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 | −3.30817 | 1.00000 | −1.00000 | −3.30817 | −2.76226 | 1.00000 | 7.94397 | −1.00000 | ||||||||||||||||||
1.2 | 1.00000 | −2.90289 | 1.00000 | −1.00000 | −2.90289 | 4.43766 | 1.00000 | 5.42676 | −1.00000 | ||||||||||||||||||
1.3 | 1.00000 | −2.88307 | 1.00000 | −1.00000 | −2.88307 | −0.867210 | 1.00000 | 5.31208 | −1.00000 | ||||||||||||||||||
1.4 | 1.00000 | −2.52966 | 1.00000 | −1.00000 | −2.52966 | 2.69506 | 1.00000 | 3.39917 | −1.00000 | ||||||||||||||||||
1.5 | 1.00000 | −1.85051 | 1.00000 | −1.00000 | −1.85051 | −3.61831 | 1.00000 | 0.424381 | −1.00000 | ||||||||||||||||||
1.6 | 1.00000 | −1.60919 | 1.00000 | −1.00000 | −1.60919 | 3.90377 | 1.00000 | −0.410498 | −1.00000 | ||||||||||||||||||
1.7 | 1.00000 | −1.34023 | 1.00000 | −1.00000 | −1.34023 | 0.629438 | 1.00000 | −1.20379 | −1.00000 | ||||||||||||||||||
1.8 | 1.00000 | −1.31697 | 1.00000 | −1.00000 | −1.31697 | −3.01403 | 1.00000 | −1.26559 | −1.00000 | ||||||||||||||||||
1.9 | 1.00000 | −1.18460 | 1.00000 | −1.00000 | −1.18460 | 4.68706 | 1.00000 | −1.59673 | −1.00000 | ||||||||||||||||||
1.10 | 1.00000 | −0.310341 | 1.00000 | −1.00000 | −0.310341 | 1.61370 | 1.00000 | −2.90369 | −1.00000 | ||||||||||||||||||
1.11 | 1.00000 | −0.165649 | 1.00000 | −1.00000 | −0.165649 | −1.98987 | 1.00000 | −2.97256 | −1.00000 | ||||||||||||||||||
1.12 | 1.00000 | 0.417458 | 1.00000 | −1.00000 | 0.417458 | −4.20952 | 1.00000 | −2.82573 | −1.00000 | ||||||||||||||||||
1.13 | 1.00000 | 0.529822 | 1.00000 | −1.00000 | 0.529822 | −4.65586 | 1.00000 | −2.71929 | −1.00000 | ||||||||||||||||||
1.14 | 1.00000 | 0.642301 | 1.00000 | −1.00000 | 0.642301 | 4.88072 | 1.00000 | −2.58745 | −1.00000 | ||||||||||||||||||
1.15 | 1.00000 | 1.74418 | 1.00000 | −1.00000 | 1.74418 | 3.19985 | 1.00000 | 0.0421511 | −1.00000 | ||||||||||||||||||
1.16 | 1.00000 | 1.77730 | 1.00000 | −1.00000 | 1.77730 | 1.31771 | 1.00000 | 0.158778 | −1.00000 | ||||||||||||||||||
1.17 | 1.00000 | 1.82747 | 1.00000 | −1.00000 | 1.82747 | 0.568203 | 1.00000 | 0.339635 | −1.00000 | ||||||||||||||||||
1.18 | 1.00000 | 2.45072 | 1.00000 | −1.00000 | 2.45072 | 4.38820 | 1.00000 | 3.00604 | −1.00000 | ||||||||||||||||||
1.19 | 1.00000 | 2.51296 | 1.00000 | −1.00000 | 2.51296 | −2.03791 | 1.00000 | 3.31499 | −1.00000 | ||||||||||||||||||
1.20 | 1.00000 | 3.00389 | 1.00000 | −1.00000 | 3.00389 | −1.21989 | 1.00000 | 6.02333 | −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.o | ✓ | 22 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4010.2.a.o | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):
\( T_{3}^{22} - 2 T_{3}^{21} - 47 T_{3}^{20} + 88 T_{3}^{19} + 936 T_{3}^{18} - 1612 T_{3}^{17} + \cdots - 3520 \) |
\( T_{7}^{22} - 13 T_{7}^{21} - 23 T_{7}^{20} + 947 T_{7}^{19} - 1876 T_{7}^{18} - 26849 T_{7}^{17} + \cdots - 182495808 \) |
\( T_{11}^{22} + 3 T_{11}^{21} - 152 T_{11}^{20} - 380 T_{11}^{19} + 9909 T_{11}^{18} + 18631 T_{11}^{17} + \cdots - 13105691136 \) |