Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6223,2,Mod(1,6223)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6223, 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("6223.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6223 = 7^{2} \cdot 127 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6223.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: | \(49.6909051778\) |
Analytic rank: | \(1\) |
Dimension: | \(40\) |
Twist minimal: | no (minimal twist has level 889) |
Fricke sign: | \(+1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | −2.73783 | −3.09982 | 5.49572 | 2.24078 | 8.48678 | 0 | −9.57068 | 6.60889 | −6.13487 | ||||||||||||||||||
1.2 | −2.68664 | −0.656933 | 5.21805 | −1.45033 | 1.76494 | 0 | −8.64575 | −2.56844 | 3.89652 | ||||||||||||||||||
1.3 | −2.67543 | 2.21467 | 5.15792 | −3.77035 | −5.92518 | 0 | −8.44878 | 1.90474 | 10.0873 | ||||||||||||||||||
1.4 | −2.60936 | −2.27618 | 4.80875 | −0.761982 | 5.93936 | 0 | −7.32905 | 2.18098 | 1.98829 | ||||||||||||||||||
1.5 | −2.24401 | 2.80584 | 3.03558 | 1.28157 | −6.29634 | 0 | −2.32386 | 4.87274 | −2.87586 | ||||||||||||||||||
1.6 | −2.15797 | 1.79179 | 2.65682 | −0.285167 | −3.86663 | 0 | −1.41741 | 0.210520 | 0.615381 | ||||||||||||||||||
1.7 | −1.89455 | −1.68668 | 1.58933 | −0.447365 | 3.19551 | 0 | 0.778030 | −0.155104 | 0.847556 | ||||||||||||||||||
1.8 | −1.82190 | 1.06573 | 1.31931 | 1.64925 | −1.94165 | 0 | 1.24014 | −1.86422 | −3.00476 | ||||||||||||||||||
1.9 | −1.79801 | −0.144112 | 1.23286 | −4.04161 | 0.259116 | 0 | 1.37934 | −2.97923 | 7.26687 | ||||||||||||||||||
1.10 | −1.77970 | −3.07662 | 1.16735 | −3.89409 | 5.47548 | 0 | 1.48188 | 6.46560 | 6.93033 | ||||||||||||||||||
1.11 | −1.73616 | −2.58223 | 1.01424 | 1.04688 | 4.48316 | 0 | 1.71143 | 3.66791 | −1.81755 | ||||||||||||||||||
1.12 | −1.60118 | 2.10158 | 0.563789 | 3.69203 | −3.36501 | 0 | 2.29964 | 1.41662 | −5.91162 | ||||||||||||||||||
1.13 | −1.28300 | 2.26059 | −0.353920 | −1.66860 | −2.90033 | 0 | 3.02007 | 2.11025 | 2.14080 | ||||||||||||||||||
1.14 | −1.14887 | 0.105959 | −0.680100 | 0.464834 | −0.121733 | 0 | 3.07908 | −2.98877 | −0.534033 | ||||||||||||||||||
1.15 | −1.09299 | −0.661876 | −0.805363 | −0.633566 | 0.723427 | 0 | 3.06625 | −2.56192 | 0.692484 | ||||||||||||||||||
1.16 | −1.01110 | 0.720309 | −0.977683 | 2.17925 | −0.728302 | 0 | 3.01073 | −2.48115 | −2.20344 | ||||||||||||||||||
1.17 | −0.783388 | −2.52435 | −1.38630 | −3.31043 | 1.97755 | 0 | 2.65279 | 3.37237 | 2.59335 | ||||||||||||||||||
1.18 | −0.628033 | −1.21971 | −1.60557 | 2.87801 | 0.766017 | 0 | 2.26442 | −1.51232 | −1.80749 | ||||||||||||||||||
1.19 | −0.156733 | −3.33897 | −1.97543 | 0.705840 | 0.523328 | 0 | 0.623083 | 8.14873 | −0.110629 | ||||||||||||||||||
1.20 | −0.138409 | −0.525573 | −1.98084 | −2.27925 | 0.0727438 | 0 | 0.550983 | −2.72377 | 0.315468 | ||||||||||||||||||
See all 40 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(7\) | \( +1 \) |
\(127\) | \( +1 \) |
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 | 6223.2.a.q | 40 | |
7.b | odd | 2 | 1 | 6223.2.a.r | 40 | ||
7.c | even | 3 | 2 | 889.2.f.d | ✓ | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
889.2.f.d | ✓ | 80 | 7.c | even | 3 | 2 | |
6223.2.a.q | 40 | 1.a | even | 1 | 1 | trivial | |
6223.2.a.r | 40 | 7.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6223))\):
\( T_{2}^{40} + 3 T_{2}^{39} - 55 T_{2}^{38} - 167 T_{2}^{37} + 1381 T_{2}^{36} + 4255 T_{2}^{35} + \cdots + 789 \) |
\( T_{3}^{40} + 6 T_{3}^{39} - 63 T_{3}^{38} - 420 T_{3}^{37} + 1764 T_{3}^{36} + 13397 T_{3}^{35} + \cdots - 16527 \) |
\( T_{5}^{40} + 24 T_{5}^{39} + 164 T_{5}^{38} - 556 T_{5}^{37} - 12110 T_{5}^{36} - 31411 T_{5}^{35} + \cdots + 25258724 \) |