Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1161,2,Mod(1,1161)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1161, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1161.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1161 = 3^{3} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1161.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: | \(9.27063167467\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.675097.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 6x^{3} + 5x^{2} + 5x - 3 \)
|
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 6x^{3} + 5x^{2} + 5x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 6\nu^{2} - 4\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 6\beta_{2} + 14 \)
|
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 |
|
−2.18386 | 0 | 2.76924 | −1.18985 | 0 | 4.05364 | −1.67992 | 0 | 2.59848 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.925580 | 0 | −1.14330 | −4.31563 | 0 | 1.33184 | 2.90938 | 0 | 3.99446 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.505911 | 0 | −1.74405 | 3.42399 | 0 | −3.03575 | −1.89416 | 0 | 1.73223 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.24257 | 0 | −0.456025 | −0.828213 | 0 | 1.63742 | −3.05178 | 0 | −1.02911 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.36096 | 0 | 3.57414 | −3.09029 | 0 | −3.98715 | 3.71648 | 0 | −7.29606 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(43\) | \(-1\) |
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 | 1161.2.a.i | yes | 5 |
| 3.b | odd | 2 | 1 | 1161.2.a.h | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1161.2.a.h | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 1161.2.a.i | yes | 5 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1161))\):
|
\( T_{2}^{5} - T_{2}^{4} - 6T_{2}^{3} + 5T_{2}^{2} + 5T_{2} - 3 \)
|
|
\( T_{5}^{5} + 6T_{5}^{4} - 3T_{5}^{3} - 66T_{5}^{2} - 104T_{5} - 45 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - T^{4} - 6 T^{3} + \cdots - 3 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + 6 T^{4} + \cdots - 45 \)
|
| $7$ |
\( T^{5} - 23 T^{3} + \cdots - 107 \)
|
| $11$ |
\( T^{5} + 7 T^{4} + \cdots - 3 \)
|
| $13$ |
\( T^{5} + 6 T^{4} + \cdots + 365 \)
|
| $17$ |
\( T^{5} - 3 T^{4} + \cdots - 39 \)
|
| $19$ |
\( T^{5} + 13 T^{4} + \cdots - 59 \)
|
| $23$ |
\( T^{5} + 17 T^{4} + \cdots - 81 \)
|
| $29$ |
\( T^{5} + 3 T^{4} + \cdots + 729 \)
|
| $31$ |
\( T^{5} + 2 T^{4} + \cdots + 7875 \)
|
| $37$ |
\( T^{5} + 3 T^{4} + \cdots - 9 \)
|
| $41$ |
\( T^{5} + T^{4} + \cdots + 8355 \)
|
| $43$ |
\( (T - 1)^{5} \)
|
| $47$ |
\( T^{5} + 40 T^{4} + \cdots + 8559 \)
|
| $53$ |
\( T^{5} + 7 T^{4} + \cdots + 10221 \)
|
| $59$ |
\( T^{5} + 6 T^{4} + \cdots - 21 \)
|
| $61$ |
\( T^{5} - 11 T^{4} + \cdots + 1157 \)
|
| $67$ |
\( T^{5} - T^{4} + \cdots + 14211 \)
|
| $71$ |
\( T^{5} + 24 T^{4} + \cdots - 55587 \)
|
| $73$ |
\( T^{5} - 3 T^{4} + \cdots - 28035 \)
|
| $79$ |
\( T^{5} + 19 T^{4} + \cdots - 2701 \)
|
| $83$ |
\( T^{5} - 20 T^{4} + \cdots - 244113 \)
|
| $89$ |
\( T^{5} + 9 T^{4} + \cdots - 2187 \)
|
| $97$ |
\( T^{5} + 2 T^{4} + \cdots + 487 \)
|
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