Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4010,2,Mod(1,4010)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4010, 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("4010.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4010 = 2 \cdot 5 \cdot 401 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4010.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: | \(32.0200112105\) |
Analytic rank: | \(0\) |
Dimension: | \(22\) |
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 | −3.32876 | 1.00000 | 1.00000 | −3.32876 | 2.40644 | 1.00000 | 8.08065 | 1.00000 | ||||||||||||||||||
1.2 | 1.00000 | −3.24170 | 1.00000 | 1.00000 | −3.24170 | 1.32182 | 1.00000 | 7.50859 | 1.00000 | ||||||||||||||||||
1.3 | 1.00000 | −3.04466 | 1.00000 | 1.00000 | −3.04466 | −0.894428 | 1.00000 | 6.26997 | 1.00000 | ||||||||||||||||||
1.4 | 1.00000 | −2.94924 | 1.00000 | 1.00000 | −2.94924 | −5.23818 | 1.00000 | 5.69803 | 1.00000 | ||||||||||||||||||
1.5 | 1.00000 | −1.93387 | 1.00000 | 1.00000 | −1.93387 | 3.26149 | 1.00000 | 0.739836 | 1.00000 | ||||||||||||||||||
1.6 | 1.00000 | −1.81588 | 1.00000 | 1.00000 | −1.81588 | −3.01167 | 1.00000 | 0.297426 | 1.00000 | ||||||||||||||||||
1.7 | 1.00000 | −1.81499 | 1.00000 | 1.00000 | −1.81499 | −3.30313 | 1.00000 | 0.294193 | 1.00000 | ||||||||||||||||||
1.8 | 1.00000 | −1.38202 | 1.00000 | 1.00000 | −1.38202 | 3.56623 | 1.00000 | −1.09003 | 1.00000 | ||||||||||||||||||
1.9 | 1.00000 | −1.18240 | 1.00000 | 1.00000 | −1.18240 | −2.63299 | 1.00000 | −1.60192 | 1.00000 | ||||||||||||||||||
1.10 | 1.00000 | −0.591795 | 1.00000 | 1.00000 | −0.591795 | −0.164076 | 1.00000 | −2.64978 | 1.00000 | ||||||||||||||||||
1.11 | 1.00000 | −0.103543 | 1.00000 | 1.00000 | −0.103543 | 2.39084 | 1.00000 | −2.98928 | 1.00000 | ||||||||||||||||||
1.12 | 1.00000 | 0.470815 | 1.00000 | 1.00000 | 0.470815 | 2.26316 | 1.00000 | −2.77833 | 1.00000 | ||||||||||||||||||
1.13 | 1.00000 | 0.822612 | 1.00000 | 1.00000 | 0.822612 | 0.710831 | 1.00000 | −2.32331 | 1.00000 | ||||||||||||||||||
1.14 | 1.00000 | 0.825899 | 1.00000 | 1.00000 | 0.825899 | −3.88199 | 1.00000 | −2.31789 | 1.00000 | ||||||||||||||||||
1.15 | 1.00000 | 1.38303 | 1.00000 | 1.00000 | 1.38303 | 3.80913 | 1.00000 | −1.08722 | 1.00000 | ||||||||||||||||||
1.16 | 1.00000 | 2.15640 | 1.00000 | 1.00000 | 2.15640 | −4.76824 | 1.00000 | 1.65006 | 1.00000 | ||||||||||||||||||
1.17 | 1.00000 | 2.40993 | 1.00000 | 1.00000 | 2.40993 | 4.92260 | 1.00000 | 2.80778 | 1.00000 | ||||||||||||||||||
1.18 | 1.00000 | 2.44141 | 1.00000 | 1.00000 | 2.44141 | −0.373089 | 1.00000 | 2.96048 | 1.00000 | ||||||||||||||||||
1.19 | 1.00000 | 2.70959 | 1.00000 | 1.00000 | 2.70959 | 0.508884 | 1.00000 | 4.34189 | 1.00000 | ||||||||||||||||||
1.20 | 1.00000 | 2.88697 | 1.00000 | 1.00000 | 2.88697 | −3.00491 | 1.00000 | 5.33461 | 1.00000 | ||||||||||||||||||
See all 22 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(5\) | \(-1\) |
\(401\) | \(-1\) |
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 | 4010.2.a.n | ✓ | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4010.2.a.n | ✓ | 22 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):
\( T_{3}^{22} - T_{3}^{21} - 54 T_{3}^{20} + 54 T_{3}^{19} + 1244 T_{3}^{18} - 1222 T_{3}^{17} + \cdots - 26624 \) |
\( T_{7}^{22} - 93 T_{7}^{20} + 40 T_{7}^{19} + 3574 T_{7}^{18} - 3033 T_{7}^{17} - 73467 T_{7}^{16} + \cdots + 524288 \) |
\( T_{11}^{22} - 12 T_{11}^{21} - 75 T_{11}^{20} + 1370 T_{11}^{19} + 557 T_{11}^{18} - 62480 T_{11}^{17} + \cdots + 155582464 \) |