Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1380,4,Mod(1,1380)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1380, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1380.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1380.a (trivial) |
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: | yes |
| Analytic conductor: | \(81.4226358079\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 278x^{3} + 216x^{2} + 14064x - 33408 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 278x^{3} + 216x^{2} + 14064x - 33408 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{4} - 25\nu^{3} - 1026\nu^{2} + 2280\nu + 16848 ) / 1584 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 5\nu^{3} + 522\nu^{2} - 456\nu - 38376 ) / 792 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{4} - 13\nu^{3} + 1065\nu^{2} + 3126\nu - 39852 ) / 396 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{3} + 2\beta_{2} + 221 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -24\beta_{4} + 37\beta_{3} - 62\beta_{2} + 150\beta _1 + 37 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -120\beta_{4} + 1211\beta_{3} + 734\beta_{2} + 294\beta _1 + 38795 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −3.00000 | 0 | −5.00000 | 0 | −28.3931 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | −5.00000 | 0 | −10.8260 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | −5.00000 | 0 | 5.33445 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −3.00000 | 0 | −5.00000 | 0 | 5.38683 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −3.00000 | 0 | −5.00000 | 0 | 21.4978 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(23\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1380.4.a.e | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1380.4.a.e | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{5} + 7T_{7}^{4} - 697T_{7}^{3} - 355T_{7}^{2} + 55452T_{7} - 189888 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1380))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( (T + 3)^{5} \)
|
| $5$ |
\( (T + 5)^{5} \)
|
| $7$ |
\( T^{5} + 7 T^{4} + \cdots - 189888 \)
|
| $11$ |
\( T^{5} - 64 T^{4} + \cdots - 99130368 \)
|
| $13$ |
\( T^{5} - 80 T^{4} + \cdots + 64527168 \)
|
| $17$ |
\( T^{5} + \cdots + 5954066460 \)
|
| $19$ |
\( T^{5} + 52 T^{4} + \cdots - 86489600 \)
|
| $23$ |
\( (T - 23)^{5} \)
|
| $29$ |
\( T^{5} + \cdots - 23912382996 \)
|
| $31$ |
\( T^{5} + \cdots + 1652313280 \)
|
| $37$ |
\( T^{5} + \cdots - 83346019308 \)
|
| $41$ |
\( T^{5} + \cdots - 528639433236 \)
|
| $43$ |
\( T^{5} + \cdots + 141065779200 \)
|
| $47$ |
\( T^{5} + \cdots + 1722888418560 \)
|
| $53$ |
\( T^{5} + \cdots + 9953509572 \)
|
| $59$ |
\( T^{5} + \cdots + 174390112800 \)
|
| $61$ |
\( T^{5} + \cdots + 1096477507536 \)
|
| $67$ |
\( T^{5} + \cdots + 1735107350656 \)
|
| $71$ |
\( T^{5} + \cdots - 35762779826496 \)
|
| $73$ |
\( T^{5} + \cdots - 2448550788896 \)
|
| $79$ |
\( T^{5} + \cdots + 4222672896 \)
|
| $83$ |
\( T^{5} + \cdots + 15605029425600 \)
|
| $89$ |
\( T^{5} + \cdots - 4785875661312 \)
|
| $97$ |
\( T^{5} + \cdots + 3328399016384 \)
|
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