Newspace parameters
| Level: | \( N \) | \(=\) | \( 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 19.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(1.12103629011\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.3144.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 16x - 8 \)
|
| 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 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
\( x^{3} - x^{2} - 16x - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 10 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + \beta _1 + 10 \)
|
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 |
|
−3.96257 | 6.71610 | 7.70200 | 18.1342 | −26.6130 | −25.8362 | 1.18085 | 18.1060 | −71.8581 | |||||||||||||||||||||||||||
| 1.2 | 1.89080 | 2.95388 | −4.42486 | −1.51710 | 5.58521 | 5.94196 | −23.4930 | −18.2746 | −2.86853 | ||||||||||||||||||||||||||||
| 1.3 | 5.07177 | −8.66998 | 17.7229 | −2.61710 | −43.9722 | −15.1058 | 49.3121 | 48.1686 | −13.2733 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(19\) | \(1\) |
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 | 19.4.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 171.4.a.f | 3 | ||
| 4.b | odd | 2 | 1 | 304.4.a.i | 3 | ||
| 5.b | even | 2 | 1 | 475.4.a.f | 3 | ||
| 5.c | odd | 4 | 2 | 475.4.b.f | 6 | ||
| 7.b | odd | 2 | 1 | 931.4.a.c | 3 | ||
| 8.b | even | 2 | 1 | 1216.4.a.s | 3 | ||
| 8.d | odd | 2 | 1 | 1216.4.a.u | 3 | ||
| 11.b | odd | 2 | 1 | 2299.4.a.h | 3 | ||
| 19.b | odd | 2 | 1 | 361.4.a.i | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 19.4.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 171.4.a.f | 3 | 3.b | odd | 2 | 1 | ||
| 304.4.a.i | 3 | 4.b | odd | 2 | 1 | ||
| 361.4.a.i | 3 | 19.b | odd | 2 | 1 | ||
| 475.4.a.f | 3 | 5.b | even | 2 | 1 | ||
| 475.4.b.f | 6 | 5.c | odd | 4 | 2 | ||
| 931.4.a.c | 3 | 7.b | odd | 2 | 1 | ||
| 1216.4.a.s | 3 | 8.b | even | 2 | 1 | ||
| 1216.4.a.u | 3 | 8.d | odd | 2 | 1 | ||
| 2299.4.a.h | 3 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 3T_{2}^{2} - 18T_{2} + 38 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(19))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - 3 T^{2} - 18 T + 38 \)
$3$
\( T^{3} - T^{2} - 64 T + 172 \)
$5$
\( T^{3} - 14 T^{2} - 71 T - 72 \)
$7$
\( T^{3} + 35 T^{2} + 147 T - 2319 \)
$11$
\( T^{3} - 16 T^{2} - 51 T + 1182 \)
$13$
\( T^{3} - 65 T^{2} + 744 T + 4848 \)
$17$
\( T^{3} - 29 T^{2} - 9225 T - 218619 \)
$19$
\( (T + 19)^{3} \)
$23$
\( T^{3} + 101 T^{2} - 4624 T - 378176 \)
$29$
\( T^{3} - 377 T^{2} + 8768 T + 4544396 \)
$31$
\( T^{3} + 140 T^{2} - 37616 T - 2444352 \)
$37$
\( T^{3} + 290 T^{2} + \cdots - 10001448 \)
$41$
\( T^{3} - 956 T^{2} + \cdots - 31578144 \)
$43$
\( T^{3} + 570 T^{2} + \cdots - 65963504 \)
$47$
\( T^{3} - 66 T^{2} - 31311 T + 2940624 \)
$53$
\( T^{3} - 817 T^{2} + \cdots - 16824816 \)
$59$
\( T^{3} - 265 T^{2} + \cdots + 31557612 \)
$61$
\( T^{3} - 988 T^{2} + \cdots + 76875874 \)
$67$
\( T^{3} + 207 T^{2} - 59928 T - 7515248 \)
$71$
\( T^{3} - 846 T^{2} + 172860 T + 1727928 \)
$73$
\( T^{3} - 627 T^{2} + \cdots + 145581839 \)
$79$
\( T^{3} - 382 T^{2} + \cdots - 56023488 \)
$83$
\( T^{3} + 766 T^{2} + \cdots - 78728352 \)
$89$
\( T^{3} + 172 T^{2} + \cdots - 76923456 \)
$97$
\( T^{3} + 2450 T^{2} + \cdots + 196438912 \)
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