Newspace parameters
| Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 225.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(1.79663404548\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $N(\mathrm{U}(1))$ |
$q$-expansion
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 | −2.00000 | 0 | 0 | 5.00000 | 0 | 0 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 225.2.a.d | yes | 1 |
| 3.b | odd | 2 | 1 | CM | 225.2.a.d | yes | 1 |
| 4.b | odd | 2 | 1 | 3600.2.a.b | 1 | ||
| 5.b | even | 2 | 1 | 225.2.a.c | ✓ | 1 | |
| 5.c | odd | 4 | 2 | 225.2.b.c | 2 | ||
| 12.b | even | 2 | 1 | 3600.2.a.b | 1 | ||
| 15.d | odd | 2 | 1 | 225.2.a.c | ✓ | 1 | |
| 15.e | even | 4 | 2 | 225.2.b.c | 2 | ||
| 20.d | odd | 2 | 1 | 3600.2.a.br | 1 | ||
| 20.e | even | 4 | 2 | 3600.2.f.k | 2 | ||
| 60.h | even | 2 | 1 | 3600.2.a.br | 1 | ||
| 60.l | odd | 4 | 2 | 3600.2.f.k | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 225.2.a.c | ✓ | 1 | 5.b | even | 2 | 1 | |
| 225.2.a.c | ✓ | 1 | 15.d | odd | 2 | 1 | |
| 225.2.a.d | yes | 1 | 1.a | even | 1 | 1 | trivial |
| 225.2.a.d | yes | 1 | 3.b | odd | 2 | 1 | CM |
| 225.2.b.c | 2 | 5.c | odd | 4 | 2 | ||
| 225.2.b.c | 2 | 15.e | even | 4 | 2 | ||
| 3600.2.a.b | 1 | 4.b | odd | 2 | 1 | ||
| 3600.2.a.b | 1 | 12.b | even | 2 | 1 | ||
| 3600.2.a.br | 1 | 20.d | odd | 2 | 1 | ||
| 3600.2.a.br | 1 | 60.h | even | 2 | 1 | ||
| 3600.2.f.k | 2 | 20.e | even | 4 | 2 | ||
| 3600.2.f.k | 2 | 60.l | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(225))\):
|
\( T_{2} \)
|
|
\( T_{7} - 5 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T \)
|
| $5$ |
\( T \)
|
| $7$ |
\( T - 5 \)
|
| $11$ |
\( T \)
|
| $13$ |
\( T - 5 \)
|
| $17$ |
\( T \)
|
| $19$ |
\( T + 1 \)
|
| $23$ |
\( T \)
|
| $29$ |
\( T \)
|
| $31$ |
\( T + 7 \)
|
| $37$ |
\( T + 10 \)
|
| $41$ |
\( T \)
|
| $43$ |
\( T - 5 \)
|
| $47$ |
\( T \)
|
| $53$ |
\( T \)
|
| $59$ |
\( T \)
|
| $61$ |
\( T + 13 \)
|
| $67$ |
\( T - 5 \)
|
| $71$ |
\( T \)
|
| $73$ |
\( T + 10 \)
|
| $79$ |
\( T + 4 \)
|
| $83$ |
\( T \)
|
| $89$ |
\( T \)
|
| $97$ |
\( T - 5 \)
|
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