Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [162,11,Mod(53,162)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(162, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("162.53");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 162.d (of order \(6\), degree \(2\), not 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: | \(102.927874933\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 54) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).
| \(n\) | \(83\) |
| \(\chi(n)\) | \(1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 53.1 |
|
−19.5959 | + | 11.3137i | 0 | 256.000 | − | 443.405i | −433.560 | − | 250.316i | 0 | 1557.50 | + | 2697.67i | 11585.2i | 0 | 11328.0 | ||||||||||||||||||||||
| 53.2 | 19.5959 | − | 11.3137i | 0 | 256.000 | − | 443.405i | 433.560 | + | 250.316i | 0 | 1557.50 | + | 2697.67i | − | 11585.2i | 0 | 11328.0 | ||||||||||||||||||||||
| 107.1 | −19.5959 | − | 11.3137i | 0 | 256.000 | + | 443.405i | −433.560 | + | 250.316i | 0 | 1557.50 | − | 2697.67i | − | 11585.2i | 0 | 11328.0 | ||||||||||||||||||||||
| 107.2 | 19.5959 | + | 11.3137i | 0 | 256.000 | + | 443.405i | 433.560 | − | 250.316i | 0 | 1557.50 | − | 2697.67i | 11585.2i | 0 | 11328.0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 9.c | even | 3 | 1 | inner |
| 9.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 162.11.d.b | 4 | |
| 3.b | odd | 2 | 1 | inner | 162.11.d.b | 4 | |
| 9.c | even | 3 | 1 | 54.11.b.a | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 162.11.d.b | 4 | |
| 9.d | odd | 6 | 1 | 54.11.b.a | ✓ | 2 | |
| 9.d | odd | 6 | 1 | inner | 162.11.d.b | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 54.11.b.a | ✓ | 2 | 9.c | even | 3 | 1 | |
| 54.11.b.a | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 162.11.d.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 162.11.d.b | 4 | 3.b | odd | 2 | 1 | inner | |
| 162.11.d.b | 4 | 9.c | even | 3 | 1 | inner | |
| 162.11.d.b | 4 | 9.d | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 250632T_{5}^{2} + 62816399424 \)
acting on \(S_{11}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 512 T^{2} + 262144 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 62816399424 \)
|
| $7$ |
\( (T^{2} - 3115 T + 9703225)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 11\!\cdots\!84 \)
|
| $13$ |
\( (T^{2} + 64319 T + 4136933761)^{2} \)
|
| $17$ |
\( (T^{2} + 7060997448)^{2} \)
|
| $19$ |
\( (T - 1675469)^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 46\!\cdots\!04 \)
|
| $29$ |
\( T^{4} + \cdots + 80\!\cdots\!44 \)
|
| $31$ |
\( (T^{2} + \cdots + 124631100268900)^{2} \)
|
| $37$ |
\( (T + 13147225)^{4} \)
|
| $41$ |
\( T^{4} + \cdots + 32\!\cdots\!24 \)
|
| $43$ |
\( (T^{2} + \cdots + 79307110946116)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 19\!\cdots\!64 \)
|
| $53$ |
\( (T^{2} + 20\!\cdots\!12)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 13\!\cdots\!84 \)
|
| $61$ |
\( (T^{2} + \cdots + 11\!\cdots\!89)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots + 19\!\cdots\!01)^{2} \)
|
| $71$ |
\( (T^{2} + 94\!\cdots\!08)^{2} \)
|
| $73$ |
\( (T - 900207263)^{4} \)
|
| $79$ |
\( (T^{2} + \cdots + 22\!\cdots\!49)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 59\!\cdots\!44 \)
|
| $89$ |
\( (T^{2} + 55\!\cdots\!32)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots + 24\!\cdots\!01)^{2} \)
|
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