Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6010,2,Mod(1,6010)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6010, 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("6010.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6010 = 2 \cdot 5 \cdot 601 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6010.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: | \(47.9900916148\) |
Analytic rank: | \(1\) |
Dimension: | \(29\) |
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 | −3.35701 | 1.00000 | −1.00000 | 3.35701 | 1.93749 | −1.00000 | 8.26955 | 1.00000 | ||||||||||||||||||
1.2 | −1.00000 | −3.28561 | 1.00000 | −1.00000 | 3.28561 | −1.57080 | −1.00000 | 7.79525 | 1.00000 | ||||||||||||||||||
1.3 | −1.00000 | −3.13073 | 1.00000 | −1.00000 | 3.13073 | 1.27896 | −1.00000 | 6.80148 | 1.00000 | ||||||||||||||||||
1.4 | −1.00000 | −3.11389 | 1.00000 | −1.00000 | 3.11389 | −3.05168 | −1.00000 | 6.69634 | 1.00000 | ||||||||||||||||||
1.5 | −1.00000 | −2.56402 | 1.00000 | −1.00000 | 2.56402 | 3.60827 | −1.00000 | 3.57419 | 1.00000 | ||||||||||||||||||
1.6 | −1.00000 | −2.49423 | 1.00000 | −1.00000 | 2.49423 | −3.71754 | −1.00000 | 3.22119 | 1.00000 | ||||||||||||||||||
1.7 | −1.00000 | −2.19430 | 1.00000 | −1.00000 | 2.19430 | −4.31508 | −1.00000 | 1.81494 | 1.00000 | ||||||||||||||||||
1.8 | −1.00000 | −1.98688 | 1.00000 | −1.00000 | 1.98688 | 4.00516 | −1.00000 | 0.947682 | 1.00000 | ||||||||||||||||||
1.9 | −1.00000 | −1.83240 | 1.00000 | −1.00000 | 1.83240 | 1.88309 | −1.00000 | 0.357691 | 1.00000 | ||||||||||||||||||
1.10 | −1.00000 | −1.48145 | 1.00000 | −1.00000 | 1.48145 | 4.09967 | −1.00000 | −0.805307 | 1.00000 | ||||||||||||||||||
1.11 | −1.00000 | −1.46035 | 1.00000 | −1.00000 | 1.46035 | −4.07873 | −1.00000 | −0.867392 | 1.00000 | ||||||||||||||||||
1.12 | −1.00000 | −1.21765 | 1.00000 | −1.00000 | 1.21765 | 0.502953 | −1.00000 | −1.51732 | 1.00000 | ||||||||||||||||||
1.13 | −1.00000 | −0.796611 | 1.00000 | −1.00000 | 0.796611 | −0.409995 | −1.00000 | −2.36541 | 1.00000 | ||||||||||||||||||
1.14 | −1.00000 | −0.774267 | 1.00000 | −1.00000 | 0.774267 | −0.164549 | −1.00000 | −2.40051 | 1.00000 | ||||||||||||||||||
1.15 | −1.00000 | −0.302339 | 1.00000 | −1.00000 | 0.302339 | 2.70786 | −1.00000 | −2.90859 | 1.00000 | ||||||||||||||||||
1.16 | −1.00000 | −0.296108 | 1.00000 | −1.00000 | 0.296108 | −4.13577 | −1.00000 | −2.91232 | 1.00000 | ||||||||||||||||||
1.17 | −1.00000 | −0.0910206 | 1.00000 | −1.00000 | 0.0910206 | 3.58336 | −1.00000 | −2.99172 | 1.00000 | ||||||||||||||||||
1.18 | −1.00000 | 0.225876 | 1.00000 | −1.00000 | −0.225876 | −5.00303 | −1.00000 | −2.94898 | 1.00000 | ||||||||||||||||||
1.19 | −1.00000 | 0.876968 | 1.00000 | −1.00000 | −0.876968 | 0.528754 | −1.00000 | −2.23093 | 1.00000 | ||||||||||||||||||
1.20 | −1.00000 | 0.937103 | 1.00000 | −1.00000 | −0.937103 | −1.92879 | −1.00000 | −2.12184 | 1.00000 | ||||||||||||||||||
See all 29 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(5\) | \(1\) |
\(601\) | \(1\) |
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 | 6010.2.a.i | ✓ | 29 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6010.2.a.i | ✓ | 29 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{29} + 10 T_{3}^{28} - 8 T_{3}^{27} - 379 T_{3}^{26} - 662 T_{3}^{25} + 5897 T_{3}^{24} + \cdots + 5392 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6010))\).