Newspace parameters
| Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 33.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(5.29266605383\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 |
|
−2.00000 | −9.00000 | −28.0000 | 46.0000 | 18.0000 | 148.000 | 120.000 | 81.0000 | −92.0000 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(11\) | \(-1\) |
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 | 33.6.a.a | ✓ | 1 |
| 3.b | odd | 2 | 1 | 99.6.a.b | 1 | ||
| 4.b | odd | 2 | 1 | 528.6.a.i | 1 | ||
| 5.b | even | 2 | 1 | 825.6.a.b | 1 | ||
| 11.b | odd | 2 | 1 | 363.6.a.c | 1 | ||
| 33.d | even | 2 | 1 | 1089.6.a.d | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.6.a.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
| 99.6.a.b | 1 | 3.b | odd | 2 | 1 | ||
| 363.6.a.c | 1 | 11.b | odd | 2 | 1 | ||
| 528.6.a.i | 1 | 4.b | odd | 2 | 1 | ||
| 825.6.a.b | 1 | 5.b | even | 2 | 1 | ||
| 1089.6.a.d | 1 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 2 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(33))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 2 \)
$3$
\( T + 9 \)
$5$
\( T - 46 \)
$7$
\( T - 148 \)
$11$
\( T - 121 \)
$13$
\( T - 574 \)
$17$
\( T + 722 \)
$19$
\( T - 2160 \)
$23$
\( T + 2536 \)
$29$
\( T - 4650 \)
$31$
\( T - 5032 \)
$37$
\( T - 8118 \)
$41$
\( T + 5138 \)
$43$
\( T - 8304 \)
$47$
\( T - 24728 \)
$53$
\( T + 28746 \)
$59$
\( T + 5860 \)
$61$
\( T + 53658 \)
$67$
\( T - 30908 \)
$71$
\( T + 69648 \)
$73$
\( T + 18446 \)
$79$
\( T + 25300 \)
$83$
\( T + 17556 \)
$89$
\( T - 132570 \)
$97$
\( T - 70658 \)
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