Newspace parameters
| Level: | \( N \) | \(=\) | \( 65 = 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 65.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(0.519027613138\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 |
|
−1.00000 | −2.00000 | −1.00000 | −1.00000 | 2.00000 | −4.00000 | 3.00000 | 1.00000 | 1.00000 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(13\) | \(1\) |
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 | 65.2.a.a | ✓ | 1 |
| 3.b | odd | 2 | 1 | 585.2.a.h | 1 | ||
| 4.b | odd | 2 | 1 | 1040.2.a.f | 1 | ||
| 5.b | even | 2 | 1 | 325.2.a.d | 1 | ||
| 5.c | odd | 4 | 2 | 325.2.b.b | 2 | ||
| 7.b | odd | 2 | 1 | 3185.2.a.e | 1 | ||
| 8.b | even | 2 | 1 | 4160.2.a.q | 1 | ||
| 8.d | odd | 2 | 1 | 4160.2.a.f | 1 | ||
| 11.b | odd | 2 | 1 | 7865.2.a.c | 1 | ||
| 12.b | even | 2 | 1 | 9360.2.a.ca | 1 | ||
| 13.b | even | 2 | 1 | 845.2.a.a | 1 | ||
| 13.c | even | 3 | 2 | 845.2.e.b | 2 | ||
| 13.d | odd | 4 | 2 | 845.2.c.a | 2 | ||
| 13.e | even | 6 | 2 | 845.2.e.a | 2 | ||
| 13.f | odd | 12 | 4 | 845.2.m.b | 4 | ||
| 15.d | odd | 2 | 1 | 2925.2.a.f | 1 | ||
| 15.e | even | 4 | 2 | 2925.2.c.h | 2 | ||
| 20.d | odd | 2 | 1 | 5200.2.a.d | 1 | ||
| 39.d | odd | 2 | 1 | 7605.2.a.f | 1 | ||
| 65.d | even | 2 | 1 | 4225.2.a.g | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.2.a.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
| 325.2.a.d | 1 | 5.b | even | 2 | 1 | ||
| 325.2.b.b | 2 | 5.c | odd | 4 | 2 | ||
| 585.2.a.h | 1 | 3.b | odd | 2 | 1 | ||
| 845.2.a.a | 1 | 13.b | even | 2 | 1 | ||
| 845.2.c.a | 2 | 13.d | odd | 4 | 2 | ||
| 845.2.e.a | 2 | 13.e | even | 6 | 2 | ||
| 845.2.e.b | 2 | 13.c | even | 3 | 2 | ||
| 845.2.m.b | 4 | 13.f | odd | 12 | 4 | ||
| 1040.2.a.f | 1 | 4.b | odd | 2 | 1 | ||
| 2925.2.a.f | 1 | 15.d | odd | 2 | 1 | ||
| 2925.2.c.h | 2 | 15.e | even | 4 | 2 | ||
| 3185.2.a.e | 1 | 7.b | odd | 2 | 1 | ||
| 4160.2.a.f | 1 | 8.d | odd | 2 | 1 | ||
| 4160.2.a.q | 1 | 8.b | even | 2 | 1 | ||
| 4225.2.a.g | 1 | 65.d | even | 2 | 1 | ||
| 5200.2.a.d | 1 | 20.d | odd | 2 | 1 | ||
| 7605.2.a.f | 1 | 39.d | odd | 2 | 1 | ||
| 7865.2.a.c | 1 | 11.b | odd | 2 | 1 | ||
| 9360.2.a.ca | 1 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(65))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 1 \)
$3$
\( T + 2 \)
$5$
\( T + 1 \)
$7$
\( T + 4 \)
$11$
\( T - 2 \)
$13$
\( T + 1 \)
$17$
\( T - 2 \)
$19$
\( T + 6 \)
$23$
\( T + 6 \)
$29$
\( T - 2 \)
$31$
\( T + 10 \)
$37$
\( T + 2 \)
$41$
\( T + 6 \)
$43$
\( T - 10 \)
$47$
\( T - 4 \)
$53$
\( T - 2 \)
$59$
\( T - 6 \)
$61$
\( T - 2 \)
$67$
\( T + 4 \)
$71$
\( T - 6 \)
$73$
\( T + 6 \)
$79$
\( T + 12 \)
$83$
\( T + 16 \)
$89$
\( T - 2 \)
$97$
\( T + 2 \)
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