Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [112,12,Mod(1,112)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(112, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("112.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 112.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: | \(86.0544362227\) |
| 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: |
\( x^{3} - x^{2} - 818x - 4704 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{9} \) |
| Twist minimal: | no (minimal twist has level 7) |
| 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
\( x^{3} - x^{2} - 818x - 4704 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{2} - 33\nu + 557 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{2} + 91\nu + 2697 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{2} - 5\beta _1 + 88 ) / 256 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -99\beta_{2} - 91\beta _1 + 139688 ) / 256 \)
|
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 |
|
0 | −800.902 | 0 | −7066.04 | 0 | 16807.0 | 0 | 464297. | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 240.378 | 0 | 12842.0 | 0 | 16807.0 | 0 | −119365. | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 700.524 | 0 | −749.998 | 0 | 16807.0 | 0 | 313587. | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 112.12.a.h | 3 | |
| 4.b | odd | 2 | 1 | 7.12.a.b | ✓ | 3 | |
| 12.b | even | 2 | 1 | 63.12.a.d | 3 | ||
| 20.d | odd | 2 | 1 | 175.12.a.b | 3 | ||
| 20.e | even | 4 | 2 | 175.12.b.b | 6 | ||
| 28.d | even | 2 | 1 | 49.12.a.d | 3 | ||
| 28.f | even | 6 | 2 | 49.12.c.f | 6 | ||
| 28.g | odd | 6 | 2 | 49.12.c.g | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.12.a.b | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 49.12.a.d | 3 | 28.d | even | 2 | 1 | ||
| 49.12.c.f | 6 | 28.f | even | 6 | 2 | ||
| 49.12.c.g | 6 | 28.g | odd | 6 | 2 | ||
| 63.12.a.d | 3 | 12.b | even | 2 | 1 | ||
| 112.12.a.h | 3 | 1.a | even | 1 | 1 | trivial | |
| 175.12.a.b | 3 | 20.d | odd | 2 | 1 | ||
| 175.12.b.b | 6 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 140T_{3}^{2} - 585180T_{3} + 134864352 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(112))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 140 T^{2} + \cdots + 134864352 \)
|
| $5$ |
\( T^{3} + \cdots - 68056486400 \)
|
| $7$ |
\( (T - 16807)^{3} \)
|
| $11$ |
\( T^{3} + \cdots + 33\!\cdots\!52 \)
|
| $13$ |
\( T^{3} + \cdots - 91\!\cdots\!44 \)
|
| $17$ |
\( T^{3} + \cdots - 62\!\cdots\!52 \)
|
| $19$ |
\( T^{3} + \cdots + 43\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots - 42\!\cdots\!72 \)
|
| $29$ |
\( T^{3} + \cdots - 25\!\cdots\!60 \)
|
| $31$ |
\( T^{3} + \cdots - 14\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots + 13\!\cdots\!84 \)
|
| $41$ |
\( T^{3} + \cdots + 46\!\cdots\!72 \)
|
| $43$ |
\( T^{3} + \cdots - 22\!\cdots\!64 \)
|
| $47$ |
\( T^{3} + \cdots + 21\!\cdots\!48 \)
|
| $53$ |
\( T^{3} + \cdots - 36\!\cdots\!88 \)
|
| $59$ |
\( T^{3} + \cdots - 66\!\cdots\!60 \)
|
| $61$ |
\( T^{3} + \cdots + 35\!\cdots\!68 \)
|
| $67$ |
\( T^{3} + \cdots - 23\!\cdots\!64 \)
|
| $71$ |
\( T^{3} + \cdots - 61\!\cdots\!92 \)
|
| $73$ |
\( T^{3} + \cdots - 15\!\cdots\!96 \)
|
| $79$ |
\( T^{3} + \cdots - 29\!\cdots\!20 \)
|
| $83$ |
\( T^{3} + \cdots + 57\!\cdots\!72 \)
|
| $89$ |
\( T^{3} + \cdots - 89\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 27\!\cdots\!84 \)
|
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