bs.b ✓ 96 77.j odd 10 1 inner
693.2.bs.b ✓ 96 231.u even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 16 T_{2}^{46} + 164 T_{2}^{44} - 1402 T_{2}^{42} + 10676 T_{2}^{40} - 62792 T_{2}^{38} + \cdots + 14641 \) acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\). Copy content Toggle raw display