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Properties

Label 3.69.b.a
Level $3$
Weight $69$
Character orbit 3.b
Analytic conductor $87.852$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3,69,Mod(2,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 69, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 69);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 69 \)
Character orbit: \([\chi]\) \(=\) 3.b (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: \(87.8517980619\)
Analytic rank: \(0\)
Dimension: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 18\!\cdots\!78 q^{3}+ \cdots + 84\!\cdots\!78 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 18\!\cdots\!78 q^{3}+ \cdots - 11\!\cdots\!00 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 − 3.12148e10i −1.31335e16 − 1.02781e16i −6.79217e20 1.89531e23i −3.20828e26 + 4.09961e26i 1.94003e27 1.19887e31i 6.68516e31 + 2.69975e32i 5.91618e33
2.2 − 3.11028e10i 9.90505e15 + 1.34171e16i −6.72234e20 − 1.00008e24i 4.17309e26 − 3.08075e26i −7.71928e28 1.17284e31i −8.19083e31 + 2.65794e32i −3.11052e34
2.3 − 2.79362e10i 1.66655e16 + 6.25230e14i −4.85282e20 1.09211e24i 1.74665e25 − 4.65569e26i 3.32586e27 5.31162e30i 2.77347e32 + 2.08395e31i 3.05093e34
2.4 − 2.53895e10i −5.69070e15 + 1.56762e16i −3.49477e20 2.74876e23i 3.98011e26 + 1.44484e26i 9.96672e28 1.37939e30i −2.13360e32 − 1.78417e32i 6.97895e33
2.5 − 2.34876e10i 6.92552e15 − 1.51712e16i −2.56517e20 − 4.36463e23i −3.56335e26 − 1.62663e26i 1.55709e28 − 9.07342e29i −1.82203e32 − 2.10137e32i −1.02514e34
2.6 − 1.98633e10i −1.52299e16 + 6.79538e15i −9.94018e19 − 1.15188e23i 1.34979e26 + 3.02517e26i −6.02651e28 − 3.88816e30i 1.85774e32 − 2.06987e32i −2.28800e33
2.7 − 1.12696e10i 1.62840e16 − 3.59996e15i 1.68144e20 − 1.29685e23i −4.05701e25 − 1.83514e26i −3.11566e28 − 5.22111e30i 2.52209e32 − 1.17244e32i −1.46150e33
2.8 − 9.15147e9i 1.20188e16 + 1.15619e16i 2.11398e20 − 4.85869e23i 1.05808e26 − 1.09989e26i 8.15893e28 − 4.63565e30i 1.07731e31 + 2.77920e32i −4.44641e33
2.9 − 9.08296e9i −6.37287e15 − 1.54115e16i 2.12648e20 8.19633e23i −1.39982e26 + 5.78846e25i −1.46000e28 − 4.61229e30i −1.96901e32 + 1.96431e32i 7.44469e33
2.10 − 8.19205e9i 3.18264e15 + 1.63707e16i 2.28038e20 5.93220e23i 1.34109e26 − 2.60723e25i −6.83050e28 − 4.28597e30i −2.57870e32 + 1.04204e32i 4.85968e33
2.11 − 6.69746e9i −1.51595e16 − 6.95107e15i 2.50292e20 − 7.66457e23i −4.65545e25 + 1.01530e26i 4.76233e28 − 3.65306e30i 1.81494e32 + 2.10750e32i −5.13331e33
2.12 6.69746e9i −1.51595e16 + 6.95107e15i 2.50292e20 7.66457e23i −4.65545e25 − 1.01530e26i 4.76233e28 3.65306e30i 1.81494e32 − 2.10750e32i −5.13331e33
2.13 8.19205e9i 3.18264e15 − 1.63707e16i 2.28038e20 − 5.93220e23i 1.34109e26 + 2.60723e25i −6.83050e28 4.28597e30i −2.57870e32 − 1.04204e32i 4.85968e33
2.14 9.08296e9i −6.37287e15 + 1.54115e16i 2.12648e20 − 8.19633e23i −1.39982e26 − 5.78846e25i −1.46000e28 4.61229e30i −1.96901e32 − 1.96431e32i 7.44469e33
2.15 9.15147e9i 1.20188e16 − 1.15619e16i 2.11398e20 4.85869e23i 1.05808e26 + 1.09989e26i 8.15893e28 4.63565e30i 1.07731e31 − 2.77920e32i −4.44641e33
2.16 1.12696e10i 1.62840e16 + 3.59996e15i 1.68144e20 1.29685e23i −4.05701e25 + 1.83514e26i −3.11566e28 5.22111e30i 2.52209e32 + 1.17244e32i −1.46150e33
2.17 1.98633e10i −1.52299e16 − 6.79538e15i −9.94018e19 1.15188e23i 1.34979e26 − 3.02517e26i −6.02651e28 3.88816e30i 1.85774e32 + 2.06987e32i −2.28800e33
2.18 2.34876e10i 6.92552e15 + 1.51712e16i −2.56517e20 4.36463e23i −3.56335e26 + 1.62663e26i 1.55709e28 9.07342e29i −1.82203e32 + 2.10137e32i −1.02514e34
2.19 2.53895e10i −5.69070e15 − 1.56762e16i −3.49477e20 − 2.74876e23i 3.98011e26 − 1.44484e26i 9.96672e28 − 1.37939e30i −2.13360e32 + 1.78417e32i 6.97895e33
2.20 2.79362e10i 1.66655e16 − 6.25230e14i −4.85282e20 − 1.09211e24i 1.74665e25 + 4.65569e26i 3.32586e27 − 5.31162e30i 2.77347e32 − 2.08395e31i 3.05093e34
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.22
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.69.b.a ✓ 22
3.b odd 2 1 inner 3.69.b.a ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.69.b.a ✓ 22 1.a even 1 1 trivial
3.69.b.a ✓ 22 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{69}^{\mathrm{new}}(3, [\chi])\).