Newspace parameters
Level: | \( N \) | = | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | = | \( 2 \) |
Character orbit: | \([\chi]\) | = | 336.s (of order \(4\) and degree \(2\)) |
Newform invariants
Self dual: | No |
Analytic conductor: | \(2.68297350792\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(i)\) |
Coefficient field: | \(\Q(\zeta_{8})\) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2 \) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring:
\(\beta_{0}\) | \(=\) | \( 1 \) |
\(\beta_{1}\) | \(=\) | \( \zeta_{8}^{2} \) |
\(\beta_{2}\) | \(=\) | \( \zeta_{8}^{3} + \zeta_{8} \) |
\(\beta_{3}\) | \(=\) | \( -\zeta_{8}^{3} + \zeta_{8} \) |
\(1\) | \(=\) | \(\beta_0\) |
\(\zeta_{8}\) | \(=\) | \((\)\(\beta_{3} + \beta_{2}\)\()/2\) |
\(\zeta_{8}^{2}\) | \(=\) | \(\beta_{1}\) |
\(\zeta_{8}^{3}\) | \(=\) | \((\)\(-\beta_{3} + \beta_{2}\)\()/2\) |
Character Values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).
\(n\) | \(85\) | \(113\) | \(127\) | \(241\) |
\(\chi(n)\) | \(\beta_{1}\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
155.1 |
|
1.00000 | + | 1.00000i | −1.41421 | − | 1.00000i | 2.00000i | −0.414214 | + | 0.414214i | −0.414214 | − | 2.41421i | −1.00000 | −2.00000 | + | 2.00000i | 1.00000 | + | 2.82843i | −0.828427 | ||||||||||||||||||
155.2 | 1.00000 | + | 1.00000i | 1.41421 | − | 1.00000i | 2.00000i | 2.41421 | − | 2.41421i | 2.41421 | + | 0.414214i | −1.00000 | −2.00000 | + | 2.00000i | 1.00000 | − | 2.82843i | 4.82843 | |||||||||||||||||||
323.1 | 1.00000 | − | 1.00000i | −1.41421 | + | 1.00000i | − | 2.00000i | −0.414214 | − | 0.414214i | −0.414214 | + | 2.41421i | −1.00000 | −2.00000 | − | 2.00000i | 1.00000 | − | 2.82843i | −0.828427 | ||||||||||||||||||
323.2 | 1.00000 | − | 1.00000i | 1.41421 | + | 1.00000i | − | 2.00000i | 2.41421 | + | 2.41421i | 2.41421 | − | 0.414214i | −1.00000 | −2.00000 | − | 2.00000i | 1.00000 | + | 2.82843i | 4.82843 |
Inner twists
Char. orbit | Parity | Mult. | Self Twist | Proved |
---|---|---|---|---|
1.a | Even | 1 | trivial | yes |
48.k | Even | 1 | yes |
Hecke kernels
This newform can be constructed as the kernel of the linear operator \( T_{5}^{4} - 4 T_{5}^{3} + 8 T_{5}^{2} + 8 T_{5} + 4 \) acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\).