Newspace parameters
Level: | \( N \) | \(=\) | \( 38 = 2 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 38.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(6.09458515289\) |
Analytic rank: | \(1\) |
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 |
|
−4.00000 | −6.00000 | 16.0000 | 31.0000 | 24.0000 | −27.0000 | −64.0000 | −207.000 | −124.000 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(19\) | \(1\) |
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 | 38.6.a.a | ✓ | 1 |
3.b | odd | 2 | 1 | 342.6.a.e | 1 | ||
4.b | odd | 2 | 1 | 304.6.a.c | 1 | ||
5.b | even | 2 | 1 | 950.6.a.b | 1 | ||
19.b | odd | 2 | 1 | 722.6.a.b | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
38.6.a.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
304.6.a.c | 1 | 4.b | odd | 2 | 1 | ||
342.6.a.e | 1 | 3.b | odd | 2 | 1 | ||
722.6.a.b | 1 | 19.b | odd | 2 | 1 | ||
950.6.a.b | 1 | 5.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} + 6 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(38))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 4 \)
$3$
\( T + 6 \)
$5$
\( T - 31 \)
$7$
\( T + 27 \)
$11$
\( T + 323 \)
$13$
\( T + 676 \)
$17$
\( T + 1107 \)
$19$
\( T + 361 \)
$23$
\( T - 1384 \)
$29$
\( T - 2870 \)
$31$
\( T - 1372 \)
$37$
\( T + 7982 \)
$41$
\( T - 1202 \)
$43$
\( T + 9911 \)
$47$
\( T - 3463 \)
$53$
\( T - 17764 \)
$59$
\( T - 27270 \)
$61$
\( T - 20867 \)
$67$
\( T - 15228 \)
$71$
\( T - 40642 \)
$73$
\( T + 66711 \)
$79$
\( T - 68960 \)
$83$
\( T + 12396 \)
$89$
\( T - 41220 \)
$97$
\( T + 113432 \)
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