Newspace parameters
| Level: | \( N \) | \(=\) | \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 180.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(10.6203438010\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 60) |
| 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 | 5.00000 | 0 | −28.0000 | 0 | 0 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
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 | 180.4.a.d | 1 | |
| 3.b | odd | 2 | 1 | 60.4.a.a | ✓ | 1 | |
| 4.b | odd | 2 | 1 | 720.4.a.bb | 1 | ||
| 5.b | even | 2 | 1 | 900.4.a.q | 1 | ||
| 5.c | odd | 4 | 2 | 900.4.d.h | 2 | ||
| 9.c | even | 3 | 2 | 1620.4.i.f | 2 | ||
| 9.d | odd | 6 | 2 | 1620.4.i.l | 2 | ||
| 12.b | even | 2 | 1 | 240.4.a.i | 1 | ||
| 15.d | odd | 2 | 1 | 300.4.a.i | 1 | ||
| 15.e | even | 4 | 2 | 300.4.d.b | 2 | ||
| 24.f | even | 2 | 1 | 960.4.a.r | 1 | ||
| 24.h | odd | 2 | 1 | 960.4.a.bc | 1 | ||
| 60.h | even | 2 | 1 | 1200.4.a.a | 1 | ||
| 60.l | odd | 4 | 2 | 1200.4.f.n | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 60.4.a.a | ✓ | 1 | 3.b | odd | 2 | 1 | |
| 180.4.a.d | 1 | 1.a | even | 1 | 1 | trivial | |
| 240.4.a.i | 1 | 12.b | even | 2 | 1 | ||
| 300.4.a.i | 1 | 15.d | odd | 2 | 1 | ||
| 300.4.d.b | 2 | 15.e | even | 4 | 2 | ||
| 720.4.a.bb | 1 | 4.b | odd | 2 | 1 | ||
| 900.4.a.q | 1 | 5.b | even | 2 | 1 | ||
| 900.4.d.h | 2 | 5.c | odd | 4 | 2 | ||
| 960.4.a.r | 1 | 24.f | even | 2 | 1 | ||
| 960.4.a.bc | 1 | 24.h | odd | 2 | 1 | ||
| 1200.4.a.a | 1 | 60.h | even | 2 | 1 | ||
| 1200.4.f.n | 2 | 60.l | odd | 4 | 2 | ||
| 1620.4.i.f | 2 | 9.c | even | 3 | 2 | ||
| 1620.4.i.l | 2 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(180))\):
|
\( T_{7} + 28 \)
|
|
\( T_{11} - 24 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T \)
|
| $5$ |
\( T - 5 \)
|
| $7$ |
\( T + 28 \)
|
| $11$ |
\( T - 24 \)
|
| $13$ |
\( T + 70 \)
|
| $17$ |
\( T + 102 \)
|
| $19$ |
\( T - 20 \)
|
| $23$ |
\( T - 72 \)
|
| $29$ |
\( T + 306 \)
|
| $31$ |
\( T + 136 \)
|
| $37$ |
\( T + 214 \)
|
| $41$ |
\( T - 150 \)
|
| $43$ |
\( T + 292 \)
|
| $47$ |
\( T - 72 \)
|
| $53$ |
\( T - 414 \)
|
| $59$ |
\( T - 744 \)
|
| $61$ |
\( T + 418 \)
|
| $67$ |
\( T - 188 \)
|
| $71$ |
\( T + 480 \)
|
| $73$ |
\( T - 434 \)
|
| $79$ |
\( T - 1352 \)
|
| $83$ |
\( T - 612 \)
|
| $89$ |
\( T - 30 \)
|
| $97$ |
\( T + 286 \)
|
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