Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1089,4,Mod(1,1089)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1089, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1089.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1089 = 3^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1089.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: | \(64.2530799963\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.5225.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 8x^{2} + x + 11 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 33) |
| 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
\( x^{4} - x^{3} - 8x^{2} + x + 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 4\nu + 3 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 4\nu^{2} + 2\nu - 11 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 4\beta_{2} + 6\beta _1 + 5 \)
|
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 |
|
−2.50105 | 0 | −1.74477 | −10.4342 | 0 | −32.7556 | 24.3721 | 0 | 26.0964 | ||||||||||||||||||||||||||||||
| 1.2 | 0.130849 | 0 | −7.98288 | −6.68911 | 0 | −14.3420 | −2.09135 | 0 | −0.875265 | |||||||||||||||||||||||||||||||
| 1.3 | 0.646943 | 0 | −7.58146 | 11.1424 | 0 | 26.1376 | −10.0803 | 0 | 7.20851 | |||||||||||||||||||||||||||||||
| 1.4 | 4.72325 | 0 | 14.3091 | −6.01909 | 0 | 9.96005 | 29.7996 | 0 | −28.4297 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(11\) | \(1\) |
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 | 1089.4.a.bg | 4 | |
| 3.b | odd | 2 | 1 | 363.4.a.p | 4 | ||
| 11.b | odd | 2 | 1 | 1089.4.a.z | 4 | ||
| 11.c | even | 5 | 2 | 99.4.f.b | 8 | ||
| 33.d | even | 2 | 1 | 363.4.a.t | 4 | ||
| 33.h | odd | 10 | 2 | 33.4.e.b | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.4.e.b | ✓ | 8 | 33.h | odd | 10 | 2 | |
| 99.4.f.b | 8 | 11.c | even | 5 | 2 | ||
| 363.4.a.p | 4 | 3.b | odd | 2 | 1 | ||
| 363.4.a.t | 4 | 33.d | even | 2 | 1 | ||
| 1089.4.a.z | 4 | 11.b | odd | 2 | 1 | ||
| 1089.4.a.bg | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1089))\):
|
\( T_{2}^{4} - 3T_{2}^{3} - 10T_{2}^{2} + 9T_{2} - 1 \)
|
|
\( T_{5}^{4} + 12T_{5}^{3} - 85T_{5}^{2} - 1506T_{5} - 4681 \)
|
|
\( T_{7}^{4} + 11T_{7}^{3} - 970T_{7}^{2} - 4697T_{7} + 122299 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 3 T^{3} + \cdots - 1 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 12 T^{3} + \cdots - 4681 \)
|
| $7$ |
\( T^{4} + 11 T^{3} + \cdots + 122299 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + 182 T^{3} + \cdots - 12384284 \)
|
| $17$ |
\( T^{4} - 57 T^{3} + \cdots + 4061936 \)
|
| $19$ |
\( T^{4} + 173 T^{3} + \cdots - 48930764 \)
|
| $23$ |
\( T^{4} - 42 T^{3} + \cdots - 46471644 \)
|
| $29$ |
\( T^{4} - 651 T^{3} + \cdots + 515311796 \)
|
| $31$ |
\( T^{4} + 170 T^{3} + \cdots + 1691239 \)
|
| $37$ |
\( T^{4} - 244 T^{3} + \cdots - 141225104 \)
|
| $41$ |
\( T^{4} + 102 T^{3} + \cdots - 108182044 \)
|
| $43$ |
\( T^{4} + \cdots + 5520039844 \)
|
| $47$ |
\( T^{4} + \cdots - 4530163504 \)
|
| $53$ |
\( T^{4} - 468 T^{3} + \cdots - 525556039 \)
|
| $59$ |
\( T^{4} - 996 T^{3} + \cdots + 192157031 \)
|
| $61$ |
\( T^{4} + \cdots + 4008371596 \)
|
| $67$ |
\( T^{4} + \cdots + 1798706704 \)
|
| $71$ |
\( T^{4} + \cdots + 357347490524 \)
|
| $73$ |
\( T^{4} + \cdots + 52618219084 \)
|
| $79$ |
\( T^{4} + \cdots - 10584399491 \)
|
| $83$ |
\( T^{4} + \cdots + 90842593811 \)
|
| $89$ |
\( T^{4} + \cdots - 245710544796 \)
|
| $97$ |
\( T^{4} + \cdots - 31330298009 \)
|
| show more | |
| show less |