Newspace parameters
| Level: | \( N \) | \(=\) | \( 16 = 2^{4} \) |
| Weight: | \( k \) | \(=\) | \( 42 \) |
| Character orbit: | \([\chi]\) | \(=\) | 16.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(170.354672730\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 2784108376x + 1945534874860 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{22}\cdot 3^{3}\cdot 5\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 1) |
| 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
\( x^{3} - x^{2} - 2784108376x + 1945534874860 \)
:
| \(\beta_{1}\) | \(=\) |
\( 98304\nu - 32768 \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{2} - 9756\nu - 7424285752 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 32768 ) / 98304 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 8192\beta_{2} + 813\beta _1 + 60819775520768 ) / 32768 \)
|
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 |
|
0 | −3.82217e9 | 0 | 9.10518e13 | 0 | 1.69829e16 | 0 | −2.18640e19 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 6.65953e9 | 0 | −3.38080e14 | 0 | 9.19453e16 | 0 | 7.87633e18 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 7.98359e9 | 0 | 3.47255e13 | 0 | −1.66807e17 | 0 | 2.72647e19 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
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 | 16.42.a.c | 3 | |
| 4.b | odd | 2 | 1 | 1.42.a.a | ✓ | 3 | |
| 12.b | even | 2 | 1 | 9.42.a.b | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.42.a.a | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 9.42.a.b | 3 | 12.b | even | 2 | 1 | ||
| 16.42.a.c | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 10820953044T_{3}^{2} - 2801484230995686096T_{3} + 203212914080074439921138332224 \)
acting on \(S_{42}^{\mathrm{new}}(\Gamma_0(16))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + \cdots + 20\!\cdots\!24 \)
|
| $5$ |
\( T^{3} + \cdots + 10\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots + 26\!\cdots\!12 \)
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| $11$ |
\( T^{3} + \cdots + 13\!\cdots\!28 \)
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| $13$ |
\( T^{3} + \cdots - 82\!\cdots\!64 \)
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| $17$ |
\( T^{3} + \cdots - 10\!\cdots\!12 \)
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| $19$ |
\( T^{3} + \cdots + 72\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots - 30\!\cdots\!96 \)
|
| $29$ |
\( T^{3} + \cdots - 41\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots - 30\!\cdots\!32 \)
|
| $37$ |
\( T^{3} + \cdots - 39\!\cdots\!12 \)
|
| $41$ |
\( T^{3} + \cdots + 94\!\cdots\!12 \)
|
| $43$ |
\( T^{3} + \cdots + 19\!\cdots\!84 \)
|
| $47$ |
\( T^{3} + \cdots - 64\!\cdots\!88 \)
|
| $53$ |
\( T^{3} + \cdots - 82\!\cdots\!24 \)
|
| $59$ |
\( T^{3} + \cdots + 28\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 48\!\cdots\!72 \)
|
| $67$ |
\( T^{3} + \cdots - 27\!\cdots\!88 \)
|
| $71$ |
\( T^{3} + \cdots + 39\!\cdots\!48 \)
|
| $73$ |
\( T^{3} + \cdots + 17\!\cdots\!96 \)
|
| $79$ |
\( T^{3} + \cdots - 16\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 69\!\cdots\!56 \)
|
| $89$ |
\( T^{3} + \cdots + 29\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 86\!\cdots\!12 \)
|
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