Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6,26,Mod(1,6)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 26, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6.1");
S:= CuspForms(chi, 26);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6 = 2 \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 26 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6.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: | \(23.7598067971\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| 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)
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 |
|
−4096.00 | −531441. | 1.67772e7 | −2.92755e8 | 2.17678e9 | 3.58064e9 | −6.87195e10 | 2.82430e11 | 1.19912e12 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
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 | 6.26.a.a | ✓ | 1 |
| 3.b | odd | 2 | 1 | 18.26.a.d | 1 | ||
| 4.b | odd | 2 | 1 | 48.26.a.c | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6.26.a.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
| 18.26.a.d | 1 | 3.b | odd | 2 | 1 | ||
| 48.26.a.c | 1 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 292754850 \)
acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(6))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T + 4096 \)
|
| $3$ |
\( T + 531441 \)
|
| $5$ |
\( T + 292754850 \)
|
| $7$ |
\( T - 3580644032 \)
|
| $11$ |
\( T - 15111573238212 \)
|
| $13$ |
\( T - 1221071681246 \)
|
| $17$ |
\( T - 2518250853863682 \)
|
| $19$ |
\( T + 7992693407413060 \)
|
| $23$ |
\( T + 99\!\cdots\!24 \)
|
| $29$ |
\( T + 20\!\cdots\!90 \)
|
| $31$ |
\( T + 49\!\cdots\!08 \)
|
| $37$ |
\( T - 19\!\cdots\!82 \)
|
| $41$ |
\( T - 22\!\cdots\!42 \)
|
| $43$ |
\( T + 72\!\cdots\!84 \)
|
| $47$ |
\( T - 18\!\cdots\!92 \)
|
| $53$ |
\( T + 26\!\cdots\!74 \)
|
| $59$ |
\( T + 16\!\cdots\!40 \)
|
| $61$ |
\( T + 35\!\cdots\!38 \)
|
| $67$ |
\( T - 10\!\cdots\!92 \)
|
| $71$ |
\( T - 73\!\cdots\!12 \)
|
| $73$ |
\( T + 26\!\cdots\!74 \)
|
| $79$ |
\( T + 10\!\cdots\!40 \)
|
| $83$ |
\( T - 15\!\cdots\!96 \)
|
| $89$ |
\( T - 21\!\cdots\!90 \)
|
| $97$ |
\( T + 44\!\cdots\!58 \)
|
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