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 5) |
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 |
|
4.00000 | 0 | 8.00000 | 5.00000 | 0 | 6.00000 | 0 | 0 | 20.0000 | |||||||||||||||||||||
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.d | 1 | |
3.b | odd | 2 | 1 | 5.4.a.a | ✓ | 1 | |
4.b | odd | 2 | 1 | 720.4.a.u | 1 | ||
5.b | even | 2 | 1 | 225.4.a.b | 1 | ||
5.c | odd | 4 | 2 | 225.4.b.c | 2 | ||
7.b | odd | 2 | 1 | 2205.4.a.q | 1 | ||
9.c | even | 3 | 2 | 405.4.e.c | 2 | ||
9.d | odd | 6 | 2 | 405.4.e.l | 2 | ||
12.b | even | 2 | 1 | 80.4.a.d | 1 | ||
15.d | odd | 2 | 1 | 25.4.a.c | 1 | ||
15.e | even | 4 | 2 | 25.4.b.a | 2 | ||
21.c | even | 2 | 1 | 245.4.a.a | 1 | ||
21.g | even | 6 | 2 | 245.4.e.g | 2 | ||
21.h | odd | 6 | 2 | 245.4.e.f | 2 | ||
24.f | even | 2 | 1 | 320.4.a.h | 1 | ||
24.h | odd | 2 | 1 | 320.4.a.g | 1 | ||
33.d | even | 2 | 1 | 605.4.a.d | 1 | ||
39.d | odd | 2 | 1 | 845.4.a.b | 1 | ||
48.i | odd | 4 | 2 | 1280.4.d.e | 2 | ||
48.k | even | 4 | 2 | 1280.4.d.l | 2 | ||
51.c | odd | 2 | 1 | 1445.4.a.a | 1 | ||
57.d | even | 2 | 1 | 1805.4.a.h | 1 | ||
60.h | even | 2 | 1 | 400.4.a.m | 1 | ||
60.l | odd | 4 | 2 | 400.4.c.k | 2 | ||
105.g | even | 2 | 1 | 1225.4.a.k | 1 | ||
120.i | odd | 2 | 1 | 1600.4.a.bi | 1 | ||
120.m | even | 2 | 1 | 1600.4.a.s | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
5.4.a.a | ✓ | 1 | 3.b | odd | 2 | 1 | |
25.4.a.c | 1 | 15.d | odd | 2 | 1 | ||
25.4.b.a | 2 | 15.e | even | 4 | 2 | ||
45.4.a.d | 1 | 1.a | even | 1 | 1 | trivial | |
80.4.a.d | 1 | 12.b | even | 2 | 1 | ||
225.4.a.b | 1 | 5.b | even | 2 | 1 | ||
225.4.b.c | 2 | 5.c | odd | 4 | 2 | ||
245.4.a.a | 1 | 21.c | even | 2 | 1 | ||
245.4.e.f | 2 | 21.h | odd | 6 | 2 | ||
245.4.e.g | 2 | 21.g | even | 6 | 2 | ||
320.4.a.g | 1 | 24.h | odd | 2 | 1 | ||
320.4.a.h | 1 | 24.f | even | 2 | 1 | ||
400.4.a.m | 1 | 60.h | even | 2 | 1 | ||
400.4.c.k | 2 | 60.l | odd | 4 | 2 | ||
405.4.e.c | 2 | 9.c | even | 3 | 2 | ||
405.4.e.l | 2 | 9.d | odd | 6 | 2 | ||
605.4.a.d | 1 | 33.d | even | 2 | 1 | ||
720.4.a.u | 1 | 4.b | odd | 2 | 1 | ||
845.4.a.b | 1 | 39.d | odd | 2 | 1 | ||
1225.4.a.k | 1 | 105.g | even | 2 | 1 | ||
1280.4.d.e | 2 | 48.i | odd | 4 | 2 | ||
1280.4.d.l | 2 | 48.k | even | 4 | 2 | ||
1445.4.a.a | 1 | 51.c | odd | 2 | 1 | ||
1600.4.a.s | 1 | 120.m | even | 2 | 1 | ||
1600.4.a.bi | 1 | 120.i | odd | 2 | 1 | ||
1805.4.a.h | 1 | 57.d | even | 2 | 1 | ||
2205.4.a.q | 1 | 7.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} - 4 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(45))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T - 4 \)
$3$
\( T \)
$5$
\( T - 5 \)
$7$
\( T - 6 \)
$11$
\( T + 32 \)
$13$
\( T + 38 \)
$17$
\( T + 26 \)
$19$
\( T - 100 \)
$23$
\( T - 78 \)
$29$
\( T - 50 \)
$31$
\( T + 108 \)
$37$
\( T - 266 \)
$41$
\( T + 22 \)
$43$
\( T - 442 \)
$47$
\( T - 514 \)
$53$
\( T + 2 \)
$59$
\( T + 500 \)
$61$
\( T + 518 \)
$67$
\( T - 126 \)
$71$
\( T + 412 \)
$73$
\( T + 878 \)
$79$
\( T - 600 \)
$83$
\( T + 282 \)
$89$
\( T - 150 \)
$97$
\( T - 386 \)
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