Newspace parameters
| Level: | \( N \) | \(=\) | \( 5004 = 2^{2} \cdot 3^{2} \cdot 139 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5004.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(39.9571411714\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1304.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - 11x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1668) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - 11x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + \beta _1 + 8 \)
|
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 | 0 | 0 | −4.22168 | 0 | 2.22168 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −1.18237 | 0 | −0.817630 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 2.40405 | 0 | −4.40405 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(139\) | \( +1 \) |
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 | 5004.2.a.h | 3 | |
| 3.b | odd | 2 | 1 | 1668.2.a.f | ✓ | 3 | |
| 12.b | even | 2 | 1 | 6672.2.a.bh | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1668.2.a.f | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 5004.2.a.h | 3 | 1.a | even | 1 | 1 | trivial | |
| 6672.2.a.bh | 3 | 12.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5004))\):
|
\( T_{5}^{3} + 3T_{5}^{2} - 8T_{5} - 12 \)
|
|
\( T_{7}^{3} + 3T_{7}^{2} - 8T_{7} - 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 3 T^{2} + \cdots - 12 \)
|
| $7$ |
\( T^{3} + 3 T^{2} + \cdots - 8 \)
|
| $11$ |
\( T^{3} - 7 T^{2} + \cdots + 9 \)
|
| $13$ |
\( (T - 1)^{3} \)
|
| $17$ |
\( T^{3} + 4 T^{2} + \cdots - 18 \)
|
| $19$ |
\( T^{3} - 11T + 2 \)
|
| $23$ |
\( (T + 2)^{3} \)
|
| $29$ |
\( T^{3} + 5 T^{2} + \cdots + 32 \)
|
| $31$ |
\( T^{3} + 13 T^{2} + \cdots - 188 \)
|
| $37$ |
\( T^{3} - 11T - 2 \)
|
| $41$ |
\( (T + 6)^{3} \)
|
| $43$ |
\( T^{3} - 2 T^{2} + \cdots - 68 \)
|
| $47$ |
\( T^{3} - 10 T^{2} + \cdots + 536 \)
|
| $53$ |
\( T^{3} + 4 T^{2} + \cdots - 102 \)
|
| $59$ |
\( T^{3} + 8 T^{2} + \cdots - 816 \)
|
| $61$ |
\( (T - 8)^{3} \)
|
| $67$ |
\( T^{3} - 7 T^{2} + \cdots + 176 \)
|
| $71$ |
\( T^{3} - 13 T^{2} + \cdots + 188 \)
|
| $73$ |
\( T^{3} - 8 T^{2} + \cdots + 64 \)
|
| $79$ |
\( T^{3} - 5 T^{2} + \cdots - 32 \)
|
| $83$ |
\( T^{3} + T^{2} + \cdots - 108 \)
|
| $89$ |
\( T^{3} - T^{2} + \cdots - 632 \)
|
| $97$ |
\( T^{3} + 10 T^{2} + \cdots - 264 \)
|
| show more | |
| show less |