Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [162,7,Mod(161,162)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(162, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("162.161");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 162.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(37.2687615464\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 4x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 4x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + 3\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + 6\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\nu^{2} + 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 6 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{2} + 2\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).
| \(n\) | \(83\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 161.1 |
|
− | 5.65685i | 0 | −32.0000 | 24.8168i | 0 | −6.19615 | 181.019i | 0 | 140.385 | |||||||||||||||||||||||||||||
| 161.2 | − | 5.65685i | 0 | −32.0000 | 208.528i | 0 | 4.19615 | 181.019i | 0 | 1179.62 | ||||||||||||||||||||||||||||||
| 161.3 | 5.65685i | 0 | −32.0000 | − | 208.528i | 0 | 4.19615 | − | 181.019i | 0 | 1179.62 | |||||||||||||||||||||||||||||
| 161.4 | 5.65685i | 0 | −32.0000 | − | 24.8168i | 0 | −6.19615 | − | 181.019i | 0 | 140.385 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 162.7.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 162.7.b.a | ✓ | 4 |
| 9.c | even | 3 | 2 | 162.7.d.f | 8 | ||
| 9.d | odd | 6 | 2 | 162.7.d.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 162.7.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 162.7.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 162.7.d.f | 8 | 9.c | even | 3 | 2 | ||
| 162.7.d.f | 8 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 44100T_{5}^{2} + 26780625 \)
acting on \(S_{7}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 32)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 44100 T^{2} + 26780625 \)
|
| $7$ |
\( (T^{2} + 2 T - 26)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 87664550724 \)
|
| $13$ |
\( (T^{2} - 1918 T - 2618507)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 44258191098489 \)
|
| $19$ |
\( (T^{2} - 5122 T - 1519922)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 16\!\cdots\!29 \)
|
| $31$ |
\( (T^{2} + 20048 T - 115148876)^{2} \)
|
| $37$ |
\( (T^{2} + 10100 T - 117700343)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 39\!\cdots\!36 \)
|
| $43$ |
\( (T^{2} - 92470 T - 3775605602)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 55\!\cdots\!24 \)
|
| $53$ |
\( T^{4} + \cdots + 33\!\cdots\!16 \)
|
| $59$ |
\( T^{4} + \cdots + 19\!\cdots\!04 \)
|
| $61$ |
\( (T^{2} - 304528 T - 5174942891)^{2} \)
|
| $67$ |
\( (T^{2} + 1004486 T + 251487596326)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 89\!\cdots\!36 \)
|
| $73$ |
\( (T^{2} - 262912 T - 22812868139)^{2} \)
|
| $79$ |
\( (T^{2} + 424358 T - 422744653442)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 10\!\cdots\!64 \)
|
| $89$ |
\( T^{4} + \cdots + 64\!\cdots\!49 \)
|
| $97$ |
\( (T^{2} - 1810864 T + 582401158324)^{2} \)
|
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