Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [162,6,Mod(55,162)]
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([4]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("162.55");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 162.c (of order \(3\), 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: | \(25.9821788097\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{3} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( 3\zeta_{12}^{3} + 3\zeta_{12} \)
|
| \(\beta_{3}\) | \(=\) |
\( -3\zeta_{12}^{3} + 6\zeta_{12} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 9 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} ) / 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).
| \(n\) | \(83\) |
| \(\chi(n)\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 55.1 |
|
2.00000 | − | 3.46410i | 0 | −8.00000 | − | 13.8564i | −15.9904 | − | 27.6962i | 0 | −13.1577 | + | 22.7898i | −64.0000 | 0 | −127.923 | ||||||||||||||||||||||
| 55.2 | 2.00000 | − | 3.46410i | 0 | −8.00000 | − | 13.8564i | 9.99038 | + | 17.3038i | 0 | 101.158 | − | 175.210i | −64.0000 | 0 | 79.9230 | |||||||||||||||||||||||
| 109.1 | 2.00000 | + | 3.46410i | 0 | −8.00000 | + | 13.8564i | −15.9904 | + | 27.6962i | 0 | −13.1577 | − | 22.7898i | −64.0000 | 0 | −127.923 | |||||||||||||||||||||||
| 109.2 | 2.00000 | + | 3.46410i | 0 | −8.00000 | + | 13.8564i | 9.99038 | − | 17.3038i | 0 | 101.158 | + | 175.210i | −64.0000 | 0 | 79.9230 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 162.6.c.p | 4 | |
| 3.b | odd | 2 | 1 | 162.6.c.n | 4 | ||
| 9.c | even | 3 | 1 | 162.6.a.c | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 162.6.c.p | 4 | |
| 9.d | odd | 6 | 1 | 162.6.a.g | yes | 2 | |
| 9.d | odd | 6 | 1 | 162.6.c.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 162.6.a.c | ✓ | 2 | 9.c | even | 3 | 1 | |
| 162.6.a.g | yes | 2 | 9.d | odd | 6 | 1 | |
| 162.6.c.n | 4 | 3.b | odd | 2 | 1 | ||
| 162.6.c.n | 4 | 9.d | odd | 6 | 1 | ||
| 162.6.c.p | 4 | 1.a | even | 1 | 1 | trivial | |
| 162.6.c.p | 4 | 9.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 12T_{5}^{3} + 783T_{5}^{2} - 7668T_{5} + 408321 \)
acting on \(S_{6}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 4 T + 16)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 12 T^{3} + \cdots + 408321 \)
|
| $7$ |
\( T^{4} - 176 T^{3} + \cdots + 28344976 \)
|
| $11$ |
\( T^{4} + 540 T^{3} + \cdots + 138579984 \)
|
| $13$ |
\( T^{4} + \cdots + 2462243641 \)
|
| $17$ |
\( (T^{2} - 1560 T + 332973)^{2} \)
|
| $19$ |
\( (T^{2} + 1592 T + 231748)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 1022096736144 \)
|
| $29$ |
\( T^{4} + \cdots + 862614698311809 \)
|
| $31$ |
\( T^{4} + \cdots + 82\!\cdots\!04 \)
|
| $37$ |
\( (T^{2} - 3526 T - 110359331)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 36\!\cdots\!64 \)
|
| $43$ |
\( T^{4} + \cdots + 59\!\cdots\!84 \)
|
| $47$ |
\( T^{4} + \cdots + 10\!\cdots\!24 \)
|
| $53$ |
\( (T^{2} - 45672 T + 296911584)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 89\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 19\!\cdots\!81 \)
|
| $67$ |
\( T^{4} + \cdots + 40\!\cdots\!56 \)
|
| $71$ |
\( (T^{2} - 24708 T + 57192516)^{2} \)
|
| $73$ |
\( (T^{2} - 60322 T - 207320927)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 20\!\cdots\!04 \)
|
| $83$ |
\( T^{4} + \cdots + 12\!\cdots\!04 \)
|
| $89$ |
\( (T^{2} + 110976 T - 3572073531)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 16\!\cdots\!36 \)
|
| show more | |
| show less |