Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,6,Mod(18,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, 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("49.18");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.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: | \(7.85880717084\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-13})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 13x^{2} + 169 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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 in terms of a root \(\nu\) of
\( x^{4} - 13x^{2} + 169 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 13 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{3} + 52\nu ) / 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{3} + 104\nu ) / 13 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( 13\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -13\beta_{3} + 26\beta_{2} ) / 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 18.1 |
|
1.00000 | + | 1.73205i | −12.4900 | + | 21.6333i | 14.0000 | − | 24.2487i | −37.4700 | − | 64.8999i | −49.9600 | 0 | 120.000 | −190.500 | − | 329.956i | 74.9400 | − | 129.800i | ||||||||||||||||||
| 18.2 | 1.00000 | + | 1.73205i | 12.4900 | − | 21.6333i | 14.0000 | − | 24.2487i | 37.4700 | + | 64.8999i | 49.9600 | 0 | 120.000 | −190.500 | − | 329.956i | −74.9400 | + | 129.800i | |||||||||||||||||||
| 30.1 | 1.00000 | − | 1.73205i | −12.4900 | − | 21.6333i | 14.0000 | + | 24.2487i | −37.4700 | + | 64.8999i | −49.9600 | 0 | 120.000 | −190.500 | + | 329.956i | 74.9400 | + | 129.800i | |||||||||||||||||||
| 30.2 | 1.00000 | − | 1.73205i | 12.4900 | + | 21.6333i | 14.0000 | + | 24.2487i | 37.4700 | − | 64.8999i | 49.9600 | 0 | 120.000 | −190.500 | + | 329.956i | −74.9400 | − | 129.800i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 49.6.c.g | 4 | |
| 7.b | odd | 2 | 1 | inner | 49.6.c.g | 4 | |
| 7.c | even | 3 | 1 | 49.6.a.c | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 49.6.c.g | 4 | |
| 7.d | odd | 6 | 1 | 49.6.a.c | ✓ | 2 | |
| 7.d | odd | 6 | 1 | inner | 49.6.c.g | 4 | |
| 21.g | even | 6 | 1 | 441.6.a.u | 2 | ||
| 21.h | odd | 6 | 1 | 441.6.a.u | 2 | ||
| 28.f | even | 6 | 1 | 784.6.a.z | 2 | ||
| 28.g | odd | 6 | 1 | 784.6.a.z | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 49.6.a.c | ✓ | 2 | 7.c | even | 3 | 1 | |
| 49.6.a.c | ✓ | 2 | 7.d | odd | 6 | 1 | |
| 49.6.c.g | 4 | 1.a | even | 1 | 1 | trivial | |
| 49.6.c.g | 4 | 7.b | odd | 2 | 1 | inner | |
| 49.6.c.g | 4 | 7.c | even | 3 | 1 | inner | |
| 49.6.c.g | 4 | 7.d | odd | 6 | 1 | inner | |
| 441.6.a.u | 2 | 21.g | even | 6 | 1 | ||
| 441.6.a.u | 2 | 21.h | odd | 6 | 1 | ||
| 784.6.a.z | 2 | 28.f | even | 6 | 1 | ||
| 784.6.a.z | 2 | 28.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(49, [\chi])\):
|
\( T_{2}^{2} - 2T_{2} + 4 \)
|
|
\( T_{3}^{4} + 624T_{3}^{2} + 389376 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 4)^{2} \)
|
| $3$ |
\( T^{4} + 624 T^{2} + 389376 \)
|
| $5$ |
\( T^{4} + 5616 T^{2} + 31539456 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} - 284 T + 80656)^{2} \)
|
| $13$ |
\( (T^{2} - 275184)^{2} \)
|
| $17$ |
\( T^{4} + 22464 T^{2} + 504631296 \)
|
| $19$ |
\( T^{4} + \cdots + 22307257979136 \)
|
| $23$ |
\( (T^{2} + 1496 T + 2238016)^{2} \)
|
| $29$ |
\( (T + 4366)^{4} \)
|
| $31$ |
\( T^{4} + \cdots + 17\!\cdots\!96 \)
|
| $37$ |
\( (T^{2} - 12630 T + 159516900)^{2} \)
|
| $41$ |
\( (T^{2} - 89159616)^{2} \)
|
| $43$ |
\( (T + 1356)^{4} \)
|
| $47$ |
\( T^{4} + \cdots + 10\!\cdots\!16 \)
|
| $53$ |
\( (T^{2} + 14150 T + 200222500)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 19\!\cdots\!56 \)
|
| $61$ |
\( T^{4} + \cdots + 16\!\cdots\!00 \)
|
| $67$ |
\( (T^{2} - 3644 T + 13278736)^{2} \)
|
| $71$ |
\( (T - 35632)^{4} \)
|
| $73$ |
\( T^{4} + \cdots + 27\!\cdots\!76 \)
|
| $79$ |
\( (T^{2} - 54616 T + 2982907456)^{2} \)
|
| $83$ |
\( (T^{2} - 275184)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 17\!\cdots\!36 \)
|
| $97$ |
\( (T^{2} - 33710040000)^{2} \)
|
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