Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [338,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("338.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 338 = 2 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(19.9426455819\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\Q(\zeta_{14})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 2x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
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 |
|
2.00000 | −8.74094 | 4.00000 | −14.1685 | −17.4819 | 28.6504 | 8.00000 | 49.4040 | −28.3370 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | −3.66487 | 4.00000 | 8.53079 | −7.32975 | −4.20105 | 8.00000 | −13.5687 | 17.0616 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 0.405813 | 4.00000 | −6.36227 | 0.811626 | 2.55065 | 8.00000 | −26.8353 | −12.7245 | ||||||||||||||||||||||||||||
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 | 338.4.a.k | yes | 3 |
| 13.b | even | 2 | 1 | 338.4.a.j | ✓ | 3 | |
| 13.c | even | 3 | 2 | 338.4.c.k | 6 | ||
| 13.d | odd | 4 | 2 | 338.4.b.f | 6 | ||
| 13.e | even | 6 | 2 | 338.4.c.l | 6 | ||
| 13.f | odd | 12 | 4 | 338.4.e.h | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 338.4.a.j | ✓ | 3 | 13.b | even | 2 | 1 | |
| 338.4.a.k | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 338.4.b.f | 6 | 13.d | odd | 4 | 2 | ||
| 338.4.c.k | 6 | 13.c | even | 3 | 2 | ||
| 338.4.c.l | 6 | 13.e | even | 6 | 2 | ||
| 338.4.e.h | 12 | 13.f | odd | 12 | 4 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(338))\):
|
\( T_{3}^{3} + 12T_{3}^{2} + 27T_{3} - 13 \)
|
|
\( T_{5}^{3} + 12T_{5}^{2} - 85T_{5} - 769 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
|
| $3$ |
\( T^{3} + 12 T^{2} + \cdots - 13 \)
|
| $5$ |
\( T^{3} + 12 T^{2} + \cdots - 769 \)
|
| $7$ |
\( T^{3} - 27 T^{2} + \cdots + 307 \)
|
| $11$ |
\( T^{3} + 82 T^{2} + \cdots - 16211 \)
|
| $13$ |
\( T^{3} \)
|
| $17$ |
\( T^{3} + 90 T^{2} + \cdots - 77167 \)
|
| $19$ |
\( T^{3} + 130 T^{2} + \cdots - 76609 \)
|
| $23$ |
\( T^{3} - 19 T^{2} + \cdots - 604157 \)
|
| $29$ |
\( T^{3} + 101 T^{2} + \cdots - 3703349 \)
|
| $31$ |
\( T^{3} + 519 T^{2} + \cdots + 3401957 \)
|
| $37$ |
\( T^{3} + 84 T^{2} + \cdots + 30919 \)
|
| $41$ |
\( T^{3} + 187 T^{2} + \cdots - 20172347 \)
|
| $43$ |
\( T^{3} + 1205 T^{2} + \cdots + 64126453 \)
|
| $47$ |
\( T^{3} - 536 T^{2} + \cdots + 26128271 \)
|
| $53$ |
\( T^{3} + 1095 T^{2} + \cdots + 45870749 \)
|
| $59$ |
\( T^{3} - 1413 T^{2} + \cdots - 72818971 \)
|
| $61$ |
\( T^{3} + 1108 T^{2} + \cdots + 28377551 \)
|
| $67$ |
\( T^{3} - 1605 T^{2} + \cdots - 110488951 \)
|
| $71$ |
\( T^{3} - 909 T^{2} + \cdots + 7534771 \)
|
| $73$ |
\( T^{3} + 287 T^{2} + \cdots - 178534237 \)
|
| $79$ |
\( T^{3} + 1961 T^{2} + \cdots + 214064899 \)
|
| $83$ |
\( T^{3} + 191 T^{2} + \cdots - 29986853 \)
|
| $89$ |
\( T^{3} + 1091 T^{2} + \cdots + 10734907 \)
|
| $97$ |
\( T^{3} - 947 T^{2} + \cdots + 228842741 \)
|
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