Newspace parameters
| Level: | \( N \) | \(=\) | \( 16 = 2^{4} \) |
| Weight: | \( k \) | \(=\) | \( 30 \) |
| Character orbit: | \([\chi]\) | \(=\) | 16.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(85.2448678129\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{3} - x^{2} - 97534740x + 130260063600 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{22}\cdot 3^{3}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 8) |
| 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
\( x^{3} - x^{2} - 97534740x + 130260063600 \)
:
| \(\beta_{1}\) | \(=\) |
\( 1152\nu - 384 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8192\nu^{2} + 16403968\nu - 532675197440 ) / 25 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 384 ) / 1152 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 225\beta_{2} - 128156\beta _1 + 4794027565056 ) / 73728 \)
|
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 | −1.27749e7 | 0 | −2.40170e10 | 0 | −6.66574e11 | 0 | 9.45684e13 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −3.83101e6 | 0 | 1.31219e10 | 0 | 3.69184e11 | 0 | −5.39537e13 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 9.81666e6 | 0 | −6.33613e9 | 0 | −1.81947e11 | 0 | 2.77365e13 | 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 | 16.30.a.d | 3 | |
| 4.b | odd | 2 | 1 | 8.30.a.a | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.30.a.a | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 16.30.a.d | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 6789276T_{3}^{2} - 114073987833936T_{3} - 480436149154912885440 \)
acting on \(S_{30}^{\mathrm{new}}(\Gamma_0(16))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + \cdots - 48\!\cdots\!40 \)
|
| $5$ |
\( T^{3} + \cdots - 19\!\cdots\!00 \)
|
| $7$ |
\( T^{3} + \cdots - 44\!\cdots\!84 \)
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| $11$ |
\( T^{3} + \cdots - 98\!\cdots\!16 \)
|
| $13$ |
\( T^{3} + \cdots + 60\!\cdots\!48 \)
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| $17$ |
\( T^{3} + \cdots + 55\!\cdots\!00 \)
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| $19$ |
\( T^{3} + \cdots + 36\!\cdots\!04 \)
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| $23$ |
\( T^{3} + \cdots + 69\!\cdots\!24 \)
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| $29$ |
\( T^{3} + \cdots - 29\!\cdots\!52 \)
|
| $31$ |
\( T^{3} + \cdots + 11\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots + 46\!\cdots\!76 \)
|
| $41$ |
\( T^{3} + \cdots + 17\!\cdots\!44 \)
|
| $43$ |
\( T^{3} + \cdots - 10\!\cdots\!92 \)
|
| $47$ |
\( T^{3} + \cdots - 70\!\cdots\!84 \)
|
| $53$ |
\( T^{3} + \cdots - 68\!\cdots\!24 \)
|
| $59$ |
\( T^{3} + \cdots + 36\!\cdots\!24 \)
|
| $61$ |
\( T^{3} + \cdots + 61\!\cdots\!00 \)
|
| $67$ |
\( T^{3} + \cdots + 18\!\cdots\!92 \)
|
| $71$ |
\( T^{3} + \cdots + 57\!\cdots\!80 \)
|
| $73$ |
\( T^{3} + \cdots + 96\!\cdots\!76 \)
|
| $79$ |
\( T^{3} + \cdots + 22\!\cdots\!80 \)
|
| $83$ |
\( T^{3} + \cdots - 50\!\cdots\!36 \)
|
| $89$ |
\( T^{3} + \cdots - 17\!\cdots\!84 \)
|
| $97$ |
\( T^{3} + \cdots + 18\!\cdots\!36 \)
|
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