Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [378,3,Mod(53,378)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(378, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("378.53");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 378.s (of order \(6\), degree \(2\), 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: | \(10.2997539928\) |
| 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: | \(\Z[a_1, \ldots, a_{25}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(325\) |
| \(\chi(n)\) | \(-1\) | \(-1 + \beta_{2}\) |
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 |
|
−1.22474 | + | 0.707107i | 0 | 1.00000 | − | 1.73205i | 2.44949 | − | 1.41421i | 0 | −7.00000 | 2.82843i | 0 | −2.00000 | + | 3.46410i | ||||||||||||||||||||||
| 53.2 | 1.22474 | − | 0.707107i | 0 | 1.00000 | − | 1.73205i | −2.44949 | + | 1.41421i | 0 | −7.00000 | − | 2.82843i | 0 | −2.00000 | + | 3.46410i | ||||||||||||||||||||||
| 107.1 | −1.22474 | − | 0.707107i | 0 | 1.00000 | + | 1.73205i | 2.44949 | + | 1.41421i | 0 | −7.00000 | − | 2.82843i | 0 | −2.00000 | − | 3.46410i | ||||||||||||||||||||||
| 107.2 | 1.22474 | + | 0.707107i | 0 | 1.00000 | + | 1.73205i | −2.44949 | − | 1.41421i | 0 | −7.00000 | 2.82843i | 0 | −2.00000 | − | 3.46410i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 21.h | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 378.3.s.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 378.3.s.a | ✓ | 4 |
| 7.c | even | 3 | 1 | inner | 378.3.s.a | ✓ | 4 |
| 21.h | odd | 6 | 1 | inner | 378.3.s.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 378.3.s.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 378.3.s.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 378.3.s.a | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 378.3.s.a | ✓ | 4 | 21.h | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 8T_{5}^{2} + 64 \)
acting on \(S_{3}^{\mathrm{new}}(378, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2T^{2} + 4 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 8T^{2} + 64 \)
|
| $7$ |
\( (T + 7)^{4} \)
|
| $11$ |
\( T^{4} - 50T^{2} + 2500 \)
|
| $13$ |
\( (T - 3)^{4} \)
|
| $17$ |
\( T^{4} - 578 T^{2} + 334084 \)
|
| $19$ |
\( (T^{2} + 20 T + 400)^{2} \)
|
| $23$ |
\( T^{4} - 1250 T^{2} + 1562500 \)
|
| $29$ |
\( (T^{2} + 722)^{2} \)
|
| $31$ |
\( (T^{2} - 51 T + 2601)^{2} \)
|
| $37$ |
\( (T^{2} - 7 T + 49)^{2} \)
|
| $41$ |
\( (T^{2} + 882)^{2} \)
|
| $43$ |
\( (T - 47)^{4} \)
|
| $47$ |
\( T^{4} - 4050 T^{2} + 16402500 \)
|
| $53$ |
\( T^{4} - 450 T^{2} + 202500 \)
|
| $59$ |
\( T^{4} - 722 T^{2} + 521284 \)
|
| $61$ |
\( (T^{2} - 39 T + 1521)^{2} \)
|
| $67$ |
\( (T^{2} - 17 T + 289)^{2} \)
|
| $71$ |
\( (T^{2} + 2312)^{2} \)
|
| $73$ |
\( (T^{2} - 20 T + 400)^{2} \)
|
| $79$ |
\( (T^{2} + 9 T + 81)^{2} \)
|
| $83$ |
\( (T^{2} + 4418)^{2} \)
|
| $89$ |
\( T^{4} - 16200 T^{2} + 262440000 \)
|
| $97$ |
\( (T + 23)^{4} \)
|
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