Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1872,4,Mod(1,1872)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1872, 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("1872.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1872.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: | \(110.451575531\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.6390848.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 58x^{2} - 152x - 7 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 936) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 58x^{2} - 152x - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 5\nu^{2} + 39\nu - 31 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 51\nu - 56 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 2\beta _1 + 29 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 10\beta_{3} + 4\beta_{2} + 59\beta _1 + 228 ) / 2 \)
|
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 | 0 | 0 | −19.3870 | 0 | −15.6343 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −1.90622 | 0 | 4.93923 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 4.18128 | 0 | 18.7509 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 9.11194 | 0 | −32.0558 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(13\) | \(-1\) |
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 | 1872.4.a.bn | 4 | |
| 3.b | odd | 2 | 1 | 1872.4.a.bq | 4 | ||
| 4.b | odd | 2 | 1 | 936.4.a.n | ✓ | 4 | |
| 12.b | even | 2 | 1 | 936.4.a.o | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 936.4.a.n | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 936.4.a.o | yes | 4 | 12.b | even | 2 | 1 | |
| 1872.4.a.bn | 4 | 1.a | even | 1 | 1 | trivial | |
| 1872.4.a.bq | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1872))\):
|
\( T_{5}^{4} + 8T_{5}^{3} - 208T_{5}^{2} + 320T_{5} + 1408 \)
|
|
\( T_{7}^{4} + 24T_{7}^{3} - 536T_{7}^{2} - 7456T_{7} + 46416 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 8 T^{3} + \cdots + 1408 \)
|
| $7$ |
\( T^{4} + 24 T^{3} + \cdots + 46416 \)
|
| $11$ |
\( T^{4} - 64 T^{3} + \cdots - 102384 \)
|
| $13$ |
\( (T - 13)^{4} \)
|
| $17$ |
\( T^{4} + 80 T^{3} + \cdots + 5727488 \)
|
| $19$ |
\( T^{4} + 24 T^{3} + \cdots - 11611312 \)
|
| $23$ |
\( T^{4} - 16 T^{3} + \cdots - 29332224 \)
|
| $29$ |
\( T^{4} + 240 T^{3} + \cdots - 859951872 \)
|
| $31$ |
\( T^{4} + 24 T^{3} + \cdots + 426232656 \)
|
| $37$ |
\( T^{4} + 152 T^{3} + \cdots + 580012304 \)
|
| $41$ |
\( T^{4} + 136 T^{3} + \cdots + 146583168 \)
|
| $43$ |
\( T^{4} + \cdots - 3794221056 \)
|
| $47$ |
\( T^{4} - 528 T^{3} + \cdots - 61976304 \)
|
| $53$ |
\( T^{4} + \cdots + 44477724672 \)
|
| $59$ |
\( T^{4} + \cdots + 40181983248 \)
|
| $61$ |
\( T^{4} + \cdots + 2559945488 \)
|
| $67$ |
\( T^{4} + \cdots + 169607975248 \)
|
| $71$ |
\( T^{4} + \cdots - 1971142128 \)
|
| $73$ |
\( T^{4} + \cdots + 262835399952 \)
|
| $79$ |
\( T^{4} + 464 T^{3} + \cdots - 59551488 \)
|
| $83$ |
\( T^{4} + \cdots - 70224262896 \)
|
| $89$ |
\( T^{4} + \cdots - 213851488128 \)
|
| $97$ |
\( T^{4} + \cdots - 115241661168 \)
|
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