Newspace parameters
| Level: | \( N \) | \(=\) | \( 513 = 3^{3} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 513.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(4.09632562369\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{4 + \sqrt{3}})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 8x^{2} + 13 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{4} - 8x^{2} + 13 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
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 |
|
−2.39417 | 0 | 3.73205 | 1.75265 | 0 | 4.73205 | −4.14682 | 0 | −4.19615 | ||||||||||||||||||||||||||||||
| 1.2 | −1.50597 | 0 | 0.267949 | −4.11439 | 0 | 1.26795 | 2.60842 | 0 | 6.19615 | |||||||||||||||||||||||||||||||
| 1.3 | 1.50597 | 0 | 0.267949 | 4.11439 | 0 | 1.26795 | −2.60842 | 0 | 6.19615 | |||||||||||||||||||||||||||||||
| 1.4 | 2.39417 | 0 | 3.73205 | −1.75265 | 0 | 4.73205 | 4.14682 | 0 | −4.19615 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(19\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 513.2.a.i | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 513.2.a.i | ✓ | 4 |
| 4.b | odd | 2 | 1 | 8208.2.a.bt | 4 | ||
| 12.b | even | 2 | 1 | 8208.2.a.bt | 4 | ||
| 19.b | odd | 2 | 1 | 9747.2.a.bf | 4 | ||
| 57.d | even | 2 | 1 | 9747.2.a.bf | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 513.2.a.i | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 513.2.a.i | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 8208.2.a.bt | 4 | 4.b | odd | 2 | 1 | ||
| 8208.2.a.bt | 4 | 12.b | even | 2 | 1 | ||
| 9747.2.a.bf | 4 | 19.b | odd | 2 | 1 | ||
| 9747.2.a.bf | 4 | 57.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 8T_{2}^{2} + 13 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(513))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 8T^{2} + 13 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 20T^{2} + 52 \)
|
| $7$ |
\( (T^{2} - 6 T + 6)^{2} \)
|
| $11$ |
\( T^{4} - 24T^{2} + 117 \)
|
| $13$ |
\( (T^{2} - 12)^{2} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( (T + 1)^{4} \)
|
| $23$ |
\( T^{4} - 44T^{2} + 52 \)
|
| $29$ |
\( T^{4} - 104T^{2} + 2197 \)
|
| $31$ |
\( (T^{2} - 4 T - 71)^{2} \)
|
| $37$ |
\( (T^{2} + 2 T - 26)^{2} \)
|
| $41$ |
\( T^{4} - 128T^{2} + 1573 \)
|
| $43$ |
\( (T^{2} + 2 T - 2)^{2} \)
|
| $47$ |
\( T^{4} - 72T^{2} + 1053 \)
|
| $53$ |
\( T^{4} - 72T^{2} + 1053 \)
|
| $59$ |
\( T^{4} - 60T^{2} + 468 \)
|
| $61$ |
\( (T^{2} + 4 T - 23)^{2} \)
|
| $67$ |
\( (T^{2} + 8 T + 13)^{2} \)
|
| $71$ |
\( T^{4} - 240T^{2} + 7488 \)
|
| $73$ |
\( (T^{2} - 8 T - 131)^{2} \)
|
| $79$ |
\( (T + 9)^{4} \)
|
| $83$ |
\( T^{4} - 200T^{2} + 8125 \)
|
| $89$ |
\( T^{4} - 512 T^{2} + 65533 \)
|
| $97$ |
\( (T^{2} - 2 T - 74)^{2} \)
|
| show more | |
| show less |