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Properties

Label 3.71.b.a
Level $3$
Weight $71$
Character orbit 3.b
Analytic conductor $93.095$
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] = [3,71,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, 71, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 71);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 71 \)
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: \(93.0951693564\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 57\!\cdots\!10 q^{3}+ \cdots - 31\!\cdots\!82 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 57\!\cdots\!10 q^{3}+ \cdots - 21\!\cdots\!80 q^{99}+O(q^{100}) \) 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 − 6.63817e10i 2.36972e16 − 4.40636e16i −3.22594e21 6.32737e23i −2.92502e27 − 1.57306e27i −2.54650e29 1.35774e32i −1.38004e33 − 2.08836e33i 4.20021e34
2.2 − 6.15211e10i −4.79288e16 + 1.43522e16i −2.60426e21 − 5.00764e24i 8.82962e26 + 2.94863e27i 5.12912e29 8.75855e31i 2.09119e33 − 1.37577e33i −3.08075e35
2.3 − 5.67567e10i 2.16075e15 + 4.99849e16i −2.04073e21 2.61507e24i 2.83697e27 − 1.22637e26i −2.08517e29 4.88185e31i −2.49382e33 + 2.16009e32i 1.48422e35
2.4 − 4.67615e10i −4.93039e16 − 8.50161e15i −1.00605e21 4.28418e24i −3.97548e26 + 2.30553e27i −1.75590e29 − 8.16202e30i 2.35860e33 + 8.38326e32i 2.00335e35
2.5 − 4.57279e10i 4.91081e16 + 9.56812e15i −9.10451e20 − 1.29561e24i 4.37530e26 − 2.24561e27i 1.42646e29 − 1.23530e31i 2.32006e33 + 9.39745e32i −5.92457e34
2.6 − 3.74673e10i −7.56180e15 − 4.94568e16i −2.23206e20 1.07403e24i −1.85301e27 + 2.83320e26i 6.73954e29 − 3.58706e31i −2.38879e33 + 7.47965e32i 4.02409e34
2.7 − 3.59410e10i −2.70537e16 − 4.20862e16i −1.11161e20 − 4.52111e24i −1.51262e27 + 9.72336e26i −7.40509e29 − 3.84364e31i −1.03935e33 + 2.27718e33i −1.62493e35
2.8 − 2.60562e10i −2.92624e15 + 4.99459e16i 5.01667e20 − 2.60973e24i 1.30140e27 + 7.62467e25i −1.01844e29 − 4.38332e31i −2.48603e33 − 2.92308e32i −6.79996e34
2.9 − 1.54812e10i 3.84365e16 − 3.20279e16i 9.40925e20 2.49363e24i −4.95830e26 − 5.95043e26i −4.12745e29 − 3.28436e31i 4.51579e32 − 2.46209e33i 3.86044e34
2.10 − 1.12481e10i −4.22339e16 + 2.68226e16i 1.05407e21 2.30186e23i 3.01705e26 + 4.75053e26i 8.16038e28 − 2.51358e31i 1.06425e33 − 2.26565e33i 2.58917e33
2.11 − 4.22234e9i 3.47254e16 + 3.60181e16i 1.16276e21 5.48695e24i 1.52081e26 − 1.46622e26i 4.84471e29 − 9.89443e30i −9.14522e31 + 2.50148e33i 2.31678e34
2.12 4.22234e9i 3.47254e16 − 3.60181e16i 1.16276e21 − 5.48695e24i 1.52081e26 + 1.46622e26i 4.84471e29 9.89443e30i −9.14522e31 − 2.50148e33i 2.31678e34
2.13 1.12481e10i −4.22339e16 − 2.68226e16i 1.05407e21 − 2.30186e23i 3.01705e26 − 4.75053e26i 8.16038e28 2.51358e31i 1.06425e33 + 2.26565e33i 2.58917e33
2.14 1.54812e10i 3.84365e16 + 3.20279e16i 9.40925e20 − 2.49363e24i −4.95830e26 + 5.95043e26i −4.12745e29 3.28436e31i 4.51579e32 + 2.46209e33i 3.86044e34
2.15 2.60562e10i −2.92624e15 − 4.99459e16i 5.01667e20 2.60973e24i 1.30140e27 − 7.62467e25i −1.01844e29 4.38332e31i −2.48603e33 + 2.92308e32i −6.79996e34
2.16 3.59410e10i −2.70537e16 + 4.20862e16i −1.11161e20 4.52111e24i −1.51262e27 − 9.72336e26i −7.40509e29 3.84364e31i −1.03935e33 − 2.27718e33i −1.62493e35
2.17 3.74673e10i −7.56180e15 + 4.94568e16i −2.23206e20 − 1.07403e24i −1.85301e27 − 2.83320e26i 6.73954e29 3.58706e31i −2.38879e33 − 7.47965e32i 4.02409e34
2.18 4.57279e10i 4.91081e16 − 9.56812e15i −9.10451e20 1.29561e24i 4.37530e26 + 2.24561e27i 1.42646e29 1.23530e31i 2.32006e33 − 9.39745e32i −5.92457e34
2.19 4.67615e10i −4.93039e16 + 8.50161e15i −1.00605e21 − 4.28418e24i −3.97548e26 − 2.30553e27i −1.75590e29 8.16202e30i 2.35860e33 − 8.38326e32i 2.00335e35
2.20 5.67567e10i 2.16075e15 − 4.99849e16i −2.04073e21 − 2.61507e24i 2.83697e27 + 1.22637e26i −2.08517e29 − 4.88185e31i −2.49382e33 − 2.16009e32i 1.48422e35
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 3.71.b.a ✓ 22
3.b odd 2 1 inner 3.71.b.a ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.71.b.a ✓ 22 1.a even 1 1 trivial
3.71.b.a ✓ 22 3.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{71}^{\mathrm{new}}(3, [\chi])\).