Newspace parameters
| Level: | \( N \) | \(=\) | \( 121 = 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 121.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(172.659141171\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1) |
| 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 |
|
−216.000 | −3348.00 | 13888.0 | 52110.0 | 723168. | −2.82246e6 | 4.07808e6 | −3.13980e6 | −1.12558e7 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(-1\) |
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 | 121.16.a.a | 1 | |
| 11.b | odd | 2 | 1 | 1.16.a.a | ✓ | 1 | |
| 33.d | even | 2 | 1 | 9.16.a.a | 1 | ||
| 44.c | even | 2 | 1 | 16.16.a.d | 1 | ||
| 55.d | odd | 2 | 1 | 25.16.a.a | 1 | ||
| 55.e | even | 4 | 2 | 25.16.b.a | 2 | ||
| 77.b | even | 2 | 1 | 49.16.a.a | 1 | ||
| 77.h | odd | 6 | 2 | 49.16.c.c | 2 | ||
| 77.i | even | 6 | 2 | 49.16.c.b | 2 | ||
| 88.b | odd | 2 | 1 | 64.16.a.i | 1 | ||
| 88.g | even | 2 | 1 | 64.16.a.c | 1 | ||
| 132.d | odd | 2 | 1 | 144.16.a.f | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.16.a.a | ✓ | 1 | 11.b | odd | 2 | 1 | |
| 9.16.a.a | 1 | 33.d | even | 2 | 1 | ||
| 16.16.a.d | 1 | 44.c | even | 2 | 1 | ||
| 25.16.a.a | 1 | 55.d | odd | 2 | 1 | ||
| 25.16.b.a | 2 | 55.e | even | 4 | 2 | ||
| 49.16.a.a | 1 | 77.b | even | 2 | 1 | ||
| 49.16.c.b | 2 | 77.i | even | 6 | 2 | ||
| 49.16.c.c | 2 | 77.h | odd | 6 | 2 | ||
| 64.16.a.c | 1 | 88.g | even | 2 | 1 | ||
| 64.16.a.i | 1 | 88.b | odd | 2 | 1 | ||
| 121.16.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
| 144.16.a.f | 1 | 132.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 216 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(121))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 216 \)
$3$
\( T + 3348 \)
$5$
\( T - 52110 \)
$7$
\( T + 2822456 \)
$11$
\( T \)
$13$
\( T - 190073338 \)
$17$
\( T + 1646527986 \)
$19$
\( T + 1563257180 \)
$23$
\( T - 9451116072 \)
$29$
\( T - 36902568330 \)
$31$
\( T - 71588483552 \)
$37$
\( T + 1033652081554 \)
$41$
\( T + 1641974018202 \)
$43$
\( T - 492403109308 \)
$47$
\( T + 3410684952624 \)
$53$
\( T - 6797151655902 \)
$59$
\( T - 9858856815540 \)
$61$
\( T + 4931842626902 \)
$67$
\( T + 28837826625364 \)
$71$
\( T - 125050114914552 \)
$73$
\( T - 82171455513478 \)
$79$
\( T - 25413078694480 \)
$83$
\( T - 281736730890468 \)
$89$
\( T - 715618564776810 \)
$97$
\( T - 612786136081826 \)
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