Newspace parameters
| Level: | \( N \) | \(=\) | \( 304 = 2^{4} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 304.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(48.7566812231\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.272193.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 74x + 168 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 76) |
| 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
\( x^{3} - x^{2} - 74x + 168 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + \nu - 50 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{2} - \beta _1 + 99 ) / 2 \)
|
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 | −14.2417 | 0 | −11.7413 | 0 | 252.153 | 0 | −40.1733 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −4.17335 | 0 | −47.6772 | 0 | −121.691 | 0 | −225.583 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 26.4151 | 0 | 50.4185 | 0 | −117.462 | 0 | 454.756 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(19\) | \( +1 \) |
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 | 304.6.a.j | 3 | |
| 4.b | odd | 2 | 1 | 76.6.a.a | ✓ | 3 | |
| 12.b | even | 2 | 1 | 684.6.a.b | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 76.6.a.a | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 304.6.a.j | 3 | 1.a | even | 1 | 1 | trivial | |
| 684.6.a.b | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 8T_{3}^{2} - 427T_{3} - 1570 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(304))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 8 T^{2} + \cdots - 1570 \)
|
| $5$ |
\( T^{3} + 9 T^{2} + \cdots - 28224 \)
|
| $7$ |
\( T^{3} - 13 T^{2} + \cdots - 3604283 \)
|
| $11$ |
\( T^{3} - 1229 T^{2} + \cdots - 31125372 \)
|
| $13$ |
\( T^{3} + 56 T^{2} + \cdots + 72234732 \)
|
| $17$ |
\( T^{3} + 3149 T^{2} + \cdots + 280662027 \)
|
| $19$ |
\( (T + 361)^{3} \)
|
| $23$ |
\( T^{3} + \cdots + 3000249264 \)
|
| $29$ |
\( T^{3} + \cdots + 128846557278 \)
|
| $31$ |
\( T^{3} + \cdots - 16959632256 \)
|
| $37$ |
\( T^{3} + \cdots - 172285751608 \)
|
| $41$ |
\( T^{3} + \cdots - 421643137536 \)
|
| $43$ |
\( T^{3} + \cdots + 1399322853024 \)
|
| $47$ |
\( T^{3} + \cdots + 1755590562048 \)
|
| $53$ |
\( T^{3} + \cdots + 572042936316 \)
|
| $59$ |
\( T^{3} + \cdots - 18472105095894 \)
|
| $61$ |
\( T^{3} + \cdots + 7996188319084 \)
|
| $67$ |
\( T^{3} + \cdots - 4818723981540 \)
|
| $71$ |
\( T^{3} + \cdots + 25351387808040 \)
|
| $73$ |
\( T^{3} + \cdots + 14324767129415 \)
|
| $79$ |
\( T^{3} + \cdots + 27708927088096 \)
|
| $83$ |
\( T^{3} + \cdots - 318522133126128 \)
|
| $89$ |
\( T^{3} + \cdots + 57207326543232 \)
|
| $97$ |
\( T^{3} + \cdots + 7731995443552 \)
|
| show more | |
| show less |