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: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{1441}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 360 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{1441})\). We also show the integral \(q\)-expansion of the trace form.
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 | −17.4803 | 16.0000 | −79.4408 | 69.9210 | 132.921 | −64.0000 | 62.5592 | 317.763 | ||||||||||||||||||||||||
| 1.2 | −4.00000 | 20.4803 | 16.0000 | 34.4408 | −81.9210 | −18.9210 | −64.0000 | 176.441 | −137.763 | |||||||||||||||||||||||||
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.c | ✓ | 2 |
| 3.b | odd | 2 | 1 | 342.6.a.i | 2 | ||
| 4.b | odd | 2 | 1 | 304.6.a.f | 2 | ||
| 5.b | even | 2 | 1 | 950.6.a.d | 2 | ||
| 19.b | odd | 2 | 1 | 722.6.a.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.6.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 304.6.a.f | 2 | 4.b | odd | 2 | 1 | ||
| 342.6.a.i | 2 | 3.b | odd | 2 | 1 | ||
| 722.6.a.c | 2 | 19.b | odd | 2 | 1 | ||
| 950.6.a.d | 2 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 3T_{3} - 358 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(38))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T + 4)^{2} \)
$3$
\( T^{2} - 3T - 358 \)
$5$
\( T^{2} + 45T - 2736 \)
$7$
\( T^{2} - 114T - 2515 \)
$11$
\( T^{2} - 661T + 108870 \)
$13$
\( T^{2} - 1613 T + 641436 \)
$17$
\( T^{2} - 64T - 35001 \)
$19$
\( (T - 361)^{2} \)
$23$
\( T^{2} + 3185 T + 1671096 \)
$29$
\( T^{2} + 2481 T - 34206966 \)
$31$
\( T^{2} + 1180 T - 35625024 \)
$37$
\( T^{2} - 10488 T + 16841900 \)
$41$
\( T^{2} - 16630 T - 61416816 \)
$43$
\( T^{2} - 11303 T + 3493752 \)
$47$
\( T^{2} + 12155 T - 410935800 \)
$53$
\( T^{2} - 20585 T - 24187104 \)
$59$
\( T^{2} + 78581 T + 1541823618 \)
$61$
\( T^{2} - 43621 T + 230502754 \)
$67$
\( T^{2} - 7805 T - 3449006904 \)
$71$
\( T^{2} + 62488 T + 396968940 \)
$73$
\( T^{2} - 16218 T - 3142002343 \)
$79$
\( T^{2} - 67122 T + 1119872072 \)
$83$
\( T^{2} + 10714 T - 2126648040 \)
$89$
\( T^{2} - 128188 T - 2478833568 \)
$97$
\( T^{2} - 178558 T + 5494062880 \)
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