Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [414,4,Mod(1,414)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(414, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("414.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 414 = 2 \cdot 3^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 414.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: | \(24.4267907424\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{4} - 2x^{3} - 219x^{2} - 468x + 3240 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 219x^{2} - 468x + 3240 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 5\nu^{2} - 168\nu - 54 ) / 18 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{2} - 5\nu - 108 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 12\beta_{3} + 5\beta _1 + 221 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 60\beta_{3} + 36\beta_{2} + 193\beta _1 + 1381 ) / 2 \)
|
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 | 0 | 4.00000 | −19.0098 | 0 | −26.4014 | 8.00000 | 0 | −38.0197 | ||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | 0 | 4.00000 | −7.28088 | 0 | 35.9873 | 8.00000 | 0 | −14.5618 | |||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 0 | 4.00000 | 13.0601 | 0 | 18.7067 | 8.00000 | 0 | 26.1202 | |||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 0 | 4.00000 | 13.2306 | 0 | −10.2926 | 8.00000 | 0 | 26.4612 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(23\) | \( -1 \) |
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 | 414.4.a.n | yes | 4 |
| 3.b | odd | 2 | 1 | 414.4.a.m | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 414.4.a.m | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 414.4.a.n | yes | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(414))\):
|
\( T_{5}^{4} - 380T_{5}^{2} + 904T_{5} + 23916 \)
|
|
\( T_{7}^{4} - 18T_{7}^{3} - 1062T_{7}^{2} + 9840T_{7} + 182936 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 380 T^{2} + \cdots + 23916 \)
|
| $7$ |
\( T^{4} - 18 T^{3} + \cdots + 182936 \)
|
| $11$ |
\( T^{4} - 42 T^{3} + \cdots - 676368 \)
|
| $13$ |
\( T^{4} - 108 T^{3} + \cdots - 1528096 \)
|
| $17$ |
\( T^{4} - 26 T^{3} + \cdots - 160272 \)
|
| $19$ |
\( T^{4} - 132 T^{3} + \cdots - 144765684 \)
|
| $23$ |
\( (T - 23)^{4} \)
|
| $29$ |
\( T^{4} - 252 T^{3} + \cdots + 152282592 \)
|
| $31$ |
\( T^{4} - 428 T^{3} + \cdots + 189998432 \)
|
| $37$ |
\( T^{4} - 278 T^{3} + \cdots - 627096 \)
|
| $41$ |
\( T^{4} + 4 T^{3} + \cdots + 59580480 \)
|
| $43$ |
\( T^{4} + \cdots + 4885011324 \)
|
| $47$ |
\( T^{4} + \cdots - 3721395696 \)
|
| $53$ |
\( T^{4} + \cdots + 28977105756 \)
|
| $59$ |
\( T^{4} + \cdots + 18288334080 \)
|
| $61$ |
\( T^{4} + \cdots - 33871710232 \)
|
| $67$ |
\( T^{4} + \cdots - 5950476756 \)
|
| $71$ |
\( T^{4} + \cdots + 47749357440 \)
|
| $73$ |
\( T^{4} + \cdots - 13734027872 \)
|
| $79$ |
\( T^{4} + \cdots - 102430136424 \)
|
| $83$ |
\( T^{4} + \cdots - 16894365072 \)
|
| $89$ |
\( T^{4} + \cdots + 815730086592 \)
|
| $97$ |
\( T^{4} + \cdots + 411321121872 \)
|
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