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: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 3) |
| 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 |
|
−1728.00 | 59049.0 | 888832. | 0 | −1.02037e8 | −5.38430e8 | 2.08798e9 | 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.a | 1 | |
| 5.b | even | 2 | 1 | 3.22.a.b | ✓ | 1 | |
| 5.c | odd | 4 | 2 | 75.22.b.b | 2 | ||
| 15.d | odd | 2 | 1 | 9.22.a.a | 1 | ||
| 20.d | odd | 2 | 1 | 48.22.a.d | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.22.a.b | ✓ | 1 | 5.b | even | 2 | 1 | |
| 9.22.a.a | 1 | 15.d | odd | 2 | 1 | ||
| 48.22.a.d | 1 | 20.d | odd | 2 | 1 | ||
| 75.22.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
| 75.22.b.b | 2 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 1728 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 1728 \)
$3$
\( T - 59049 \)
$5$
\( T \)
$7$
\( T + 538429808 \)
$11$
\( T + 64113040188 \)
$13$
\( T - 130980107986 \)
$17$
\( T + 8242029723618 \)
$19$
\( T - 13492101753020 \)
$23$
\( T - 233184825844776 \)
$29$
\( T + 2024562031123770 \)
$31$
\( T + 6869194988701768 \)
$37$
\( T + 3443998107027638 \)
$41$
\( T + 21\!\cdots\!58 \)
$43$
\( T - 71\!\cdots\!56 \)
$47$
\( T + 28\!\cdots\!48 \)
$53$
\( T - 21\!\cdots\!46 \)
$59$
\( T - 15\!\cdots\!60 \)
$61$
\( T - 43\!\cdots\!62 \)
$67$
\( T + 92\!\cdots\!68 \)
$71$
\( T + 20\!\cdots\!28 \)
$73$
\( T + 16\!\cdots\!74 \)
$79$
\( T - 67\!\cdots\!80 \)
$83$
\( T + 39\!\cdots\!84 \)
$89$
\( T - 41\!\cdots\!90 \)
$97$
\( T + 57\!\cdots\!98 \)
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