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: | \(3\) |
| Coefficient field: | 3.3.16372.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 40x - 84 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 138) |
| 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
\( x^{3} - 40x - 84 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4\nu - 27 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 27 \)
|
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 | −20.9800 | 0 | −20.3934 | −8.00000 | 0 | 41.9600 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | 0 | 4.00000 | −2.92753 | 0 | 21.4508 | −8.00000 | 0 | 5.85507 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0 | 4.00000 | 3.90754 | 0 | 8.94262 | −8.00000 | 0 | −7.81508 | ||||||||||||||||||||||||||||
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.l | 3 | |
| 3.b | odd | 2 | 1 | 138.4.a.f | ✓ | 3 | |
| 12.b | even | 2 | 1 | 1104.4.a.r | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 138.4.a.f | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 414.4.a.l | 3 | 1.a | even | 1 | 1 | trivial | |
| 1104.4.a.r | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(414))\):
|
\( T_{5}^{3} + 20T_{5}^{2} - 32T_{5} - 240 \)
|
|
\( T_{7}^{3} - 10T_{7}^{2} - 428T_{7} + 3912 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 20 T^{2} + \cdots - 240 \)
|
| $7$ |
\( T^{3} - 10 T^{2} + \cdots + 3912 \)
|
| $11$ |
\( T^{3} + 20 T^{2} + \cdots - 76320 \)
|
| $13$ |
\( T^{3} + 2 T^{2} + \cdots + 34456 \)
|
| $17$ |
\( T^{3} + 32 T^{2} + \cdots - 321648 \)
|
| $19$ |
\( T^{3} + 26 T^{2} + \cdots + 124552 \)
|
| $23$ |
\( (T - 23)^{3} \)
|
| $29$ |
\( T^{3} - 50 T^{2} + \cdots - 2646168 \)
|
| $31$ |
\( T^{3} - 12 T^{2} + \cdots + 4079040 \)
|
| $37$ |
\( T^{3} + 90 T^{2} + \cdots - 235496 \)
|
| $41$ |
\( T^{3} - 466 T^{2} + \cdots + 66217896 \)
|
| $43$ |
\( T^{3} + 274 T^{2} + \cdots - 25943800 \)
|
| $47$ |
\( T^{3} - 680 T^{2} + \cdots + 574848 \)
|
| $53$ |
\( T^{3} - 408 T^{2} + \cdots + 117418800 \)
|
| $59$ |
\( T^{3} - 1244 T^{2} + \cdots - 7860672 \)
|
| $61$ |
\( T^{3} + 130 T^{2} + \cdots + 87594872 \)
|
| $67$ |
\( T^{3} - 450 T^{2} + \cdots + 15728056 \)
|
| $71$ |
\( T^{3} - 1512 T^{2} + \cdots - 125266176 \)
|
| $73$ |
\( T^{3} - 710 T^{2} + \cdots + 12034296 \)
|
| $79$ |
\( T^{3} - 478 T^{2} + \cdots + 175419096 \)
|
| $83$ |
\( T^{3} - 1092 T^{2} + \cdots + 152516256 \)
|
| $89$ |
\( T^{3} - 372 T^{2} + \cdots + 139730832 \)
|
| $97$ |
\( T^{3} - 2106 T^{2} + \cdots - 304805880 \)
|
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