Newspace parameters
| Level: | \( N \) | \(=\) | \( 272 = 2^{4} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 272.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(43.6243989891\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1505580.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 200x - 480 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 34) |
| 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
\( x^{3} - x^{2} - 200x - 480 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 132 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 8\beta_{2} + \beta _1 + 265 ) / 2 \)
|
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 | −25.6466 | 0 | 30.0934 | 0 | 245.353 | 0 | 414.747 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −8.14993 | 0 | 9.33483 | 0 | −162.407 | 0 | −176.579 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 29.7965 | 0 | 104.572 | 0 | −64.9455 | 0 | 644.832 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(17\) | \( +1 \) |
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 | 272.6.a.i | 3 | |
| 4.b | odd | 2 | 1 | 34.6.a.d | ✓ | 3 | |
| 8.b | even | 2 | 1 | 1088.6.a.r | 3 | ||
| 8.d | odd | 2 | 1 | 1088.6.a.q | 3 | ||
| 12.b | even | 2 | 1 | 306.6.a.o | 3 | ||
| 20.d | odd | 2 | 1 | 850.6.a.g | 3 | ||
| 68.d | odd | 2 | 1 | 578.6.a.d | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 34.6.a.d | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 272.6.a.i | 3 | 1.a | even | 1 | 1 | trivial | |
| 306.6.a.o | 3 | 12.b | even | 2 | 1 | ||
| 578.6.a.d | 3 | 68.d | odd | 2 | 1 | ||
| 850.6.a.g | 3 | 20.d | odd | 2 | 1 | ||
| 1088.6.a.q | 3 | 8.d | odd | 2 | 1 | ||
| 1088.6.a.r | 3 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 4T_{3}^{2} - 798T_{3} - 6228 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(272))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 4 T^{2} + \cdots - 6228 \)
|
| $5$ |
\( T^{3} - 144 T^{2} + \cdots - 29376 \)
|
| $7$ |
\( T^{3} - 18 T^{2} + \cdots - 2587888 \)
|
| $11$ |
\( T^{3} - 180 T^{2} + \cdots + 1592028 \)
|
| $13$ |
\( T^{3} - 318 T^{2} + \cdots - 1444016 \)
|
| $17$ |
\( (T + 289)^{3} \)
|
| $19$ |
\( T^{3} + \cdots - 5820446752 \)
|
| $23$ |
\( T^{3} + \cdots - 1848978072 \)
|
| $29$ |
\( T^{3} + \cdots + 36826311744 \)
|
| $31$ |
\( T^{3} + \cdots - 3248846848 \)
|
| $37$ |
\( T^{3} + \cdots - 261944912624 \)
|
| $41$ |
\( T^{3} + \cdots + 922961585448 \)
|
| $43$ |
\( T^{3} + \cdots + 370663566656 \)
|
| $47$ |
\( T^{3} + \cdots - 711662408064 \)
|
| $53$ |
\( T^{3} + \cdots + 6151268662200 \)
|
| $59$ |
\( T^{3} + \cdots + 3293925494400 \)
|
| $61$ |
\( T^{3} + \cdots - 1449557264240 \)
|
| $67$ |
\( T^{3} + \cdots + 89837382361472 \)
|
| $71$ |
\( T^{3} + \cdots - 67543901263656 \)
|
| $73$ |
\( T^{3} + \cdots + 74998640041720 \)
|
| $79$ |
\( T^{3} + \cdots + 50826829505720 \)
|
| $83$ |
\( T^{3} + \cdots - 182533520848896 \)
|
| $89$ |
\( T^{3} + \cdots - 735405794207856 \)
|
| $97$ |
\( T^{3} + \cdots + 68821299416296 \)
|
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