Newspace parameters
| Level: | \( N \) | \(=\) | \( 38 = 2 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 38.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(11.8706309684\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{17953}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 4488 \)
|
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{17953})\). 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 |
|
−8.00000 | −72.4944 | 64.0000 | 166.483 | 579.955 | 763.922 | −512.000 | 3068.44 | −1331.87 | ||||||||||||||||||||||||
| 1.2 | −8.00000 | 61.4944 | 64.0000 | −235.483 | −491.955 | −1111.92 | −512.000 | 1594.56 | 1883.87 | |||||||||||||||||||||||||
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 | 38.8.a.c | ✓ | 2 |
| 3.b | odd | 2 | 1 | 342.8.a.i | 2 | ||
| 4.b | odd | 2 | 1 | 304.8.a.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.8.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 304.8.a.b | 2 | 4.b | odd | 2 | 1 | ||
| 342.8.a.i | 2 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 11T_{3} - 4458 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(38))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T + 8)^{2} \)
$3$
\( T^{2} + 11T - 4458 \)
$5$
\( T^{2} + 69T - 39204 \)
$7$
\( T^{2} + 348T - 849421 \)
$11$
\( T^{2} - 2723 T + 1849194 \)
$13$
\( T^{2} + 14113 T + 47419908 \)
$17$
\( T^{2} + 38560 T + 304915287 \)
$19$
\( (T - 6859)^{2} \)
$23$
\( T^{2} + 73897 T + 727281168 \)
$29$
\( T^{2} - 159813 T - 10315913526 \)
$31$
\( T^{2} + 259468 T + 16830838944 \)
$37$
\( T^{2} + 528168 T + 17489301548 \)
$41$
\( T^{2} - 1005650 T + 249116260800 \)
$43$
\( T^{2} - 286217 T - 224831764452 \)
$47$
\( T^{2} + 1397509 T + 485859505512 \)
$53$
\( T^{2} + 1385969 T - 1418308915764 \)
$59$
\( T^{2} - 2700953 T - 405173436054 \)
$61$
\( T^{2} + 3975947 T + 1928935887694 \)
$67$
\( T^{2} + 134557 T - 12718841223612 \)
$71$
\( T^{2} - 4202740 T + 1157380019748 \)
$73$
\( T^{2} - 900498 T - 4028508966511 \)
$79$
\( T^{2} + 6893730 T + 11081929595552 \)
$83$
\( T^{2} + 2465330 T - 991765457112 \)
$89$
\( T^{2} - 17431724 T + 64480164517536 \)
$97$
\( T^{2} - 6351934 T - 97107139603184 \)
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