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: | \(\Q(\sqrt{633}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 158 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{633})\). 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 | −72.2392 | 64.0000 | 467.472 | −577.914 | −1471.23 | 512.000 | 3031.51 | 3739.78 | ||||||||||||||||||||||||
| 1.2 | 8.00000 | 3.23924 | 64.0000 | −312.472 | 25.9139 | −766.767 | 512.000 | −2176.51 | −2499.78 | |||||||||||||||||||||||||
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.d | ✓ | 2 |
| 3.b | odd | 2 | 1 | 342.8.a.g | 2 | ||
| 4.b | odd | 2 | 1 | 304.8.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.8.a.d | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 304.8.a.d | 2 | 4.b | odd | 2 | 1 | ||
| 342.8.a.g | 2 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 69T_{3} - 234 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(38))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T - 8)^{2} \)
$3$
\( T^{2} + 69T - 234 \)
$5$
\( T^{2} - 155T - 146072 \)
$7$
\( T^{2} + 2238 T + 1128093 \)
$11$
\( T^{2} + 3295 T - 894002 \)
$13$
\( T^{2} + 13427 T + 13167724 \)
$17$
\( T^{2} + 32256 T + 253133559 \)
$19$
\( (T + 6859)^{2} \)
$23$
\( T^{2} + 82525 T - 1087606952 \)
$29$
\( T^{2} + 12749 T - 1552615514 \)
$31$
\( T^{2} - 258944 T + 9936311536 \)
$37$
\( T^{2} + 149260 T - 6156318428 \)
$41$
\( T^{2} - 339130 T - 393872642768 \)
$43$
\( T^{2} + 83869 T - 268023173408 \)
$47$
\( T^{2} - 1471025 T + 474895142800 \)
$53$
\( T^{2} + 945643 T + 79392286228 \)
$59$
\( T^{2} + 969009 T - 1837525132614 \)
$61$
\( T^{2} + 1506755 T - 359679230318 \)
$67$
\( T^{2} + 1848219 T - 1107042934188 \)
$71$
\( T^{2} + 3417184 T - 10503963815108 \)
$73$
\( T^{2} + 2499822 T - 4520032488687 \)
$79$
\( T^{2} - 2636926 T - 5162668519808 \)
$83$
\( T^{2} + 10059354 T + 23086028557656 \)
$89$
\( T^{2} + 3506160 T - 29977064763600 \)
$97$
\( T^{2} - 5893526 T - 88352296135184 \)
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