Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6014,2,Mod(1,6014)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6014, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6014.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6014 = 2 \cdot 31 \cdot 97 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6014.a (trivial) |
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: | yes |
Analytic conductor: | \(48.0220317756\) |
Analytic rank: | \(0\) |
Dimension: | \(38\) |
Twist minimal: | yes |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −1.00000 | −3.36850 | 1.00000 | 3.93449 | 3.36850 | −2.34410 | −1.00000 | 8.34681 | −3.93449 | ||||||||||||||||||
1.2 | −1.00000 | −3.27880 | 1.00000 | 0.401675 | 3.27880 | −4.68111 | −1.00000 | 7.75051 | −0.401675 | ||||||||||||||||||
1.3 | −1.00000 | −3.21269 | 1.00000 | −0.168256 | 3.21269 | 2.46391 | −1.00000 | 7.32140 | 0.168256 | ||||||||||||||||||
1.4 | −1.00000 | −3.06494 | 1.00000 | −3.85153 | 3.06494 | 4.92968 | −1.00000 | 6.39384 | 3.85153 | ||||||||||||||||||
1.5 | −1.00000 | −2.46616 | 1.00000 | −2.24161 | 2.46616 | 4.72952 | −1.00000 | 3.08193 | 2.24161 | ||||||||||||||||||
1.6 | −1.00000 | −2.41930 | 1.00000 | 0.0104516 | 2.41930 | −3.11450 | −1.00000 | 2.85300 | −0.0104516 | ||||||||||||||||||
1.7 | −1.00000 | −2.33846 | 1.00000 | 2.68670 | 2.33846 | 5.16448 | −1.00000 | 2.46841 | −2.68670 | ||||||||||||||||||
1.8 | −1.00000 | −2.25991 | 1.00000 | 3.07445 | 2.25991 | −1.73147 | −1.00000 | 2.10720 | −3.07445 | ||||||||||||||||||
1.9 | −1.00000 | −2.22677 | 1.00000 | −4.13450 | 2.22677 | −4.78775 | −1.00000 | 1.95850 | 4.13450 | ||||||||||||||||||
1.10 | −1.00000 | −2.20978 | 1.00000 | −2.38189 | 2.20978 | −2.04649 | −1.00000 | 1.88314 | 2.38189 | ||||||||||||||||||
1.11 | −1.00000 | −1.71746 | 1.00000 | −1.04814 | 1.71746 | 0.0655046 | −1.00000 | −0.0503220 | 1.04814 | ||||||||||||||||||
1.12 | −1.00000 | −1.56405 | 1.00000 | 2.83927 | 1.56405 | 2.44268 | −1.00000 | −0.553753 | −2.83927 | ||||||||||||||||||
1.13 | −1.00000 | −1.31913 | 1.00000 | 1.08741 | 1.31913 | −0.837035 | −1.00000 | −1.25989 | −1.08741 | ||||||||||||||||||
1.14 | −1.00000 | −1.26759 | 1.00000 | −3.06753 | 1.26759 | 1.65725 | −1.00000 | −1.39322 | 3.06753 | ||||||||||||||||||
1.15 | −1.00000 | −1.26563 | 1.00000 | 1.94989 | 1.26563 | −2.66768 | −1.00000 | −1.39817 | −1.94989 | ||||||||||||||||||
1.16 | −1.00000 | −0.885413 | 1.00000 | −0.534430 | 0.885413 | 1.09138 | −1.00000 | −2.21604 | 0.534430 | ||||||||||||||||||
1.17 | −1.00000 | −0.717443 | 1.00000 | 3.51838 | 0.717443 | −3.26736 | −1.00000 | −2.48528 | −3.51838 | ||||||||||||||||||
1.18 | −1.00000 | −0.679880 | 1.00000 | −4.32143 | 0.679880 | 1.08152 | −1.00000 | −2.53776 | 4.32143 | ||||||||||||||||||
1.19 | −1.00000 | −0.181456 | 1.00000 | 3.41962 | 0.181456 | 3.19929 | −1.00000 | −2.96707 | −3.41962 | ||||||||||||||||||
1.20 | −1.00000 | −0.173832 | 1.00000 | −1.31890 | 0.173832 | −4.23696 | −1.00000 | −2.96978 | 1.31890 | ||||||||||||||||||
See all 38 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(31\) | \(-1\) |
\(97\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 6014.2.a.l | ✓ | 38 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6014.2.a.l | ✓ | 38 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{38} + 2 T_{3}^{37} - 82 T_{3}^{36} - 161 T_{3}^{35} + 3068 T_{3}^{34} + 5915 T_{3}^{33} + \cdots - 8280704 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6014))\).