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: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 6981x^{2} - 35424x + 7188480 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - 6981x^{2} - 35424x + 7188480 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 12\nu^{2} - 5073\nu - 68328 ) / 252 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 72\nu^{2} + 4317\nu - 225000 ) / 504 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 6\beta_{3} + 3\beta_{2} + 9\beta _1 + 3492 \)
|
| \(\nu^{3}\) | \(=\) |
\( -72\beta_{3} + 216\beta_{2} + 4965\beta _1 + 26424 \)
|
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 | −71.1584 | 64.0000 | −523.489 | −569.267 | 416.801 | 512.000 | 2876.52 | −4187.91 | ||||||||||||||||||||||||||||||
| 1.2 | 8.00000 | −39.9844 | 64.0000 | 323.700 | −319.875 | −568.591 | 512.000 | −588.246 | 2589.60 | |||||||||||||||||||||||||||||||
| 1.3 | 8.00000 | 31.8716 | 64.0000 | 54.4265 | 254.973 | 1112.71 | 512.000 | −1171.20 | 435.412 | |||||||||||||||||||||||||||||||
| 1.4 | 8.00000 | 79.2712 | 64.0000 | −132.638 | 634.170 | −1508.92 | 512.000 | 4096.92 | −1061.10 | |||||||||||||||||||||||||||||||
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.j | 4 | |
| 13.b | even | 2 | 1 | 338.8.a.i | 4 | ||
| 13.d | odd | 4 | 2 | 338.8.b.h | 8 | ||
| 13.e | even | 6 | 2 | 26.8.c.b | ✓ | 8 | |
| 39.h | odd | 6 | 2 | 234.8.h.b | 8 | ||
| 52.i | odd | 6 | 2 | 208.8.i.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.8.c.b | ✓ | 8 | 13.e | even | 6 | 2 | |
| 208.8.i.b | 8 | 52.i | odd | 6 | 2 | ||
| 234.8.h.b | 8 | 39.h | odd | 6 | 2 | ||
| 338.8.a.i | 4 | 13.b | even | 2 | 1 | ||
| 338.8.a.j | 4 | 1.a | even | 1 | 1 | trivial | |
| 338.8.b.h | 8 | 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 \(S_{8}^{\mathrm{new}}(\Gamma_0(338))\):
|
\( T_{3}^{4} - 6981T_{3}^{2} - 35424T_{3} + 7188480 \)
|
|
\( T_{5}^{4} + 278T_{5}^{3} - 161047T_{5}^{2} - 14695460T_{5} + 1223288100 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{4} \)
|
| $3$ |
\( T^{4} - 6981 T^{2} + \cdots + 7188480 \)
|
| $5$ |
\( T^{4} + \cdots + 1223288100 \)
|
| $7$ |
\( T^{4} + \cdots + 397901702960 \)
|
| $11$ |
\( T^{4} + \cdots + 17021121355056 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + \cdots - 59\!\cdots\!71 \)
|
| $19$ |
\( T^{4} + \cdots - 40\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 80\!\cdots\!44 \)
|
| $29$ |
\( T^{4} + \cdots + 25\!\cdots\!13 \)
|
| $31$ |
\( T^{4} + \cdots + 10\!\cdots\!28 \)
|
| $37$ |
\( T^{4} + \cdots - 16\!\cdots\!79 \)
|
| $41$ |
\( T^{4} + \cdots - 27\!\cdots\!15 \)
|
| $43$ |
\( T^{4} + \cdots - 44\!\cdots\!64 \)
|
| $47$ |
\( T^{4} + \cdots + 87\!\cdots\!28 \)
|
| $53$ |
\( T^{4} + \cdots + 10\!\cdots\!32 \)
|
| $59$ |
\( T^{4} + \cdots + 93\!\cdots\!08 \)
|
| $61$ |
\( T^{4} + \cdots + 60\!\cdots\!45 \)
|
| $67$ |
\( T^{4} + \cdots - 12\!\cdots\!64 \)
|
| $71$ |
\( T^{4} + \cdots + 24\!\cdots\!88 \)
|
| $73$ |
\( T^{4} + \cdots - 21\!\cdots\!40 \)
|
| $79$ |
\( T^{4} + \cdots + 46\!\cdots\!08 \)
|
| $83$ |
\( T^{4} + \cdots - 67\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots - 52\!\cdots\!56 \)
|
| $97$ |
\( T^{4} + \cdots + 48\!\cdots\!28 \)
|
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