Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [338,8,Mod(1,338)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(338, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("338.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 338 = 2 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 338.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: | \(105.586138614\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{105}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 26 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 26) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{105}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
8.00000 | −77.7287 | 64.0000 | 441.890 | −621.829 | 1166.67 | 512.000 | 3854.74 | 3535.12 | ||||||||||||||||||||||||
| 1.2 | 8.00000 | 65.7287 | 64.0000 | −295.890 | 525.829 | 613.332 | 512.000 | 2133.26 | −2367.12 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( +1 \) |
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 | 338.8.a.f | 2 | |
| 13.b | even | 2 | 1 | 26.8.a.d | ✓ | 2 | |
| 13.d | odd | 4 | 2 | 338.8.b.e | 4 | ||
| 39.d | odd | 2 | 1 | 234.8.a.l | 2 | ||
| 52.b | odd | 2 | 1 | 208.8.a.h | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.8.a.d | ✓ | 2 | 13.b | even | 2 | 1 | |
| 208.8.a.h | 2 | 52.b | odd | 2 | 1 | ||
| 234.8.a.l | 2 | 39.d | odd | 2 | 1 | ||
| 338.8.a.f | 2 | 1.a | even | 1 | 1 | trivial | |
| 338.8.b.e | 4 | 13.d | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(338))\):
|
\( T_{3}^{2} + 12T_{3} - 5109 \)
|
|
\( T_{5}^{2} - 146T_{5} - 130751 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{2} \)
|
| $3$ |
\( T^{2} + 12T - 5109 \)
|
| $5$ |
\( T^{2} - 146T - 130751 \)
|
| $7$ |
\( T^{2} - 1780 T + 715555 \)
|
| $11$ |
\( T^{2} + 10904 T + 28873804 \)
|
| $13$ |
\( T^{2} \)
|
| $17$ |
\( T^{2} + 14118 T - 507554199 \)
|
| $19$ |
\( T^{2} + 54408 T + 538353036 \)
|
| $23$ |
\( T^{2} + \cdots + 4786525744 \)
|
| $29$ |
\( T^{2} + \cdots - 16891064924 \)
|
| $31$ |
\( T^{2} + 45800 T - 303500720 \)
|
| $37$ |
\( T^{2} + \cdots + 8162736561 \)
|
| $41$ |
\( T^{2} + \cdots - 4113607680 \)
|
| $43$ |
\( T^{2} + \cdots - 7014189629 \)
|
| $47$ |
\( T^{2} + \cdots - 109005494141 \)
|
| $53$ |
\( T^{2} + \cdots - 18298543536 \)
|
| $59$ |
\( T^{2} + \cdots - 3539035961684 \)
|
| $61$ |
\( T^{2} + \cdots + 4106598596416 \)
|
| $67$ |
\( T^{2} + \cdots + 927005771020 \)
|
| $71$ |
\( T^{2} + \cdots - 14263049562725 \)
|
| $73$ |
\( T^{2} + \cdots - 1610493427196 \)
|
| $79$ |
\( T^{2} + \cdots + 791817441136 \)
|
| $83$ |
\( T^{2} + \cdots + 15381431470080 \)
|
| $89$ |
\( T^{2} + \cdots + 69622103547396 \)
|
| $97$ |
\( T^{2} + \cdots - 53125913495324 \)
|
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