Newspace parameters
| Level: | \( N \) | \(=\) | \( 5 \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(7.13467525500\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
| Defining polynomial: |
\( x^{3} - x^{2} - 1972x + 21070 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4}\cdot 5 \) |
| 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
\( x^{3} - x^{2} - 1972x + 21070 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{2} + 78\nu + 2597 ) / 21 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{2} + 42\nu - 2645 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 7\beta _1 + 16 ) / 40 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 39\beta_{2} - 147\beta _1 + 52564 ) / 40 \)
|
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 |
|
−152.575 | −6379.22 | −9488.76 | −78125.0 | 973313. | −1.77538e6 | 6.44734e6 | 2.63456e7 | 1.19200e7 | |||||||||||||||||||||||||||
| 1.2 | −125.613 | 4146.67 | −16989.4 | −78125.0 | −520875. | 4.19779e6 | 6.25017e6 | 2.84597e6 | 9.81350e6 | ||||||||||||||||||||||||||||
| 1.3 | 282.188 | 5750.55 | 46862.2 | −78125.0 | 1.62274e6 | −3.32761e6 | 3.97721e6 | 1.87200e7 | −2.20460e7 | ||||||||||||||||||||||||||||
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 | 5.16.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 45.16.a.f | 3 | ||
| 4.b | odd | 2 | 1 | 80.16.a.g | 3 | ||
| 5.b | even | 2 | 1 | 25.16.a.c | 3 | ||
| 5.c | odd | 4 | 2 | 25.16.b.c | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5.16.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 25.16.a.c | 3 | 5.b | even | 2 | 1 | ||
| 25.16.b.c | 6 | 5.c | odd | 4 | 2 | ||
| 45.16.a.f | 3 | 3.b | odd | 2 | 1 | ||
| 80.16.a.g | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 4T_{2}^{2} - 59336T_{2} - 5408256 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(5))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - 4 T^{2} - 59336 T - 5408256 \)
$3$
\( T^{3} - 3518 T^{2} + \cdots + 152116768152 \)
$5$
\( (T + 78125)^{3} \)
$7$
\( T^{3} + 905206 T^{2} + \cdots - 24\!\cdots\!16 \)
$11$
\( T^{3} - 122875456 T^{2} + \cdots + 92\!\cdots\!92 \)
$13$
\( T^{3} + 90911522 T^{2} + \cdots - 95\!\cdots\!08 \)
$17$
\( T^{3} + 3868973426 T^{2} + \cdots + 65\!\cdots\!64 \)
$19$
\( T^{3} - 3670884220 T^{2} + \cdots + 46\!\cdots\!00 \)
$23$
\( T^{3} - 26698058238 T^{2} + \cdots - 27\!\cdots\!68 \)
$29$
\( T^{3} - 145544932730 T^{2} + \cdots + 75\!\cdots\!00 \)
$31$
\( T^{3} + 25873382644 T^{2} + \cdots - 90\!\cdots\!08 \)
$37$
\( T^{3} - 419480249934 T^{2} + \cdots + 70\!\cdots\!24 \)
$41$
\( T^{3} - 274005770306 T^{2} + \cdots - 22\!\cdots\!08 \)
$43$
\( T^{3} - 2350065869158 T^{2} + \cdots + 82\!\cdots\!12 \)
$47$
\( T^{3} + 8891070209486 T^{2} + \cdots - 20\!\cdots\!96 \)
$53$
\( T^{3} - 8749242811318 T^{2} + \cdots + 14\!\cdots\!52 \)
$59$
\( T^{3} + 14173516437140 T^{2} + \cdots + 44\!\cdots\!00 \)
$61$
\( T^{3} + 38066837721794 T^{2} + \cdots - 18\!\cdots\!08 \)
$67$
\( T^{3} - 144391638065474 T^{2} + \cdots - 97\!\cdots\!36 \)
$71$
\( T^{3} + 126512337318844 T^{2} + \cdots + 13\!\cdots\!92 \)
$73$
\( T^{3} - 199804772078038 T^{2} + \cdots + 74\!\cdots\!32 \)
$79$
\( T^{3} + 73797562093720 T^{2} + \cdots - 65\!\cdots\!00 \)
$83$
\( T^{3} + 219109046205402 T^{2} + \cdots + 10\!\cdots\!72 \)
$89$
\( T^{3} + 360765737744610 T^{2} + \cdots - 25\!\cdots\!00 \)
$97$
\( T^{3} + 509209931654586 T^{2} + \cdots - 21\!\cdots\!96 \)
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