Newspace parameters
| Level: | \( N \) | \(=\) | \( 45 = 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 45.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(2.65508595026\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 15) |
| 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 |
|
−3.00000 | 0 | 1.00000 | 5.00000 | 0 | 20.0000 | 21.0000 | 0 | −15.0000 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
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 | 45.4.a.b | 1 | |
| 3.b | odd | 2 | 1 | 15.4.a.b | ✓ | 1 | |
| 4.b | odd | 2 | 1 | 720.4.a.r | 1 | ||
| 5.b | even | 2 | 1 | 225.4.a.g | 1 | ||
| 5.c | odd | 4 | 2 | 225.4.b.d | 2 | ||
| 7.b | odd | 2 | 1 | 2205.4.a.c | 1 | ||
| 9.c | even | 3 | 2 | 405.4.e.k | 2 | ||
| 9.d | odd | 6 | 2 | 405.4.e.d | 2 | ||
| 12.b | even | 2 | 1 | 240.4.a.f | 1 | ||
| 15.d | odd | 2 | 1 | 75.4.a.a | 1 | ||
| 15.e | even | 4 | 2 | 75.4.b.a | 2 | ||
| 21.c | even | 2 | 1 | 735.4.a.i | 1 | ||
| 24.f | even | 2 | 1 | 960.4.a.l | 1 | ||
| 24.h | odd | 2 | 1 | 960.4.a.bi | 1 | ||
| 33.d | even | 2 | 1 | 1815.4.a.a | 1 | ||
| 60.h | even | 2 | 1 | 1200.4.a.o | 1 | ||
| 60.l | odd | 4 | 2 | 1200.4.f.m | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.4.a.b | ✓ | 1 | 3.b | odd | 2 | 1 | |
| 45.4.a.b | 1 | 1.a | even | 1 | 1 | trivial | |
| 75.4.a.a | 1 | 15.d | odd | 2 | 1 | ||
| 75.4.b.a | 2 | 15.e | even | 4 | 2 | ||
| 225.4.a.g | 1 | 5.b | even | 2 | 1 | ||
| 225.4.b.d | 2 | 5.c | odd | 4 | 2 | ||
| 240.4.a.f | 1 | 12.b | even | 2 | 1 | ||
| 405.4.e.d | 2 | 9.d | odd | 6 | 2 | ||
| 405.4.e.k | 2 | 9.c | even | 3 | 2 | ||
| 720.4.a.r | 1 | 4.b | odd | 2 | 1 | ||
| 735.4.a.i | 1 | 21.c | even | 2 | 1 | ||
| 960.4.a.l | 1 | 24.f | even | 2 | 1 | ||
| 960.4.a.bi | 1 | 24.h | odd | 2 | 1 | ||
| 1200.4.a.o | 1 | 60.h | even | 2 | 1 | ||
| 1200.4.f.m | 2 | 60.l | odd | 4 | 2 | ||
| 1815.4.a.a | 1 | 33.d | even | 2 | 1 | ||
| 2205.4.a.c | 1 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 3 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(45))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 3 \)
$3$
\( T \)
$5$
\( T - 5 \)
$7$
\( T - 20 \)
$11$
\( T - 24 \)
$13$
\( T - 74 \)
$17$
\( T + 54 \)
$19$
\( T + 124 \)
$23$
\( T - 120 \)
$29$
\( T - 78 \)
$31$
\( T - 200 \)
$37$
\( T + 70 \)
$41$
\( T + 330 \)
$43$
\( T - 92 \)
$47$
\( T - 24 \)
$53$
\( T + 450 \)
$59$
\( T + 24 \)
$61$
\( T + 322 \)
$67$
\( T + 196 \)
$71$
\( T - 288 \)
$73$
\( T + 430 \)
$79$
\( T + 520 \)
$83$
\( T + 156 \)
$89$
\( T + 1026 \)
$97$
\( T + 286 \)
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