Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [162,6,Mod(1,162)]
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([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("162.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 162.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: | \(25.9821788097\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{6}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 6 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 18) |
| 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 \(\beta = 6\sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−4.00000 | 0 | 16.0000 | −2.39388 | 0 | −51.6969 | −64.0000 | 0 | 9.57551 | ||||||||||||||||||||||||
| 1.2 | −4.00000 | 0 | 16.0000 | 56.3939 | 0 | −22.3031 | −64.0000 | 0 | −225.576 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
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 | 162.6.a.e | 2 | |
| 3.b | odd | 2 | 1 | 162.6.a.f | 2 | ||
| 9.c | even | 3 | 2 | 18.6.c.a | ✓ | 4 | |
| 9.d | odd | 6 | 2 | 54.6.c.a | 4 | ||
| 36.f | odd | 6 | 2 | 144.6.i.a | 4 | ||
| 36.h | even | 6 | 2 | 432.6.i.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 18.6.c.a | ✓ | 4 | 9.c | even | 3 | 2 | |
| 54.6.c.a | 4 | 9.d | odd | 6 | 2 | ||
| 144.6.i.a | 4 | 36.f | odd | 6 | 2 | ||
| 162.6.a.e | 2 | 1.a | even | 1 | 1 | trivial | |
| 162.6.a.f | 2 | 3.b | odd | 2 | 1 | ||
| 432.6.i.a | 4 | 36.h | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 54T_{5} - 135 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(162))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 54T - 135 \)
|
| $7$ |
\( T^{2} + 74T + 1153 \)
|
| $11$ |
\( T^{2} - 78T - 397863 \)
|
| $13$ |
\( T^{2} + 1106 T + 219409 \)
|
| $17$ |
\( T^{2} - 492 T - 1848060 \)
|
| $19$ |
\( T^{2} + 1640 T - 2648816 \)
|
| $23$ |
\( T^{2} - 5538 T + 7532361 \)
|
| $29$ |
\( T^{2} - 3894 T + 469593 \)
|
| $31$ |
\( T^{2} + 4718 T - 35307719 \)
|
| $37$ |
\( T^{2} + 4796 T - 141621212 \)
|
| $41$ |
\( T^{2} + 15354 T + 58893993 \)
|
| $43$ |
\( T^{2} + 32858 T + 254513617 \)
|
| $47$ |
\( T^{2} + 24954 T + 142283313 \)
|
| $53$ |
\( T^{2} + 16332 T + 59839812 \)
|
| $59$ |
\( T^{2} + 21966 T + 99734553 \)
|
| $61$ |
\( T^{2} + 3050 T - 105947399 \)
|
| $67$ |
\( T^{2} + 36758 T - 265336415 \)
|
| $71$ |
\( T^{2} + 73848 T + 665096112 \)
|
| $73$ |
\( T^{2} + 51188 T + 491562436 \)
|
| $79$ |
\( T^{2} - 14926 T - 52271015 \)
|
| $83$ |
\( T^{2} + \cdots + 2054767617 \)
|
| $89$ |
\( T^{2} + \cdots - 7978887036 \)
|
| $97$ |
\( T^{2} + \cdots - 9106027655 \)
|
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