Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [15,22,Mod(1,15)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 22, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("15.1");
S:= CuspForms(chi, 22);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 15 = 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 15.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: | \(41.9216016431\) |
| 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: |
\( x^{3} - x^{2} - 6395796x - 2792983104 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3\cdot 5\cdot 7 \) |
| 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
\( x^{3} - x^{2} - 6395796x - 2792983104 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 656\nu - 4263646 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 656\beta _1 + 4263646 \)
|
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 |
|
−2456.63 | 59049.0 | 3.93788e6 | −9.76562e6 | −1.45062e8 | −5.82386e8 | −4.52198e9 | 3.48678e9 | 2.39905e10 | |||||||||||||||||||||||||||
| 1.2 | 719.072 | 59049.0 | −1.58009e6 | −9.76562e6 | 4.24605e7 | −1.45023e9 | −2.64420e9 | 3.48678e9 | −7.02219e9 | ||||||||||||||||||||||||||||
| 1.3 | 2540.56 | 59049.0 | 4.35728e6 | −9.76562e6 | 1.50017e8 | 4.55015e8 | 5.74199e9 | 3.48678e9 | −2.48101e10 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
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 | 15.22.a.d | ✓ | 3 |
| 3.b | odd | 2 | 1 | 45.22.a.b | 3 | ||
| 5.b | even | 2 | 1 | 75.22.a.e | 3 | ||
| 5.c | odd | 4 | 2 | 75.22.b.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.22.a.d | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 45.22.a.b | 3 | 3.b | odd | 2 | 1 | ||
| 75.22.a.e | 3 | 5.b | even | 2 | 1 | ||
| 75.22.b.e | 6 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 803T_{2}^{2} - 6180860T_{2} + 4487879424 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(15))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + \cdots + 4487879424 \)
|
| $3$ |
\( (T - 59049)^{3} \)
|
| $5$ |
\( (T + 9765625)^{3} \)
|
| $7$ |
\( T^{3} + \cdots - 38\!\cdots\!80 \)
|
| $11$ |
\( T^{3} + \cdots + 25\!\cdots\!36 \)
|
| $13$ |
\( T^{3} + \cdots - 10\!\cdots\!28 \)
|
| $17$ |
\( T^{3} + \cdots + 16\!\cdots\!92 \)
|
| $19$ |
\( T^{3} + \cdots + 39\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots + 12\!\cdots\!80 \)
|
| $29$ |
\( T^{3} + \cdots - 19\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 10\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots - 18\!\cdots\!60 \)
|
| $41$ |
\( T^{3} + \cdots - 86\!\cdots\!00 \)
|
| $43$ |
\( T^{3} + \cdots - 19\!\cdots\!04 \)
|
| $47$ |
\( T^{3} + \cdots + 62\!\cdots\!68 \)
|
| $53$ |
\( T^{3} + \cdots + 96\!\cdots\!40 \)
|
| $59$ |
\( T^{3} + \cdots - 67\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 41\!\cdots\!32 \)
|
| $67$ |
\( T^{3} + \cdots - 29\!\cdots\!48 \)
|
| $71$ |
\( T^{3} + \cdots + 14\!\cdots\!52 \)
|
| $73$ |
\( T^{3} + \cdots - 10\!\cdots\!20 \)
|
| $79$ |
\( T^{3} + \cdots - 61\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 10\!\cdots\!92 \)
|
| $89$ |
\( T^{3} + \cdots + 26\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots + 16\!\cdots\!52 \)
|
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