Newspace parameters
Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 63.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.503057532734\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{3}) \) |
Defining polynomial: |
\( x^{2} - 3 \)
|
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 \(\beta = \sqrt{3}\). 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 |
|
−1.73205 | 0 | 1.00000 | 3.46410 | 0 | 1.00000 | 1.73205 | 0 | −6.00000 | ||||||||||||||||||||||||
1.2 | 1.73205 | 0 | 1.00000 | −3.46410 | 0 | 1.00000 | −1.73205 | 0 | −6.00000 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(7\) | \(-1\) |
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 | 63.2.a.b | ✓ | 2 |
3.b | odd | 2 | 1 | inner | 63.2.a.b | ✓ | 2 |
4.b | odd | 2 | 1 | 1008.2.a.n | 2 | ||
5.b | even | 2 | 1 | 1575.2.a.q | 2 | ||
5.c | odd | 4 | 2 | 1575.2.d.i | 4 | ||
7.b | odd | 2 | 1 | 441.2.a.g | 2 | ||
7.c | even | 3 | 2 | 441.2.e.j | 4 | ||
7.d | odd | 6 | 2 | 441.2.e.i | 4 | ||
8.b | even | 2 | 1 | 4032.2.a.bt | 2 | ||
8.d | odd | 2 | 1 | 4032.2.a.bq | 2 | ||
9.c | even | 3 | 2 | 567.2.f.j | 4 | ||
9.d | odd | 6 | 2 | 567.2.f.j | 4 | ||
11.b | odd | 2 | 1 | 7623.2.a.bi | 2 | ||
12.b | even | 2 | 1 | 1008.2.a.n | 2 | ||
15.d | odd | 2 | 1 | 1575.2.a.q | 2 | ||
15.e | even | 4 | 2 | 1575.2.d.i | 4 | ||
21.c | even | 2 | 1 | 441.2.a.g | 2 | ||
21.g | even | 6 | 2 | 441.2.e.i | 4 | ||
21.h | odd | 6 | 2 | 441.2.e.j | 4 | ||
24.f | even | 2 | 1 | 4032.2.a.bq | 2 | ||
24.h | odd | 2 | 1 | 4032.2.a.bt | 2 | ||
28.d | even | 2 | 1 | 7056.2.a.cm | 2 | ||
33.d | even | 2 | 1 | 7623.2.a.bi | 2 | ||
84.h | odd | 2 | 1 | 7056.2.a.cm | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
63.2.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
63.2.a.b | ✓ | 2 | 3.b | odd | 2 | 1 | inner |
441.2.a.g | 2 | 7.b | odd | 2 | 1 | ||
441.2.a.g | 2 | 21.c | even | 2 | 1 | ||
441.2.e.i | 4 | 7.d | odd | 6 | 2 | ||
441.2.e.i | 4 | 21.g | even | 6 | 2 | ||
441.2.e.j | 4 | 7.c | even | 3 | 2 | ||
441.2.e.j | 4 | 21.h | odd | 6 | 2 | ||
567.2.f.j | 4 | 9.c | even | 3 | 2 | ||
567.2.f.j | 4 | 9.d | odd | 6 | 2 | ||
1008.2.a.n | 2 | 4.b | odd | 2 | 1 | ||
1008.2.a.n | 2 | 12.b | even | 2 | 1 | ||
1575.2.a.q | 2 | 5.b | even | 2 | 1 | ||
1575.2.a.q | 2 | 15.d | odd | 2 | 1 | ||
1575.2.d.i | 4 | 5.c | odd | 4 | 2 | ||
1575.2.d.i | 4 | 15.e | even | 4 | 2 | ||
4032.2.a.bq | 2 | 8.d | odd | 2 | 1 | ||
4032.2.a.bq | 2 | 24.f | even | 2 | 1 | ||
4032.2.a.bt | 2 | 8.b | even | 2 | 1 | ||
4032.2.a.bt | 2 | 24.h | odd | 2 | 1 | ||
7056.2.a.cm | 2 | 28.d | even | 2 | 1 | ||
7056.2.a.cm | 2 | 84.h | odd | 2 | 1 | ||
7623.2.a.bi | 2 | 11.b | odd | 2 | 1 | ||
7623.2.a.bi | 2 | 33.d | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(63))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 3 \)
$3$
\( T^{2} \)
$5$
\( T^{2} - 12 \)
$7$
\( (T - 1)^{2} \)
$11$
\( T^{2} - 12 \)
$13$
\( (T - 2)^{2} \)
$17$
\( T^{2} - 12 \)
$19$
\( (T + 4)^{2} \)
$23$
\( T^{2} - 12 \)
$29$
\( T^{2} \)
$31$
\( (T + 4)^{2} \)
$37$
\( (T - 2)^{2} \)
$41$
\( T^{2} - 108 \)
$43$
\( (T + 4)^{2} \)
$47$
\( T^{2} - 48 \)
$53$
\( T^{2} - 48 \)
$59$
\( T^{2} - 48 \)
$61$
\( (T + 10)^{2} \)
$67$
\( (T + 4)^{2} \)
$71$
\( T^{2} - 108 \)
$73$
\( (T - 14)^{2} \)
$79$
\( (T - 8)^{2} \)
$83$
\( T^{2} \)
$89$
\( T^{2} - 12 \)
$97$
\( (T - 14)^{2} \)
show more
show less