Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [26,10,Mod(1,26)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(26, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("26.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 26 = 2 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 26.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: | \(13.3909317403\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 6144x - 66096 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 a root \(\nu\) of
\( x^{3} - x^{2} - 6144x - 66096 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 13\nu - 4092 ) / 24 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 48\beta_{2} + 13\beta _1 + 8197 ) / 2 \)
|
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 |
|
−16.0000 | −83.0152 | 256.000 | 921.862 | 1328.24 | −10987.6 | −4096.00 | −12791.5 | −14749.8 | |||||||||||||||||||||||||||
| 1.2 | −16.0000 | −76.4939 | 256.000 | −2441.31 | 1223.90 | 1788.13 | −4096.00 | −13831.7 | 39060.9 | ||||||||||||||||||||||||||||
| 1.3 | −16.0000 | 159.509 | 256.000 | 1767.44 | −2552.15 | 6243.46 | −4096.00 | 5760.16 | −28279.1 | ||||||||||||||||||||||||||||
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 | 26.10.a.d | ✓ | 3 |
| 3.b | odd | 2 | 1 | 234.10.a.l | 3 | ||
| 4.b | odd | 2 | 1 | 208.10.a.e | 3 | ||
| 13.b | even | 2 | 1 | 338.10.a.f | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.10.a.d | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 208.10.a.e | 3 | 4.b | odd | 2 | 1 | ||
| 234.10.a.l | 3 | 3.b | odd | 2 | 1 | ||
| 338.10.a.f | 3 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 19093T_{3} - 1012908 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(26))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16)^{3} \)
|
| $3$ |
\( T^{3} - 19093 T - 1012908 \)
|
| $5$ |
\( T^{3} + \cdots + 3977720730 \)
|
| $7$ |
\( T^{3} + \cdots + 122666634104 \)
|
| $11$ |
\( T^{3} + \cdots - 146425442149584 \)
|
| $13$ |
\( (T - 28561)^{3} \)
|
| $17$ |
\( T^{3} + \cdots - 10\!\cdots\!86 \)
|
| $19$ |
\( T^{3} + \cdots - 15\!\cdots\!48 \)
|
| $23$ |
\( T^{3} + \cdots + 11\!\cdots\!00 \)
|
| $29$ |
\( T^{3} + \cdots - 12\!\cdots\!84 \)
|
| $31$ |
\( T^{3} + \cdots - 18\!\cdots\!52 \)
|
| $37$ |
\( T^{3} + \cdots - 16\!\cdots\!58 \)
|
| $41$ |
\( T^{3} + \cdots - 73\!\cdots\!00 \)
|
| $43$ |
\( T^{3} + \cdots + 71\!\cdots\!48 \)
|
| $47$ |
\( T^{3} + \cdots - 12\!\cdots\!52 \)
|
| $53$ |
\( T^{3} + \cdots + 70\!\cdots\!00 \)
|
| $59$ |
\( T^{3} + \cdots - 11\!\cdots\!76 \)
|
| $61$ |
\( T^{3} + \cdots - 26\!\cdots\!08 \)
|
| $67$ |
\( T^{3} + \cdots - 19\!\cdots\!16 \)
|
| $71$ |
\( T^{3} + \cdots + 28\!\cdots\!84 \)
|
| $73$ |
\( T^{3} + \cdots - 52\!\cdots\!00 \)
|
| $79$ |
\( T^{3} + \cdots + 40\!\cdots\!76 \)
|
| $83$ |
\( T^{3} + \cdots - 45\!\cdots\!20 \)
|
| $89$ |
\( T^{3} + \cdots - 18\!\cdots\!44 \)
|
| $97$ |
\( T^{3} + \cdots - 31\!\cdots\!00 \)
|
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