Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5915,2,Mod(1,5915)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5915, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("5915.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5915 = 5 \cdot 7 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5915.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: | \(47.2315127956\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.81589.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 6x^{3} + 8x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 455) |
| 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
\( x^{5} - 6x^{3} + 8x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 4\nu^{2} + 3\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 4\beta_{2} + 6 \)
|
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 |
|
−2.05411 | −1.76878 | 2.21936 | 1.00000 | 3.63325 | −1.00000 | −0.450581 | 0.128565 | −2.05411 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.29150 | −2.67979 | −0.332026 | 1.00000 | 3.46095 | −1.00000 | 3.01181 | 4.18126 | −1.29150 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.126515 | 1.47996 | −1.98399 | 1.00000 | −0.187237 | −1.00000 | 0.504034 | −0.809718 | −0.126515 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.55053 | 2.07030 | 0.404133 | 1.00000 | 3.21006 | −1.00000 | −2.47443 | 1.28615 | 1.55053 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.92160 | −1.10170 | 1.69253 | 1.00000 | −2.11702 | −1.00000 | −0.590831 | −1.78626 | 1.92160 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
| \(13\) | \(-1\) |
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 | 5915.2.a.r | 5 | |
| 13.b | even | 2 | 1 | 5915.2.a.q | 5 | ||
| 13.d | odd | 4 | 2 | 455.2.d.a | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 455.2.d.a | ✓ | 10 | 13.d | odd | 4 | 2 | |
| 5915.2.a.q | 5 | 13.b | even | 2 | 1 | ||
| 5915.2.a.r | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5915))\):
|
\( T_{2}^{5} - 6T_{2}^{3} + 8T_{2} + 1 \)
|
|
\( T_{3}^{5} + 2T_{3}^{4} - 7T_{3}^{3} - 12T_{3}^{2} + 11T_{3} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 6 T^{3} + \cdots + 1 \)
|
| $3$ |
\( T^{5} + 2 T^{4} + \cdots + 16 \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( (T + 1)^{5} \)
|
| $11$ |
\( T^{5} - 24 T^{3} + \cdots - 64 \)
|
| $13$ |
\( T^{5} \)
|
| $17$ |
\( T^{5} + 2 T^{4} + \cdots - 8 \)
|
| $19$ |
\( T^{5} - 2 T^{4} + \cdots - 2 \)
|
| $23$ |
\( T^{5} - 4 T^{4} + \cdots - 32 \)
|
| $29$ |
\( T^{5} + 12 T^{4} + \cdots + 142 \)
|
| $31$ |
\( T^{5} + 12 T^{4} + \cdots - 206 \)
|
| $37$ |
\( T^{5} + 8 T^{4} + \cdots + 2122 \)
|
| $41$ |
\( T^{5} + 4 T^{4} + \cdots + 142 \)
|
| $43$ |
\( T^{5} + 10 T^{4} + \cdots + 416 \)
|
| $47$ |
\( T^{5} - 2 T^{4} + \cdots - 17216 \)
|
| $53$ |
\( T^{5} + 6 T^{4} + \cdots - 32 \)
|
| $59$ |
\( T^{5} + 16 T^{4} + \cdots - 14066 \)
|
| $61$ |
\( T^{5} + 12 T^{4} + \cdots + 1952 \)
|
| $67$ |
\( T^{5} - 16 T^{4} + \cdots - 4244 \)
|
| $71$ |
\( T^{5} - 8 T^{4} + \cdots + 256 \)
|
| $73$ |
\( T^{5} - 22 T^{4} + \cdots + 416 \)
|
| $79$ |
\( T^{5} + 32 T^{4} + \cdots - 46948 \)
|
| $83$ |
\( T^{5} - 12 T^{4} + \cdots - 16832 \)
|
| $89$ |
\( T^{5} + 10 T^{4} + \cdots + 21842 \)
|
| $97$ |
\( T^{5} + 10 T^{4} + \cdots + 12832 \)
|
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