Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2548,2,Mod(2157,2548)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2548, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2548.2157");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2548.g (of order \(2\), degree \(1\), 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: | \(20.3458824350\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.65712.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 7x^{2} + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 364) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 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
\( x^{4} + 7x^{2} + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + 6\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu + 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{2} + \nu - 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} - 8 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{3} - 3\beta_{2} + \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(885\) | \(1275\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2157.1 |
|
0 | −1.00000 | 0 | − | 3.75270i | 0 | 0 | 0 | −2.00000 | 0 | |||||||||||||||||||||||||||||
| 2157.2 | 0 | −1.00000 | 0 | − | 1.38464i | 0 | 0 | 0 | −2.00000 | 0 | ||||||||||||||||||||||||||||||
| 2157.3 | 0 | −1.00000 | 0 | 1.38464i | 0 | 0 | 0 | −2.00000 | 0 | |||||||||||||||||||||||||||||||
| 2157.4 | 0 | −1.00000 | 0 | 3.75270i | 0 | 0 | 0 | −2.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2548.2.g.a | 4 | |
| 7.b | odd | 2 | 1 | 2548.2.g.d | 4 | ||
| 7.c | even | 3 | 2 | 364.2.y.a | ✓ | 8 | |
| 7.d | odd | 6 | 2 | 2548.2.y.a | 8 | ||
| 13.b | even | 2 | 1 | inner | 2548.2.g.a | 4 | |
| 21.h | odd | 6 | 2 | 3276.2.gv.d | 8 | ||
| 91.b | odd | 2 | 1 | 2548.2.g.d | 4 | ||
| 91.r | even | 6 | 2 | 364.2.y.a | ✓ | 8 | |
| 91.s | odd | 6 | 2 | 2548.2.y.a | 8 | ||
| 273.w | odd | 6 | 2 | 3276.2.gv.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 364.2.y.a | ✓ | 8 | 7.c | even | 3 | 2 | |
| 364.2.y.a | ✓ | 8 | 91.r | even | 6 | 2 | |
| 2548.2.g.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 2548.2.g.a | 4 | 13.b | even | 2 | 1 | inner | |
| 2548.2.g.d | 4 | 7.b | odd | 2 | 1 | ||
| 2548.2.g.d | 4 | 91.b | odd | 2 | 1 | ||
| 2548.2.y.a | 8 | 7.d | odd | 6 | 2 | ||
| 2548.2.y.a | 8 | 91.s | odd | 6 | 2 | ||
| 3276.2.gv.d | 8 | 21.h | odd | 6 | 2 | ||
| 3276.2.gv.d | 8 | 273.w | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(2548, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 1)^{4} \)
|
| $5$ |
\( T^{4} + 16T^{2} + 27 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 16T^{2} + 27 \)
|
| $13$ |
\( T^{4} + 2 T^{3} + \cdots + 169 \)
|
| $17$ |
\( (T^{2} - 4 T - 33)^{2} \)
|
| $19$ |
\( T^{4} + 40T^{2} + 363 \)
|
| $23$ |
\( (T - 3)^{4} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} + 40T^{2} + 363 \)
|
| $37$ |
\( T^{4} + 48T^{2} + 243 \)
|
| $41$ |
\( T^{4} + 64T^{2} + 432 \)
|
| $43$ |
\( (T^{2} + 6 T - 28)^{2} \)
|
| $47$ |
\( T^{4} + 16T^{2} + 27 \)
|
| $53$ |
\( (T^{2} - 8 T - 21)^{2} \)
|
| $59$ |
\( T^{4} + 144T^{2} + 2187 \)
|
| $61$ |
\( (T^{2} - 37)^{2} \)
|
| $67$ |
\( T^{4} + 120T^{2} + 3267 \)
|
| $71$ |
\( T^{4} + 252T^{2} + 3888 \)
|
| $73$ |
\( T^{4} + 280 T^{2} + 15123 \)
|
| $79$ |
\( (T^{2} + 8 T - 21)^{2} \)
|
| $83$ |
\( T^{4} + 252T^{2} + 3888 \)
|
| $89$ |
\( T^{4} + 144T^{2} + 2187 \)
|
| $97$ |
\( T^{4} + 292 T^{2} + 21168 \)
|
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