Newspace parameters
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(209.608008215\) |
| Analytic rank: | \(1\) |
| 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 |
|
−544.000 | −59049.0 | −1.80122e6 | 0 | 3.21227e7 | −1.27770e9 | 2.12071e9 | 3.48678e9 | 0 | |||||||||||||||||||||
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 | 75.22.a.b | 1 | |
| 5.b | even | 2 | 1 | 15.22.a.a | ✓ | 1 | |
| 5.c | odd | 4 | 2 | 75.22.b.c | 2 | ||
| 15.d | odd | 2 | 1 | 45.22.a.a | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.22.a.a | ✓ | 1 | 5.b | even | 2 | 1 | |
| 45.22.a.a | 1 | 15.d | odd | 2 | 1 | ||
| 75.22.a.b | 1 | 1.a | even | 1 | 1 | trivial | |
| 75.22.b.c | 2 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 544 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 544 \)
$3$
\( T + 59049 \)
$5$
\( T \)
$7$
\( T + 1277698380 \)
$11$
\( T + 77585921744 \)
$13$
\( T - 434110898702 \)
$17$
\( T + 12848917115782 \)
$19$
\( T + 28605256159796 \)
$23$
\( T + 224022192208080 \)
$29$
\( T + 51676030833142 \)
$31$
\( T - 8921108838285000 \)
$37$
\( T + 43\!\cdots\!90 \)
$41$
\( T - 58\!\cdots\!70 \)
$43$
\( T - 16\!\cdots\!76 \)
$47$
\( T - 16\!\cdots\!92 \)
$53$
\( T + 22\!\cdots\!70 \)
$59$
\( T - 51\!\cdots\!16 \)
$61$
\( T - 12\!\cdots\!98 \)
$67$
\( T - 54\!\cdots\!88 \)
$71$
\( T + 11\!\cdots\!68 \)
$73$
\( T - 37\!\cdots\!50 \)
$79$
\( T - 63\!\cdots\!60 \)
$83$
\( T - 14\!\cdots\!28 \)
$89$
\( T - 13\!\cdots\!14 \)
$97$
\( T - 32\!\cdots\!18 \)
show more
show less