Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [14,16,Mod(1,14)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(14, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("14.1");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 14 = 2 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 14.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: | \(19.9770907140\) |
| 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: |
\( x^{3} - 441577x - 112695480 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3\cdot 5\cdot 7 \) |
| 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
\( x^{3} - 441577x - 112695480 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{2} + 435\nu + 294387 ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( 13\nu^{2} - 4815\nu - 3827001 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 91\beta _1 - 30 ) / 840 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 29\beta_{2} + 2247\beta _1 + 16484802 ) / 56 \)
|
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 |
|
128.000 | −4407.98 | 16384.0 | −274525. | −564221. | 823543. | 2.09715e6 | 5.08138e6 | −3.51392e7 | |||||||||||||||||||||||||||
| 1.2 | 128.000 | 625.264 | 16384.0 | 259546. | 80033.8 | 823543. | 2.09715e6 | −1.39580e7 | 3.32219e7 | ||||||||||||||||||||||||||||
| 1.3 | 128.000 | 6594.72 | 16384.0 | −32114.8 | 844124. | 823543. | 2.09715e6 | 2.91414e7 | −4.11070e6 | ||||||||||||||||||||||||||||
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 | 14.16.a.d | ✓ | 3 |
| 3.b | odd | 2 | 1 | 126.16.a.m | 3 | ||
| 4.b | odd | 2 | 1 | 112.16.a.d | 3 | ||
| 7.b | odd | 2 | 1 | 98.16.a.e | 3 | ||
| 7.c | even | 3 | 2 | 98.16.c.i | 6 | ||
| 7.d | odd | 6 | 2 | 98.16.c.j | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.16.a.d | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 98.16.a.e | 3 | 7.b | odd | 2 | 1 | ||
| 98.16.c.i | 6 | 7.c | even | 3 | 2 | ||
| 98.16.c.j | 6 | 7.d | odd | 6 | 2 | ||
| 112.16.a.d | 3 | 4.b | odd | 2 | 1 | ||
| 126.16.a.m | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 2812T_{3}^{2} - 27702084T_{3} + 18176041008 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(14))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 128)^{3} \)
|
| $3$ |
\( T^{3} + \cdots + 18176041008 \)
|
| $5$ |
\( T^{3} + \cdots - 22\!\cdots\!00 \)
|
| $7$ |
\( (T - 823543)^{3} \)
|
| $11$ |
\( T^{3} + \cdots + 12\!\cdots\!92 \)
|
| $13$ |
\( T^{3} + \cdots + 27\!\cdots\!72 \)
|
| $17$ |
\( T^{3} + \cdots + 10\!\cdots\!16 \)
|
| $19$ |
\( T^{3} + \cdots - 73\!\cdots\!20 \)
|
| $23$ |
\( T^{3} + \cdots + 35\!\cdots\!00 \)
|
| $29$ |
\( T^{3} + \cdots - 15\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 30\!\cdots\!28 \)
|
| $37$ |
\( T^{3} + \cdots + 51\!\cdots\!96 \)
|
| $41$ |
\( T^{3} + \cdots + 49\!\cdots\!08 \)
|
| $43$ |
\( T^{3} + \cdots - 11\!\cdots\!68 \)
|
| $47$ |
\( T^{3} + \cdots - 35\!\cdots\!36 \)
|
| $53$ |
\( T^{3} + \cdots - 13\!\cdots\!08 \)
|
| $59$ |
\( T^{3} + \cdots - 11\!\cdots\!20 \)
|
| $61$ |
\( T^{3} + \cdots - 17\!\cdots\!00 \)
|
| $67$ |
\( T^{3} + \cdots + 23\!\cdots\!96 \)
|
| $71$ |
\( T^{3} + \cdots - 10\!\cdots\!12 \)
|
| $73$ |
\( T^{3} + \cdots + 27\!\cdots\!00 \)
|
| $79$ |
\( T^{3} + \cdots + 11\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 21\!\cdots\!68 \)
|
| $89$ |
\( T^{3} + \cdots + 20\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots + 26\!\cdots\!00 \)
|
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