Newspace parameters
Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 144.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(1.14984578911\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 36) |
Fricke sign: | \(-1\) |
Sato-Tate group: | $N(\mathrm{U}(1))$ |
$q$-expansion
Expression as an eta quotient
\(f(z) = \dfrac{\eta(12z)^{12}}{\eta(6z)^{4}\eta(24z)^{4}}=q\prod_{n=1}^\infty(1 - q^{6n})^{-4}(1 - q^{12n})^{12}(1 - q^{24n})^{-4}\)
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 | 0 | 0 | 4.00000 | 0 | 0 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(3\) | \(1\) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(144))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T \)
$5$
\( T \)
$7$
\( T - 4 \)
$11$
\( T \)
$13$
\( T - 2 \)
$17$
\( T \)
$19$
\( T + 8 \)
$23$
\( T \)
$29$
\( T \)
$31$
\( T - 4 \)
$37$
\( T + 10 \)
$41$
\( T \)
$43$
\( T + 8 \)
$47$
\( T \)
$53$
\( T \)
$59$
\( T \)
$61$
\( T - 14 \)
$67$
\( T - 16 \)
$71$
\( T \)
$73$
\( T + 10 \)
$79$
\( T - 4 \)
$83$
\( T \)
$89$
\( T \)
$97$
\( T - 14 \)
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