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: | \(0\) |
| 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: |
\( x^{5} - 116x^{3} - 22x^{2} + 2859x - 2034 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 308) |
| 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
\( x^{5} - 116x^{3} - 22x^{2} + 2859x - 2034 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 21\nu^{3} + 35\nu^{2} - 1313\nu + 1194 ) / 120 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{4} - 27\nu^{3} - 485\nu^{2} + 1271\nu + 1242 ) / 120 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 21\nu^{3} + 85\nu^{2} + 1313\nu - 6714 ) / 120 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{2} + 46 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + \beta_{3} + 9\beta_{2} + 66\beta _1 + 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( 77\beta_{4} + 21\beta_{3} + 104\beta_{2} + 73\beta _1 + 3056 \)
|
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 | −9.16912 | 0 | −10.7180 | 0 | −7.00000 | 0 | 57.0728 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −8.08017 | 0 | 21.9878 | 0 | −7.00000 | 0 | 38.2892 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.268648 | 0 | 5.16987 | 0 | −7.00000 | 0 | −26.9278 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 4.11231 | 0 | −17.6741 | 0 | −7.00000 | 0 | −10.0889 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 8.40564 | 0 | 16.2345 | 0 | −7.00000 | 0 | 43.6547 | 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.x | 5 | |
| 4.b | odd | 2 | 1 | 308.4.a.e | ✓ | 5 | |
| 28.d | even | 2 | 1 | 2156.4.a.g | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 308.4.a.e | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 1232.4.a.x | 5 | 1.a | even | 1 | 1 | trivial | |
| 2156.4.a.g | 5 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1232))\):
|
\( T_{3}^{5} + 5T_{3}^{4} - 106T_{3}^{3} - 360T_{3}^{2} + 2472T_{3} + 688 \)
|
|
\( T_{5}^{5} - 15T_{5}^{4} - 488T_{5}^{3} + 5680T_{5}^{2} + 52656T_{5} - 349584 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} + 5 T^{4} + \cdots + 688 \)
|
| $5$ |
\( T^{5} - 15 T^{4} + \cdots - 349584 \)
|
| $7$ |
\( (T + 7)^{5} \)
|
| $11$ |
\( (T - 11)^{5} \)
|
| $13$ |
\( T^{5} - 70 T^{4} + \cdots - 71584768 \)
|
| $17$ |
\( T^{5} - 104 T^{4} + \cdots - 762459840 \)
|
| $19$ |
\( T^{5} - 94 T^{4} + \cdots - 207309056 \)
|
| $23$ |
\( T^{5} + \cdots + 13952097024 \)
|
| $29$ |
\( T^{5} - 14 T^{4} + \cdots - 105914592 \)
|
| $31$ |
\( T^{5} + \cdots + 26839714544 \)
|
| $37$ |
\( T^{5} + \cdots + 8597743600 \)
|
| $41$ |
\( T^{5} + \cdots + 2108713005504 \)
|
| $43$ |
\( T^{5} + \cdots - 94214553600 \)
|
| $47$ |
\( T^{5} + 378 T^{4} + \cdots + 30164736 \)
|
| $53$ |
\( T^{5} + \cdots - 54102174432 \)
|
| $59$ |
\( T^{5} + \cdots - 6726646275120 \)
|
| $61$ |
\( T^{5} + \cdots - 3356972813440 \)
|
| $67$ |
\( T^{5} + \cdots + 19330693120 \)
|
| $71$ |
\( T^{5} + \cdots - 30220124785920 \)
|
| $73$ |
\( T^{5} + \cdots - 415472733120 \)
|
| $79$ |
\( T^{5} + \cdots + 41144623266816 \)
|
| $83$ |
\( T^{5} + \cdots - 1948582310400 \)
|
| $89$ |
\( T^{5} + \cdots - 227399586768 \)
|
| $97$ |
\( T^{5} + \cdots + 1065686219248 \)
|
| show more | |
| show less |