Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,3,Mod(376,525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("525.376");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 525.o (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: | \(14.3052138789\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 2\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 4\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{3} + 4\beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(176\) | \(451\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 376.1 |
|
−0.724745 | + | 1.25529i | −1.50000 | + | 0.866025i | 0.949490 | + | 1.64456i | 0 | − | 2.51059i | 6.50000 | − | 2.59808i | −8.55051 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||
| 376.2 | 1.72474 | − | 2.98735i | −1.50000 | + | 0.866025i | −3.94949 | − | 6.84072i | 0 | 5.97469i | 6.50000 | − | 2.59808i | −13.4495 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||
| 451.1 | −0.724745 | − | 1.25529i | −1.50000 | − | 0.866025i | 0.949490 | − | 1.64456i | 0 | 2.51059i | 6.50000 | + | 2.59808i | −8.55051 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||
| 451.2 | 1.72474 | + | 2.98735i | −1.50000 | − | 0.866025i | −3.94949 | + | 6.84072i | 0 | − | 5.97469i | 6.50000 | + | 2.59808i | −13.4495 | 1.50000 | + | 2.59808i | 0 | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 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 | 525.3.o.k | yes | 4 |
| 5.b | even | 2 | 1 | 525.3.o.j | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 525.3.s.g | 8 | ||
| 7.d | odd | 6 | 1 | inner | 525.3.o.k | yes | 4 |
| 35.i | odd | 6 | 1 | 525.3.o.j | ✓ | 4 | |
| 35.k | even | 12 | 2 | 525.3.s.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 525.3.o.j | ✓ | 4 | 5.b | even | 2 | 1 | |
| 525.3.o.j | ✓ | 4 | 35.i | odd | 6 | 1 | |
| 525.3.o.k | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 525.3.o.k | yes | 4 | 7.d | odd | 6 | 1 | inner |
| 525.3.s.g | 8 | 5.c | odd | 4 | 2 | ||
| 525.3.s.g | 8 | 35.k | even | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\):
|
\( T_{2}^{4} - 2T_{2}^{3} + 9T_{2}^{2} + 10T_{2} + 25 \)
|
|
\( T_{11}^{4} - 4T_{11}^{3} + 36T_{11}^{2} + 80T_{11} + 400 \)
|
|
\( T_{13}^{4} + 294T_{13}^{2} + 9 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2 T^{3} + \cdots + 25 \)
|
| $3$ |
\( (T^{2} + 3 T + 3)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} - 13 T + 49)^{2} \)
|
| $11$ |
\( T^{4} - 4 T^{3} + \cdots + 400 \)
|
| $13$ |
\( T^{4} + 294T^{2} + 9 \)
|
| $17$ |
\( T^{4} + 36 T^{3} + \cdots + 1296 \)
|
| $19$ |
\( T^{4} - 30 T^{3} + \cdots + 328329 \)
|
| $23$ |
\( (T^{2} + 20 T + 400)^{2} \)
|
| $29$ |
\( (T^{2} + 4 T - 860)^{2} \)
|
| $31$ |
\( T^{4} - 1152 T^{2} + 1327104 \)
|
| $37$ |
\( T^{4} + 50 T^{3} + \cdots + 625 \)
|
| $41$ |
\( T^{4} + 7656 T^{2} + 10419984 \)
|
| $43$ |
\( (T^{2} + 100 T + 2404)^{2} \)
|
| $47$ |
\( T^{4} - 144 T^{3} + \cdots + 1166400 \)
|
| $53$ |
\( T^{4} - 44 T^{3} + \cdots + 24167056 \)
|
| $59$ |
\( T^{4} + 24 T^{3} + \cdots + 33454656 \)
|
| $61$ |
\( T^{4} + 234 T^{3} + \cdots + 15327225 \)
|
| $67$ |
\( T^{4} + 2 T^{3} + \cdots + 75047569 \)
|
| $71$ |
\( (T^{2} + 52 T - 1724)^{2} \)
|
| $73$ |
\( T^{4} - 90 T^{3} + \cdots + 149769 \)
|
| $79$ |
\( T^{4} - 38 T^{3} + \cdots + 1380625 \)
|
| $83$ |
\( T^{4} + 7056 T^{2} + 5184 \)
|
| $89$ |
\( T^{4} + 72 T^{3} + \cdots + 1871424 \)
|
| $97$ |
\( T^{4} + 22470 T^{2} + 90269001 \)
|
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