Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6760,2,Mod(1,6760)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6760, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6760.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6760 = 2^{3} \cdot 5 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6760.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: | \(53.9788717664\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.34868.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 8x^{2} + 4x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 520) |
| 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
\( x^{4} - 2x^{3} - 8x^{2} + 4x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 8\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 12\beta _1 + 5 \)
|
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 |
|
0 | −2.80720 | 0 | −1.00000 | 0 | −3.54454 | 0 | 4.88037 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0.369700 | 0 | −1.00000 | 0 | 0.956248 | 0 | −2.86332 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.18501 | 0 | −1.00000 | 0 | −5.22025 | 0 | −1.59576 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.25249 | 0 | −1.00000 | 0 | 1.80854 | 0 | 7.57872 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(1\) |
| \(13\) | \(-1\) |
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 | 6760.2.a.bc | 4 | |
| 13.b | even | 2 | 1 | 6760.2.a.bd | 4 | ||
| 13.d | odd | 4 | 2 | 520.2.k.b | ✓ | 8 | |
| 39.f | even | 4 | 2 | 4680.2.g.k | 8 | ||
| 52.f | even | 4 | 2 | 1040.2.k.e | 8 | ||
| 65.f | even | 4 | 2 | 2600.2.f.e | 8 | ||
| 65.g | odd | 4 | 2 | 2600.2.k.c | 8 | ||
| 65.k | even | 4 | 2 | 2600.2.f.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 520.2.k.b | ✓ | 8 | 13.d | odd | 4 | 2 | |
| 1040.2.k.e | 8 | 52.f | even | 4 | 2 | ||
| 2600.2.f.e | 8 | 65.f | even | 4 | 2 | ||
| 2600.2.f.f | 8 | 65.k | even | 4 | 2 | ||
| 2600.2.k.c | 8 | 65.g | odd | 4 | 2 | ||
| 4680.2.g.k | 8 | 39.f | even | 4 | 2 | ||
| 6760.2.a.bc | 4 | 1.a | even | 1 | 1 | trivial | |
| 6760.2.a.bd | 4 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6760))\):
|
\( T_{3}^{4} - 2T_{3}^{3} - 8T_{3}^{2} + 14T_{3} - 4 \)
|
|
\( T_{7}^{4} + 6T_{7}^{3} - 4T_{7}^{2} - 36T_{7} + 32 \)
|
|
\( T_{11}^{4} - 14T_{11}^{2} + 22T_{11} - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 2 T^{3} + \cdots - 4 \)
|
| $5$ |
\( (T + 1)^{4} \)
|
| $7$ |
\( T^{4} + 6 T^{3} + \cdots + 32 \)
|
| $11$ |
\( T^{4} - 14 T^{2} + \cdots - 8 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} - 2 T^{3} + \cdots - 64 \)
|
| $19$ |
\( T^{4} + 2 T^{3} + \cdots - 136 \)
|
| $23$ |
\( T^{4} - 6 T^{3} + \cdots + 56 \)
|
| $29$ |
\( T^{4} - 8 T^{3} + \cdots - 664 \)
|
| $31$ |
\( T^{4} + 18 T^{3} + \cdots - 2476 \)
|
| $37$ |
\( T^{4} + 6 T^{3} + \cdots + 232 \)
|
| $41$ |
\( T^{4} + 2 T^{3} + \cdots + 128 \)
|
| $43$ |
\( T^{4} - 12 T^{3} + \cdots - 236 \)
|
| $47$ |
\( T^{4} - 2 T^{3} + \cdots + 2704 \)
|
| $53$ |
\( T^{4} + 2 T^{3} + \cdots + 128 \)
|
| $59$ |
\( T^{4} + 6 T^{3} + \cdots + 224 \)
|
| $61$ |
\( T^{4} - 16 T^{3} + \cdots + 2264 \)
|
| $67$ |
\( T^{4} - 2 T^{3} + \cdots + 7984 \)
|
| $71$ |
\( T^{4} + 12 T^{3} + \cdots + 1196 \)
|
| $73$ |
\( T^{4} + 2 T^{3} + \cdots - 472 \)
|
| $79$ |
\( T^{4} + 12 T^{3} + \cdots + 1024 \)
|
| $83$ |
\( T^{4} + 30 T^{3} + \cdots - 944 \)
|
| $89$ |
\( T^{4} + 28 T^{3} + \cdots + 128 \)
|
| $97$ |
\( T^{4} - 20 T^{3} + \cdots - 464 \)
|
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