Newspace parameters
| Level: | \( N \) | \(=\) | \( 1815 = 3 \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1815.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(14.4928479669\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.8112.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - 5x^{2} + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} - 5x^{2} + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 5 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 2\beta_1 \)
|
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 |
|
−2.75782 | −1.00000 | 5.60555 | 1.00000 | 2.75782 | −1.67000 | −9.94345 | 1.00000 | −2.75782 | ||||||||||||||||||||||||||||||
| 1.2 | −0.628052 | −1.00000 | −1.60555 | 1.00000 | 0.628052 | 4.14863 | 2.26447 | 1.00000 | −0.628052 | |||||||||||||||||||||||||||||||
| 1.3 | 0.628052 | −1.00000 | −1.60555 | 1.00000 | −0.628052 | −4.14863 | −2.26447 | 1.00000 | 0.628052 | |||||||||||||||||||||||||||||||
| 1.4 | 2.75782 | −1.00000 | 5.60555 | 1.00000 | −2.75782 | 1.67000 | 9.94345 | 1.00000 | 2.75782 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
| \(11\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1815.2.a.s | ✓ | 4 |
| 3.b | odd | 2 | 1 | 5445.2.a.bo | 4 | ||
| 5.b | even | 2 | 1 | 9075.2.a.dc | 4 | ||
| 11.b | odd | 2 | 1 | inner | 1815.2.a.s | ✓ | 4 |
| 33.d | even | 2 | 1 | 5445.2.a.bo | 4 | ||
| 55.d | odd | 2 | 1 | 9075.2.a.dc | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1815.2.a.s | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1815.2.a.s | ✓ | 4 | 11.b | odd | 2 | 1 | inner |
| 5445.2.a.bo | 4 | 3.b | odd | 2 | 1 | ||
| 5445.2.a.bo | 4 | 33.d | even | 2 | 1 | ||
| 9075.2.a.dc | 4 | 5.b | even | 2 | 1 | ||
| 9075.2.a.dc | 4 | 55.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1815))\):
|
\( T_{2}^{4} - 8T_{2}^{2} + 3 \)
|
|
\( T_{7}^{4} - 20T_{7}^{2} + 48 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 8T^{2} + 3 \)
|
| $3$ |
\( (T + 1)^{4} \)
|
| $5$ |
\( (T - 1)^{4} \)
|
| $7$ |
\( T^{4} - 20T^{2} + 48 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} - 44T^{2} + 432 \)
|
| $17$ |
\( T^{4} - 20T^{2} + 48 \)
|
| $19$ |
\( T^{4} - 32T^{2} + 48 \)
|
| $23$ |
\( (T^{2} + 4 T - 48)^{2} \)
|
| $29$ |
\( T^{4} - 128T^{2} + 3888 \)
|
| $31$ |
\( (T - 4)^{4} \)
|
| $37$ |
\( (T^{2} - 52)^{2} \)
|
| $41$ |
\( T^{4} - 96T^{2} + 432 \)
|
| $43$ |
\( T^{4} - 132T^{2} + 3888 \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( (T^{2} - 8 T - 36)^{2} \)
|
| $59$ |
\( (T^{2} - 4 T - 48)^{2} \)
|
| $61$ |
\( T^{4} - 80T^{2} + 768 \)
|
| $67$ |
\( (T^{2} + 12 T - 16)^{2} \)
|
| $71$ |
\( (T^{2} + 4 T - 48)^{2} \)
|
| $73$ |
\( T^{4} - 60T^{2} + 432 \)
|
| $79$ |
\( T^{4} - 32T^{2} + 48 \)
|
| $83$ |
\( T^{4} - 188T^{2} + 48 \)
|
| $89$ |
\( (T - 6)^{4} \)
|
| $97$ |
\( (T^{2} - 24 T + 92)^{2} \)
|
| show more | |
| show less |