Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1232,4,Mod(1,1232)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1232, 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("1232.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1232 = 2^{4} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1232.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: | \(72.6903531271\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.522072.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 12x^{2} + 5x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 77) |
| 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
\( x^{4} - x^{3} - 12x^{2} + 5x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 12\nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -2\nu^{3} + 2\nu^{2} + 24\nu - 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + 13 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 6\beta _1 + 3 \)
|
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 | −10.1459 | 0 | 8.69995 | 0 | 7.00000 | 0 | 75.9402 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −6.57251 | 0 | 15.5514 | 0 | 7.00000 | 0 | 16.1978 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.77399 | 0 | 1.84418 | 0 | 7.00000 | 0 | −19.3050 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 5.49244 | 0 | −16.0955 | 0 | 7.00000 | 0 | 3.16692 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 1232.4.a.s | 4 | |
| 4.b | odd | 2 | 1 | 77.4.a.d | ✓ | 4 | |
| 12.b | even | 2 | 1 | 693.4.a.l | 4 | ||
| 20.d | odd | 2 | 1 | 1925.4.a.p | 4 | ||
| 28.d | even | 2 | 1 | 539.4.a.g | 4 | ||
| 44.c | even | 2 | 1 | 847.4.a.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 77.4.a.d | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 539.4.a.g | 4 | 28.d | even | 2 | 1 | ||
| 693.4.a.l | 4 | 12.b | even | 2 | 1 | ||
| 847.4.a.d | 4 | 44.c | even | 2 | 1 | ||
| 1232.4.a.s | 4 | 1.a | even | 1 | 1 | trivial | |
| 1925.4.a.p | 4 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):
|
\( T_{3}^{4} + 14T_{3}^{3} + 6T_{3}^{2} - 436T_{3} - 1016 \)
|
|
\( T_{5}^{4} - 10T_{5}^{3} - 240T_{5}^{2} + 2648T_{5} - 4016 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 14 T^{3} + \cdots - 1016 \)
|
| $5$ |
\( T^{4} - 10 T^{3} + \cdots - 4016 \)
|
| $7$ |
\( (T - 7)^{4} \)
|
| $11$ |
\( (T - 11)^{4} \)
|
| $13$ |
\( T^{4} - 58 T^{3} + \cdots - 4947656 \)
|
| $17$ |
\( T^{4} - 4 T^{3} + \cdots - 2705024 \)
|
| $19$ |
\( T^{4} + 258 T^{3} + \cdots - 14423904 \)
|
| $23$ |
\( T^{4} + 8 T^{3} + \cdots - 17449856 \)
|
| $29$ |
\( T^{4} + 396 T^{3} + \cdots + 22336464 \)
|
| $31$ |
\( T^{4} - 56 T^{3} + \cdots - 11250248 \)
|
| $37$ |
\( T^{4} - 84 T^{3} + \cdots + 11157312 \)
|
| $41$ |
\( T^{4} - 52 T^{3} + \cdots - 659233664 \)
|
| $43$ |
\( T^{4} + \cdots + 1210397376 \)
|
| $47$ |
\( T^{4} + 8 T^{3} + \cdots - 318931592 \)
|
| $53$ |
\( T^{4} - 624 T^{3} + \cdots + 403923072 \)
|
| $59$ |
\( T^{4} + \cdots + 17599820728 \)
|
| $61$ |
\( T^{4} + \cdots + 6668930664 \)
|
| $67$ |
\( T^{4} + \cdots - 140865466496 \)
|
| $71$ |
\( T^{4} + \cdots - 72982082688 \)
|
| $73$ |
\( T^{4} + \cdots - 322052228384 \)
|
| $79$ |
\( T^{4} + \cdots - 59537293568 \)
|
| $83$ |
\( T^{4} + \cdots + 21951092064 \)
|
| $89$ |
\( T^{4} + \cdots - 109303561968 \)
|
| $97$ |
\( T^{4} + \cdots + 868634650768 \)
|
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