Newspace parameters
| Level: | \( N \) | \(=\) | \( 1728 = 2^{6} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1728.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(101.955300490\) |
| Analytic rank: | \(2\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 54) |
| 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 |
|
0 | 0 | 0 | −3.00000 | 0 | −29.0000 | 0 | 0 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
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 | 1728.4.a.k | 1 | |
| 3.b | odd | 2 | 1 | 1728.4.a.u | 1 | ||
| 4.b | odd | 2 | 1 | 1728.4.a.l | 1 | ||
| 8.b | even | 2 | 1 | 432.4.a.j | 1 | ||
| 8.d | odd | 2 | 1 | 54.4.a.c | yes | 1 | |
| 12.b | even | 2 | 1 | 1728.4.a.v | 1 | ||
| 24.f | even | 2 | 1 | 54.4.a.b | ✓ | 1 | |
| 24.h | odd | 2 | 1 | 432.4.a.e | 1 | ||
| 40.e | odd | 2 | 1 | 1350.4.a.a | 1 | ||
| 40.k | even | 4 | 2 | 1350.4.c.b | 2 | ||
| 72.l | even | 6 | 2 | 162.4.c.g | 2 | ||
| 72.p | odd | 6 | 2 | 162.4.c.b | 2 | ||
| 120.m | even | 2 | 1 | 1350.4.a.o | 1 | ||
| 120.q | odd | 4 | 2 | 1350.4.c.s | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 54.4.a.b | ✓ | 1 | 24.f | even | 2 | 1 | |
| 54.4.a.c | yes | 1 | 8.d | odd | 2 | 1 | |
| 162.4.c.b | 2 | 72.p | odd | 6 | 2 | ||
| 162.4.c.g | 2 | 72.l | even | 6 | 2 | ||
| 432.4.a.e | 1 | 24.h | odd | 2 | 1 | ||
| 432.4.a.j | 1 | 8.b | even | 2 | 1 | ||
| 1350.4.a.a | 1 | 40.e | odd | 2 | 1 | ||
| 1350.4.a.o | 1 | 120.m | even | 2 | 1 | ||
| 1350.4.c.b | 2 | 40.k | even | 4 | 2 | ||
| 1350.4.c.s | 2 | 120.q | odd | 4 | 2 | ||
| 1728.4.a.k | 1 | 1.a | even | 1 | 1 | trivial | |
| 1728.4.a.l | 1 | 4.b | odd | 2 | 1 | ||
| 1728.4.a.u | 1 | 3.b | odd | 2 | 1 | ||
| 1728.4.a.v | 1 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1728))\):
|
\( T_{5} + 3 \)
|
|
\( T_{7} + 29 \)
|
|
\( T_{11} + 57 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T \)
|
| $5$ |
\( T + 3 \)
|
| $7$ |
\( T + 29 \)
|
| $11$ |
\( T + 57 \)
|
| $13$ |
\( T + 20 \)
|
| $17$ |
\( T + 72 \)
|
| $19$ |
\( T + 106 \)
|
| $23$ |
\( T + 174 \)
|
| $29$ |
\( T - 210 \)
|
| $31$ |
\( T + 47 \)
|
| $37$ |
\( T + 2 \)
|
| $41$ |
\( T + 6 \)
|
| $43$ |
\( T - 218 \)
|
| $47$ |
\( T + 474 \)
|
| $53$ |
\( T + 81 \)
|
| $59$ |
\( T - 84 \)
|
| $61$ |
\( T + 56 \)
|
| $67$ |
\( T + 142 \)
|
| $71$ |
\( T + 360 \)
|
| $73$ |
\( T + 1159 \)
|
| $79$ |
\( T - 160 \)
|
| $83$ |
\( T - 735 \)
|
| $89$ |
\( T + 954 \)
|
| $97$ |
\( T - 191 \)
|
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