Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [162,7,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, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("162.53");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| 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: | \(37.2687615464\) |
| 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 |
|
−4.89898 | + | 2.82843i | 0 | 16.0000 | − | 27.7128i | −36.7423 | − | 21.2132i | 0 | −194.500 | − | 336.884i | 181.019i | 0 | 240.000 | ||||||||||||||||||||||
| 53.2 | 4.89898 | − | 2.82843i | 0 | 16.0000 | − | 27.7128i | 36.7423 | + | 21.2132i | 0 | −194.500 | − | 336.884i | − | 181.019i | 0 | 240.000 | ||||||||||||||||||||||
| 107.1 | −4.89898 | − | 2.82843i | 0 | 16.0000 | + | 27.7128i | −36.7423 | + | 21.2132i | 0 | −194.500 | + | 336.884i | − | 181.019i | 0 | 240.000 | ||||||||||||||||||||||
| 107.2 | 4.89898 | + | 2.82843i | 0 | 16.0000 | + | 27.7128i | 36.7423 | − | 21.2132i | 0 | −194.500 | + | 336.884i | 181.019i | 0 | 240.000 | |||||||||||||||||||||||
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 | 162.7.d.a | 4 | |
| 3.b | odd | 2 | 1 | inner | 162.7.d.a | 4 | |
| 9.c | even | 3 | 1 | 54.7.b.b | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 162.7.d.a | 4 | |
| 9.d | odd | 6 | 1 | 54.7.b.b | ✓ | 2 | |
| 9.d | odd | 6 | 1 | inner | 162.7.d.a | 4 | |
| 36.f | odd | 6 | 1 | 432.7.e.c | 2 | ||
| 36.h | even | 6 | 1 | 432.7.e.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 54.7.b.b | ✓ | 2 | 9.c | even | 3 | 1 | |
| 54.7.b.b | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 162.7.d.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 162.7.d.a | 4 | 3.b | odd | 2 | 1 | inner | |
| 162.7.d.a | 4 | 9.c | even | 3 | 1 | inner | |
| 162.7.d.a | 4 | 9.d | odd | 6 | 1 | inner | |
| 432.7.e.c | 2 | 36.f | odd | 6 | 1 | ||
| 432.7.e.c | 2 | 36.h | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 1800T_{5}^{2} + 3240000 \)
acting on \(S_{7}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 32T^{2} + 1024 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 1800 T^{2} + 3240000 \)
|
| $7$ |
\( (T^{2} + 389 T + 151321)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 18677955240000 \)
|
| $13$ |
\( (T^{2} + 1415 T + 2002225)^{2} \)
|
| $17$ |
\( (T^{2} + 5604552)^{2} \)
|
| $19$ |
\( (T + 3067)^{4} \)
|
| $23$ |
\( T^{4} + \cdots + 19\!\cdots\!64 \)
|
| $29$ |
\( T^{4} + \cdots + 28\!\cdots\!84 \)
|
| $31$ |
\( (T^{2} - 11338 T + 128550244)^{2} \)
|
| $37$ |
\( (T - 47135)^{4} \)
|
| $41$ |
\( T^{4} + \cdots + 14\!\cdots\!00 \)
|
| $43$ |
\( (T^{2} + 145118 T + 21059233924)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $53$ |
\( (T^{2} + 70654915872)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 17\!\cdots\!64 \)
|
| $61$ |
\( (T^{2} - 350305 T + 122713593025)^{2} \)
|
| $67$ |
\( (T^{2} + 120341 T + 14481956281)^{2} \)
|
| $71$ |
\( (T^{2} + 112292500608)^{2} \)
|
| $73$ |
\( (T - 175151)^{4} \)
|
| $79$ |
\( (T^{2} - 252259 T + 63634603081)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 13\!\cdots\!24 \)
|
| $89$ |
\( (T^{2} + 624054513672)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots + 1682481381025)^{2} \)
|
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