Newspace parameters
| Level: | \( N \) | \(=\) | \( 7260 = 2^{2} \cdot 3 \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7260.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(57.9713918674\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.2525.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 4x^{2} + 5x + 5 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 660) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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
\( x^{4} - 2x^{3} - 4x^{2} + 5x + 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 3 \)
|
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.
| 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 | 1.00000 | 0 | −1.00000 | 0 | −1.39552 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 1.00000 | 0 | −1.00000 | 0 | 0.151302 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.00000 | 0 | −1.00000 | 0 | 1.15945 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.00000 | 0 | −1.00000 | 0 | 4.08477 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(11\) | \( -1 \) |
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 | 7260.2.a.bg | 4 | |
| 11.b | odd | 2 | 1 | 7260.2.a.be | 4 | ||
| 11.d | odd | 10 | 2 | 660.2.y.b | ✓ | 8 | |
| 33.f | even | 10 | 2 | 1980.2.z.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 660.2.y.b | ✓ | 8 | 11.d | odd | 10 | 2 | |
| 1980.2.z.b | 8 | 33.f | even | 10 | 2 | ||
| 7260.2.a.be | 4 | 11.b | odd | 2 | 1 | ||
| 7260.2.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7260))\):
|
\( T_{7}^{4} - 4T_{7}^{3} - 2T_{7}^{2} + 7T_{7} - 1 \)
|
|
\( T_{13}^{4} + T_{13}^{3} - 32T_{13}^{2} + 72T_{13} - 41 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T - 1)^{4} \)
|
| $5$ |
\( (T + 1)^{4} \)
|
| $7$ |
\( T^{4} - 4 T^{3} + \cdots - 1 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + T^{3} + \cdots - 41 \)
|
| $17$ |
\( T^{4} - 2 T^{3} + \cdots + 55 \)
|
| $19$ |
\( T^{4} - 7 T^{3} + \cdots - 109 \)
|
| $23$ |
\( T^{4} - 15 T^{3} + \cdots - 125 \)
|
| $29$ |
\( T^{4} - 15 T^{3} + \cdots - 1375 \)
|
| $31$ |
\( T^{4} + 7 T^{3} + \cdots + 109 \)
|
| $37$ |
\( T^{4} + T^{3} - 11 T^{2} + \cdots - 5 \)
|
| $41$ |
\( T^{4} - 10 T^{3} + \cdots - 625 \)
|
| $43$ |
\( T^{4} - 4 T^{3} + \cdots + 479 \)
|
| $47$ |
\( T^{4} - 4 T^{3} + \cdots + 725 \)
|
| $53$ |
\( T^{4} + 16 T^{3} + \cdots + 1291 \)
|
| $59$ |
\( T^{4} + 5 T^{3} + \cdots + 451 \)
|
| $61$ |
\( T^{4} - 11 T^{3} + \cdots + 275 \)
|
| $67$ |
\( T^{4} - 9 T^{3} + \cdots - 29 \)
|
| $71$ |
\( T^{4} + 5 T^{3} + \cdots + 125 \)
|
| $73$ |
\( T^{4} + 5 T^{3} + \cdots + 241 \)
|
| $79$ |
\( (T^{2} + 5 T - 95)^{2} \)
|
| $83$ |
\( T^{4} - 11 T^{3} + \cdots + 319 \)
|
| $89$ |
\( T^{4} - 16 T^{3} + \cdots - 769 \)
|
| $97$ |
\( T^{4} + 2 T^{3} + \cdots + 5255 \)
|
| show more | |
| show less |