Newspace parameters
| Level: | \( N \) | \(=\) | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 336.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(104.961368563\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.2910828.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - 303x - 490 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 21) |
| 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
\( x^{3} - 303x - 490 \)
:
| \(\beta_{1}\) | \(=\) |
\( 16\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} - 4\nu - 404 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 16 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{2} + \beta _1 + 1616 ) / 8 \)
|
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 | −27.0000 | 0 | −511.401 | 0 | −343.000 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −27.0000 | 0 | 69.3548 | 0 | −343.000 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | −27.0000 | 0 | 328.047 | 0 | −343.000 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(7\) | \( +1 \) |
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 | 336.8.a.r | 3 | |
| 4.b | odd | 2 | 1 | 21.8.a.d | ✓ | 3 | |
| 12.b | even | 2 | 1 | 63.8.a.g | 3 | ||
| 20.d | odd | 2 | 1 | 525.8.a.g | 3 | ||
| 28.d | even | 2 | 1 | 147.8.a.e | 3 | ||
| 28.f | even | 6 | 2 | 147.8.e.j | 6 | ||
| 28.g | odd | 6 | 2 | 147.8.e.i | 6 | ||
| 84.h | odd | 2 | 1 | 441.8.a.n | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.8.a.d | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 63.8.a.g | 3 | 12.b | even | 2 | 1 | ||
| 147.8.a.e | 3 | 28.d | even | 2 | 1 | ||
| 147.8.e.i | 6 | 28.g | odd | 6 | 2 | ||
| 147.8.e.j | 6 | 28.f | even | 6 | 2 | ||
| 336.8.a.r | 3 | 1.a | even | 1 | 1 | trivial | |
| 441.8.a.n | 3 | 84.h | odd | 2 | 1 | ||
| 525.8.a.g | 3 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 114T_{5}^{2} - 180480T_{5} + 11635200 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(336))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T + 27)^{3} \)
|
| $5$ |
\( T^{3} + 114 T^{2} + \cdots + 11635200 \)
|
| $7$ |
\( (T + 343)^{3} \)
|
| $11$ |
\( T^{3} + \cdots - 21104274672 \)
|
| $13$ |
\( T^{3} + \cdots + 755769903784 \)
|
| $17$ |
\( T^{3} + \cdots + 12467114860032 \)
|
| $19$ |
\( T^{3} + \cdots + 5437419408320 \)
|
| $23$ |
\( T^{3} + \cdots + 102906745004448 \)
|
| $29$ |
\( T^{3} + \cdots + 208394349111720 \)
|
| $31$ |
\( T^{3} + \cdots - 24\!\cdots\!64 \)
|
| $37$ |
\( T^{3} + \cdots + 14\!\cdots\!08 \)
|
| $41$ |
\( T^{3} + \cdots + 24\!\cdots\!00 \)
|
| $43$ |
\( T^{3} + \cdots + 30\!\cdots\!36 \)
|
| $47$ |
\( T^{3} + \cdots - 22\!\cdots\!84 \)
|
| $53$ |
\( T^{3} + \cdots + 56\!\cdots\!32 \)
|
| $59$ |
\( T^{3} + \cdots + 31\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 37\!\cdots\!52 \)
|
| $67$ |
\( T^{3} + \cdots + 42\!\cdots\!36 \)
|
| $71$ |
\( T^{3} + \cdots + 56\!\cdots\!08 \)
|
| $73$ |
\( T^{3} + \cdots - 28\!\cdots\!52 \)
|
| $79$ |
\( T^{3} + \cdots - 35\!\cdots\!60 \)
|
| $83$ |
\( T^{3} + \cdots - 79\!\cdots\!52 \)
|
| $89$ |
\( T^{3} + \cdots + 26\!\cdots\!20 \)
|
| $97$ |
\( T^{3} + \cdots + 70\!\cdots\!88 \)
|
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