Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6046,2,Mod(1,6046)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6046, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6046.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 6046 = 2 \cdot 3023 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 6046.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: | \(48.2775530621\) |
Analytic rank: | \(0\) |
Dimension: | \(69\) |
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 | −3.31246 | 1.00000 | 3.61669 | 3.31246 | −0.0193368 | −1.00000 | 7.97240 | −3.61669 | ||||||||||||||||||
1.2 | −1.00000 | −3.30557 | 1.00000 | −1.09873 | 3.30557 | 3.10435 | −1.00000 | 7.92681 | 1.09873 | ||||||||||||||||||
1.3 | −1.00000 | −3.30292 | 1.00000 | −2.02929 | 3.30292 | −1.40073 | −1.00000 | 7.90925 | 2.02929 | ||||||||||||||||||
1.4 | −1.00000 | −3.27820 | 1.00000 | 0.761195 | 3.27820 | 3.33846 | −1.00000 | 7.74657 | −0.761195 | ||||||||||||||||||
1.5 | −1.00000 | −3.16263 | 1.00000 | 1.22760 | 3.16263 | −4.76545 | −1.00000 | 7.00223 | −1.22760 | ||||||||||||||||||
1.6 | −1.00000 | −3.15165 | 1.00000 | 1.27267 | 3.15165 | −3.14596 | −1.00000 | 6.93292 | −1.27267 | ||||||||||||||||||
1.7 | −1.00000 | −3.13336 | 1.00000 | −4.20234 | 3.13336 | −4.60058 | −1.00000 | 6.81797 | 4.20234 | ||||||||||||||||||
1.8 | −1.00000 | −2.76254 | 1.00000 | 2.17667 | 2.76254 | −4.35539 | −1.00000 | 4.63161 | −2.17667 | ||||||||||||||||||
1.9 | −1.00000 | −2.73247 | 1.00000 | −4.08673 | 2.73247 | −1.35525 | −1.00000 | 4.46639 | 4.08673 | ||||||||||||||||||
1.10 | −1.00000 | −2.42918 | 1.00000 | 1.49357 | 2.42918 | 3.38071 | −1.00000 | 2.90091 | −1.49357 | ||||||||||||||||||
1.11 | −1.00000 | −2.36493 | 1.00000 | −1.05232 | 2.36493 | −2.21269 | −1.00000 | 2.59290 | 1.05232 | ||||||||||||||||||
1.12 | −1.00000 | −2.32306 | 1.00000 | −1.40989 | 2.32306 | −1.24456 | −1.00000 | 2.39659 | 1.40989 | ||||||||||||||||||
1.13 | −1.00000 | −2.30469 | 1.00000 | 4.02559 | 2.30469 | −3.39321 | −1.00000 | 2.31160 | −4.02559 | ||||||||||||||||||
1.14 | −1.00000 | −2.29528 | 1.00000 | 2.67586 | 2.29528 | −0.797143 | −1.00000 | 2.26829 | −2.67586 | ||||||||||||||||||
1.15 | −1.00000 | −2.17249 | 1.00000 | −2.48902 | 2.17249 | 3.29516 | −1.00000 | 1.71973 | 2.48902 | ||||||||||||||||||
1.16 | −1.00000 | −2.10025 | 1.00000 | 4.19656 | 2.10025 | 2.72635 | −1.00000 | 1.41104 | −4.19656 | ||||||||||||||||||
1.17 | −1.00000 | −1.93704 | 1.00000 | −1.86801 | 1.93704 | 4.01320 | −1.00000 | 0.752114 | 1.86801 | ||||||||||||||||||
1.18 | −1.00000 | −1.82817 | 1.00000 | −4.13260 | 1.82817 | 2.43979 | −1.00000 | 0.342212 | 4.13260 | ||||||||||||||||||
1.19 | −1.00000 | −1.76075 | 1.00000 | 0.351086 | 1.76075 | −1.33291 | −1.00000 | 0.100238 | −0.351086 | ||||||||||||||||||
1.20 | −1.00000 | −1.66158 | 1.00000 | −1.21846 | 1.66158 | −1.75845 | −1.00000 | −0.239146 | 1.21846 | ||||||||||||||||||
See all 69 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(3023\) | \(-1\) |
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 | 6046.2.a.g | ✓ | 69 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6046.2.a.g | ✓ | 69 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6046))\):
\( T_{3}^{69} - 153 T_{3}^{67} - 5 T_{3}^{66} + 11120 T_{3}^{65} + 729 T_{3}^{64} - 510967 T_{3}^{63} + \cdots - 4657414144 \) |
\( T_{11}^{69} - 42 T_{11}^{68} + 442 T_{11}^{67} + 5724 T_{11}^{66} - 149564 T_{11}^{65} + \cdots - 19\!\cdots\!50 \) |