Newspace parameters
| Level: | \( N \) | \(=\) | \( 8330 = 2 \cdot 5 \cdot 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8330.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(66.5153848837\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.44688.2 |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 5x^{2} + 6x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1190) |
| 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
\( x^{4} - 2x^{3} - 5x^{2} + 6x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 6\beta _1 + 2 \)
|
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 |
|
−1.00000 | −2.92812 | 1.00000 | −1.00000 | 2.92812 | 0 | −1.00000 | 5.57387 | 1.00000 | ||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.27733 | 1.00000 | −1.00000 | 1.27733 | 0 | −1.00000 | −1.36842 | 1.00000 | |||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0.277334 | 1.00000 | −1.00000 | −0.277334 | 0 | −1.00000 | −2.92309 | 1.00000 | |||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.92812 | 1.00000 | −1.00000 | −1.92812 | 0 | −1.00000 | 0.717634 | 1.00000 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(7\) | \( +1 \) |
| \(17\) | \( +1 \) |
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 | 8330.2.a.bx | 4 | |
| 7.b | odd | 2 | 1 | 8330.2.a.cb | 4 | ||
| 7.c | even | 3 | 2 | 1190.2.i.l | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1190.2.i.l | ✓ | 8 | 7.c | even | 3 | 2 | |
| 8330.2.a.bx | 4 | 1.a | even | 1 | 1 | trivial | |
| 8330.2.a.cb | 4 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8330))\):
|
\( T_{3}^{4} + 2T_{3}^{3} - 5T_{3}^{2} - 6T_{3} + 2 \)
|
|
\( T_{11}^{4} + 2T_{11}^{3} - 26T_{11}^{2} - 6T_{11} + 93 \)
|
|
\( T_{13}^{4} + 6T_{13}^{3} - 10T_{13}^{2} - 106T_{13} - 139 \)
|
|
\( T_{19}^{4} + 8T_{19}^{3} + 5T_{19}^{2} - 16T_{19} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{4} \)
|
| $3$ |
\( T^{4} + 2 T^{3} + \cdots + 2 \)
|
| $5$ |
\( (T + 1)^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 2 T^{3} + \cdots + 93 \)
|
| $13$ |
\( T^{4} + 6 T^{3} + \cdots - 139 \)
|
| $17$ |
\( (T + 1)^{4} \)
|
| $19$ |
\( T^{4} + 8 T^{3} + \cdots + 4 \)
|
| $23$ |
\( T^{4} - 8 T^{3} + \cdots - 108 \)
|
| $29$ |
\( T^{4} + 2 T^{3} + \cdots - 48 \)
|
| $31$ |
\( T^{4} + 10 T^{3} + \cdots - 152 \)
|
| $37$ |
\( T^{4} - 14 T^{3} + \cdots + 548 \)
|
| $41$ |
\( T^{4} - 4 T^{3} + \cdots + 228 \)
|
| $43$ |
\( T^{4} - 12 T^{3} + \cdots - 2224 \)
|
| $47$ |
\( T^{4} - 6 T^{3} + \cdots + 1332 \)
|
| $53$ |
\( T^{4} - 10 T^{3} + \cdots - 999 \)
|
| $59$ |
\( T^{4} - 16 T^{3} + \cdots - 96 \)
|
| $61$ |
\( T^{4} + 30 T^{3} + \cdots + 128 \)
|
| $67$ |
\( T^{4} - 76 T^{2} + \cdots + 128 \)
|
| $71$ |
\( T^{4} + 16 T^{3} + \cdots + 72 \)
|
| $73$ |
\( T^{4} - 6 T^{3} + \cdots - 496 \)
|
| $79$ |
\( T^{4} - 14 T^{3} + \cdots - 278 \)
|
| $83$ |
\( T^{4} + 14 T^{3} + \cdots - 24 \)
|
| $89$ |
\( T^{4} - 10 T^{3} + \cdots - 2712 \)
|
| $97$ |
\( T^{4} - 12 T^{3} + \cdots - 9728 \)
|
| show more | |
| show less |