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: | \(1\) |
Dimension: | \(56\) |
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 | −3.35147 | 1.00000 | 0.434868 | −3.35147 | 3.30614 | 1.00000 | 8.23237 | 0.434868 | ||||||||||||||||||
1.2 | 1.00000 | −3.31079 | 1.00000 | −3.14984 | −3.31079 | −0.324393 | 1.00000 | 7.96133 | −3.14984 | ||||||||||||||||||
1.3 | 1.00000 | −3.25917 | 1.00000 | −0.219000 | −3.25917 | −4.63324 | 1.00000 | 7.62218 | −0.219000 | ||||||||||||||||||
1.4 | 1.00000 | −3.18186 | 1.00000 | 3.25833 | −3.18186 | 0.730955 | 1.00000 | 7.12426 | 3.25833 | ||||||||||||||||||
1.5 | 1.00000 | −2.93762 | 1.00000 | 1.81603 | −2.93762 | −0.742842 | 1.00000 | 5.62960 | 1.81603 | ||||||||||||||||||
1.6 | 1.00000 | −2.89764 | 1.00000 | 3.79029 | −2.89764 | −4.56649 | 1.00000 | 5.39633 | 3.79029 | ||||||||||||||||||
1.7 | 1.00000 | −2.64315 | 1.00000 | −1.33957 | −2.64315 | 3.06951 | 1.00000 | 3.98626 | −1.33957 | ||||||||||||||||||
1.8 | 1.00000 | −2.45102 | 1.00000 | 0.781132 | −2.45102 | −0.0575638 | 1.00000 | 3.00749 | 0.781132 | ||||||||||||||||||
1.9 | 1.00000 | −2.44881 | 1.00000 | 2.87933 | −2.44881 | −3.14262 | 1.00000 | 2.99666 | 2.87933 | ||||||||||||||||||
1.10 | 1.00000 | −2.43348 | 1.00000 | −1.73446 | −2.43348 | −1.67668 | 1.00000 | 2.92185 | −1.73446 | ||||||||||||||||||
1.11 | 1.00000 | −2.39212 | 1.00000 | 0.456785 | −2.39212 | 2.48835 | 1.00000 | 2.72224 | 0.456785 | ||||||||||||||||||
1.12 | 1.00000 | −2.37532 | 1.00000 | −4.29598 | −2.37532 | −4.13959 | 1.00000 | 2.64215 | −4.29598 | ||||||||||||||||||
1.13 | 1.00000 | −2.14327 | 1.00000 | 4.37018 | −2.14327 | −1.18558 | 1.00000 | 1.59362 | 4.37018 | ||||||||||||||||||
1.14 | 1.00000 | −2.05550 | 1.00000 | −1.29697 | −2.05550 | 0.414264 | 1.00000 | 1.22507 | −1.29697 | ||||||||||||||||||
1.15 | 1.00000 | −1.97472 | 1.00000 | −3.51408 | −1.97472 | 4.87260 | 1.00000 | 0.899523 | −3.51408 | ||||||||||||||||||
1.16 | 1.00000 | −1.79217 | 1.00000 | −3.46015 | −1.79217 | −4.09544 | 1.00000 | 0.211861 | −3.46015 | ||||||||||||||||||
1.17 | 1.00000 | −1.73760 | 1.00000 | −0.720137 | −1.73760 | −3.09490 | 1.00000 | 0.0192639 | −0.720137 | ||||||||||||||||||
1.18 | 1.00000 | −1.70174 | 1.00000 | −3.77151 | −1.70174 | 2.90105 | 1.00000 | −0.104070 | −3.77151 | ||||||||||||||||||
1.19 | 1.00000 | −1.47034 | 1.00000 | 2.17542 | −1.47034 | −0.538935 | 1.00000 | −0.838111 | 2.17542 | ||||||||||||||||||
1.20 | 1.00000 | −1.44356 | 1.00000 | 1.04385 | −1.44356 | 3.20508 | 1.00000 | −0.916133 | 1.04385 | ||||||||||||||||||
See all 56 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.e | ✓ | 56 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
6046.2.a.e | ✓ | 56 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6046))\):
\( T_{3}^{56} + 18 T_{3}^{55} + 61 T_{3}^{54} - 789 T_{3}^{53} - 6192 T_{3}^{52} + 7239 T_{3}^{51} + \cdots + 1185404 \) |
\( T_{11}^{56} + 53 T_{11}^{55} + 1068 T_{11}^{54} + 7444 T_{11}^{53} - 66251 T_{11}^{52} + \cdots + 97\!\cdots\!21 \) |