Newspace parameters
| Level: | \( N \) | \(=\) | \( 5 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(5.36154644760\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
| Defining polynomial: |
\( x^{3} - x^{2} - 4466x - 18720 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3}\cdot 5 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{3} - x^{2} - 4466x - 18720 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{2} - 16\nu - 5949 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} + 8\beta _1 + 5957 ) / 2 \)
|
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 |
|
−90.6415 | −1125.79 | 23.8902 | 15625.0 | 102044. | 324482. | 740370. | −326912. | −1.41627e6 | |||||||||||||||||||||||||||
| 1.2 | 56.4248 | 2114.98 | −5008.25 | 15625.0 | 119337. | 325303. | −744821. | 2.87881e6 | 881637. | ||||||||||||||||||||||||||||
| 1.3 | 176.217 | −573.185 | 22860.4 | 15625.0 | −101005. | −201493. | 2.58481e6 | −1.26578e6 | 2.75339e6 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
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 | 5.14.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 45.14.a.e | 3 | ||
| 4.b | odd | 2 | 1 | 80.14.a.g | 3 | ||
| 5.b | even | 2 | 1 | 25.14.a.b | 3 | ||
| 5.c | odd | 4 | 2 | 25.14.b.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.14.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 25.14.a.b | 3 | 5.b | even | 2 | 1 | ||
| 25.14.b.b | 6 | 5.c | odd | 4 | 2 | ||
| 45.14.a.e | 3 | 3.b | odd | 2 | 1 | ||
| 80.14.a.g | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 142T_{2}^{2} - 11144T_{2} + 901248 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(5))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - 142 T^{2} - 11144 T + 901248 \)
$3$
\( T^{3} - 416 T^{2} + \cdots - 1364770944 \)
$5$
\( (T - 15625)^{3} \)
$7$
\( T^{3} - 448292 T^{2} + \cdots + 21\!\cdots\!88 \)
$11$
\( T^{3} + 6604004 T^{2} + \cdots + 88\!\cdots\!32 \)
$13$
\( T^{3} + 33501974 T^{2} + \cdots - 15\!\cdots\!04 \)
$17$
\( T^{3} - 83129542 T^{2} + \cdots + 55\!\cdots\!68 \)
$19$
\( T^{3} - 97491100 T^{2} + \cdots - 10\!\cdots\!00 \)
$23$
\( T^{3} - 316255836 T^{2} + \cdots - 13\!\cdots\!64 \)
$29$
\( T^{3} - 2236171850 T^{2} + \cdots + 18\!\cdots\!00 \)
$31$
\( T^{3} - 7482994376 T^{2} + \cdots + 61\!\cdots\!12 \)
$37$
\( T^{3} - 31447174242 T^{2} + \cdots + 46\!\cdots\!28 \)
$41$
\( T^{3} + 10752884434 T^{2} + \cdots - 82\!\cdots\!48 \)
$43$
\( T^{3} - 16930554856 T^{2} + \cdots + 34\!\cdots\!16 \)
$47$
\( T^{3} - 31934201692 T^{2} + \cdots + 18\!\cdots\!08 \)
$53$
\( T^{3} + 221149123934 T^{2} + \cdots - 33\!\cdots\!44 \)
$59$
\( T^{3} + 55436423900 T^{2} + \cdots + 18\!\cdots\!00 \)
$61$
\( T^{3} - 496161392746 T^{2} + \cdots + 83\!\cdots\!32 \)
$67$
\( T^{3} - 459297824792 T^{2} + \cdots + 78\!\cdots\!68 \)
$71$
\( T^{3} - 521997878336 T^{2} + \cdots - 40\!\cdots\!28 \)
$73$
\( T^{3} - 2505025571086 T^{2} + \cdots + 11\!\cdots\!36 \)
$79$
\( T^{3} - 2990636883200 T^{2} + \cdots + 23\!\cdots\!00 \)
$83$
\( T^{3} - 5137135467696 T^{2} + \cdots + 18\!\cdots\!76 \)
$89$
\( T^{3} + 19423025958450 T^{2} + \cdots + 22\!\cdots\!00 \)
$97$
\( T^{3} + 11088325396458 T^{2} + \cdots - 56\!\cdots\!92 \)
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