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: | 4.4.14221152.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 17x^{2} + 22 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 17x^{2} + 22 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 14\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 14\beta_1 \)
|
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 |
|
−3.94826 | 0 | 7.58872 | −6.27269 | 0 | −6.58872 | 1.62383 | 0 | 24.7662 | ||||||||||||||||||||||||||||||
| 1.2 | −1.18797 | 0 | −6.58872 | 14.9550 | 0 | 7.58872 | 17.3310 | 0 | −17.7662 | |||||||||||||||||||||||||||||||
| 1.3 | 1.18797 | 0 | −6.58872 | −14.9550 | 0 | 7.58872 | −17.3310 | 0 | −17.7662 | |||||||||||||||||||||||||||||||
| 1.4 | 3.94826 | 0 | 7.58872 | 6.27269 | 0 | −6.58872 | −1.62383 | 0 | 24.7662 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(11\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1089.4.a.bd | yes | 4 |
| 3.b | odd | 2 | 1 | inner | 1089.4.a.bd | yes | 4 |
| 11.b | odd | 2 | 1 | 1089.4.a.bc | ✓ | 4 | |
| 33.d | even | 2 | 1 | 1089.4.a.bc | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1089.4.a.bc | ✓ | 4 | 11.b | odd | 2 | 1 | |
| 1089.4.a.bc | ✓ | 4 | 33.d | even | 2 | 1 | |
| 1089.4.a.bd | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 1089.4.a.bd | yes | 4 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1089))\):
|
\( T_{2}^{4} - 17T_{2}^{2} + 22 \)
|
|
\( T_{5}^{4} - 263T_{5}^{2} + 8800 \)
|
|
\( T_{7}^{2} - T_{7} - 50 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 17T^{2} + 22 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 263T^{2} + 8800 \)
|
| $7$ |
\( (T^{2} - T - 50)^{2} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( (T^{2} + 5 T - 1250)^{2} \)
|
| $17$ |
\( T^{4} - 7127 T^{2} + 10903552 \)
|
| $19$ |
\( (T^{2} + 21 T - 3960)^{2} \)
|
| $23$ |
\( T^{4} - 51068 T^{2} + 151478272 \)
|
| $29$ |
\( T^{4} - 6783 T^{2} + 7920000 \)
|
| $31$ |
\( (T^{2} + 85 T - 12716)^{2} \)
|
| $37$ |
\( (T^{2} + 144 T - 101145)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 25701992800 \)
|
| $43$ |
\( (T^{2} - 222 T - 60240)^{2} \)
|
| $47$ |
\( T^{4} - 292112 T^{2} + 223032832 \)
|
| $53$ |
\( T^{4} + \cdots + 9026721792 \)
|
| $59$ |
\( T^{4} - 518108 T^{2} + 192755200 \)
|
| $61$ |
\( (T^{2} + 525 T + 114)^{2} \)
|
| $67$ |
\( (T^{2} + 395 T - 22550)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 273411388800 \)
|
| $73$ |
\( (T^{2} - 1285 T + 311050)^{2} \)
|
| $79$ |
\( (T^{2} - 65 T - 173864)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 137998080000 \)
|
| $89$ |
\( T^{4} + \cdots + 291658831200 \)
|
| $97$ |
\( (T + 835)^{4} \)
|
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