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: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{2}) \) |
Defining polynomial: |
\( x^{2} - 2 \)
|
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−2.41421 | −1.41421 | 3.82843 | 1.00000 | 3.41421 | 4.82843 | −4.41421 | −1.00000 | −2.41421 | ||||||||||||||||||||||||
1.2 | 0.414214 | 1.41421 | −1.82843 | 1.00000 | 0.585786 | −0.828427 | −1.58579 | −1.00000 | 0.414214 | |||||||||||||||||||||||||
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.b | ✓ | 2 |
3.b | odd | 2 | 1 | 585.2.a.m | 2 | ||
4.b | odd | 2 | 1 | 1040.2.a.j | 2 | ||
5.b | even | 2 | 1 | 325.2.a.i | 2 | ||
5.c | odd | 4 | 2 | 325.2.b.f | 4 | ||
7.b | odd | 2 | 1 | 3185.2.a.j | 2 | ||
8.b | even | 2 | 1 | 4160.2.a.bf | 2 | ||
8.d | odd | 2 | 1 | 4160.2.a.z | 2 | ||
11.b | odd | 2 | 1 | 7865.2.a.j | 2 | ||
12.b | even | 2 | 1 | 9360.2.a.cd | 2 | ||
13.b | even | 2 | 1 | 845.2.a.g | 2 | ||
13.c | even | 3 | 2 | 845.2.e.h | 4 | ||
13.d | odd | 4 | 2 | 845.2.c.b | 4 | ||
13.e | even | 6 | 2 | 845.2.e.c | 4 | ||
13.f | odd | 12 | 4 | 845.2.m.f | 8 | ||
15.d | odd | 2 | 1 | 2925.2.a.u | 2 | ||
15.e | even | 4 | 2 | 2925.2.c.r | 4 | ||
20.d | odd | 2 | 1 | 5200.2.a.bu | 2 | ||
39.d | odd | 2 | 1 | 7605.2.a.x | 2 | ||
65.d | even | 2 | 1 | 4225.2.a.r | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
65.2.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
325.2.a.i | 2 | 5.b | even | 2 | 1 | ||
325.2.b.f | 4 | 5.c | odd | 4 | 2 | ||
585.2.a.m | 2 | 3.b | odd | 2 | 1 | ||
845.2.a.g | 2 | 13.b | even | 2 | 1 | ||
845.2.c.b | 4 | 13.d | odd | 4 | 2 | ||
845.2.e.c | 4 | 13.e | even | 6 | 2 | ||
845.2.e.h | 4 | 13.c | even | 3 | 2 | ||
845.2.m.f | 8 | 13.f | odd | 12 | 4 | ||
1040.2.a.j | 2 | 4.b | odd | 2 | 1 | ||
2925.2.a.u | 2 | 15.d | odd | 2 | 1 | ||
2925.2.c.r | 4 | 15.e | even | 4 | 2 | ||
3185.2.a.j | 2 | 7.b | odd | 2 | 1 | ||
4160.2.a.z | 2 | 8.d | odd | 2 | 1 | ||
4160.2.a.bf | 2 | 8.b | even | 2 | 1 | ||
4225.2.a.r | 2 | 65.d | even | 2 | 1 | ||
5200.2.a.bu | 2 | 20.d | odd | 2 | 1 | ||
7605.2.a.x | 2 | 39.d | odd | 2 | 1 | ||
7865.2.a.j | 2 | 11.b | odd | 2 | 1 | ||
9360.2.a.cd | 2 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 2T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(65))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 2T - 1 \)
$3$
\( T^{2} - 2 \)
$5$
\( (T - 1)^{2} \)
$7$
\( T^{2} - 4T - 4 \)
$11$
\( T^{2} - 4T + 2 \)
$13$
\( (T + 1)^{2} \)
$17$
\( T^{2} + 4T - 4 \)
$19$
\( T^{2} - 4T + 2 \)
$23$
\( T^{2} - 2 \)
$29$
\( T^{2} - 32 \)
$31$
\( T^{2} - 12T + 18 \)
$37$
\( T^{2} - 72 \)
$41$
\( T^{2} + 12T + 28 \)
$43$
\( T^{2} + 8T - 34 \)
$47$
\( T^{2} + 4T - 4 \)
$53$
\( T^{2} + 12T - 36 \)
$59$
\( T^{2} - 12T + 18 \)
$61$
\( (T + 8)^{2} \)
$67$
\( (T + 2)^{2} \)
$71$
\( T^{2} - 4T - 94 \)
$73$
\( T^{2} - 72 \)
$79$
\( T^{2} - 72 \)
$83$
\( T^{2} + 12T + 28 \)
$89$
\( (T - 6)^{2} \)
$97$
\( T^{2} + 4T - 28 \)
show more
show less