Newspace parameters
Level: | \( N \) | = | \( 65 = 5 \cdot 13 \) |
Weight: | \( k \) | = | \( 2 \) |
Character orbit: | \([\chi]\) | = | 65.l (of order \(6\) and degree \(2\)) |
Newform invariants
Self dual: | No |
Analytic conductor: | \(0.519027613138\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 8.0.49787136.1 |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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 \(x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16\):
\(\beta_{0}\) | \(=\) | \( 1 \) |
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \((\)\( -\nu^{6} + 5 \nu^{4} - 5 \nu^{2} - 12 \)\()/20\) |
\(\beta_{3}\) | \(=\) | \((\)\( -\nu^{7} + 5 \nu^{5} - 5 \nu^{3} - 12 \nu \)\()/40\) |
\(\beta_{4}\) | \(=\) | \((\)\( 3 \nu^{6} + 5 \nu^{4} + 15 \nu^{2} + 36 \)\()/20\) |
\(\beta_{5}\) | \(=\) | \((\)\( -3 \nu^{7} - 5 \nu^{5} + 5 \nu^{3} - 16 \nu \)\()/40\) |
\(\beta_{6}\) | \(=\) | \((\)\( -\nu^{6} - 7 \)\()/5\) |
\(\beta_{7}\) | \(=\) | \((\)\( \nu^{7} + 3 \nu^{5} + 5 \nu^{3} + 12 \nu \)\()/8\) |
\(1\) | \(=\) | \(\beta_0\) |
\(\nu\) | \(=\) | \(\beta_{1}\) |
\(\nu^{2}\) | \(=\) | \(\beta_{6} + \beta_{4} - \beta_{2} - 1\) |
\(\nu^{3}\) | \(=\) | \(\beta_{7} + 2 \beta_{5} - \beta_{3} - \beta_{1}\) |
\(\nu^{4}\) | \(=\) | \(\beta_{4} + 3 \beta_{2}\) |
\(\nu^{5}\) | \(=\) | \(\beta_{7} + 5 \beta_{3}\) |
\(\nu^{6}\) | \(=\) | \(-5 \beta_{6} - 7\) |
\(\nu^{7}\) | \(=\) | \(-10 \beta_{5} - 10 \beta_{3} - 7 \beta_{1}\) |
Character Values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).
\(n\) | \(27\) | \(41\) |
\(\chi(n)\) | \(-1\) | \(1 - \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 |
|
−1.09445 | + | 1.89564i | 0.866025 | + | 0.500000i | −1.39564 | − | 2.41733i | 0.456850 | + | 2.18890i | −1.89564 | + | 1.09445i | −0.866025 | − | 1.50000i | 1.73205 | −1.00000 | − | 1.73205i | −4.64938 | − | 1.52962i | ||||||||||||||||||||||||||
4.2 | −0.228425 | + | 0.395644i | −0.866025 | − | 0.500000i | 0.895644 | + | 1.55130i | 2.18890 | − | 0.456850i | 0.395644 | − | 0.228425i | 0.866025 | + | 1.50000i | −1.73205 | −1.00000 | − | 1.73205i | −0.319250 | + | 0.970381i | |||||||||||||||||||||||||||
4.3 | 0.228425 | − | 0.395644i | 0.866025 | + | 0.500000i | 0.895644 | + | 1.55130i | −2.18890 | − | 0.456850i | 0.395644 | − | 0.228425i | −0.866025 | − | 1.50000i | 1.73205 | −1.00000 | − | 1.73205i | −0.680750 | + | 0.761669i | |||||||||||||||||||||||||||
4.4 | 1.09445 | − | 1.89564i | −0.866025 | − | 0.500000i | −1.39564 | − | 2.41733i | −0.456850 | + | 2.18890i | −1.89564 | + | 1.09445i | 0.866025 | + | 1.50000i | −1.73205 | −1.00000 | − | 1.73205i | 3.64938 | + | 3.26167i | |||||||||||||||||||||||||||
49.1 | −1.09445 | − | 1.89564i | 0.866025 | − | 0.500000i | −1.39564 | + | 2.41733i | 0.456850 | − | 2.18890i | −1.89564 | − | 1.09445i | −0.866025 | + | 1.50000i | 1.73205 | −1.00000 | + | 1.73205i | −4.64938 | + | 1.52962i | |||||||||||||||||||||||||||
49.2 | −0.228425 | − | 0.395644i | −0.866025 | + | 0.500000i | 0.895644 | − | 1.55130i | 2.18890 | + | 0.456850i | 0.395644 | + | 0.228425i | 0.866025 | − | 1.50000i | −1.73205 | −1.00000 | + | 1.73205i | −0.319250 | − | 0.970381i | |||||||||||||||||||||||||||
49.3 | 0.228425 | + | 0.395644i | 0.866025 | − | 0.500000i | 0.895644 | − | 1.55130i | −2.18890 | + | 0.456850i | 0.395644 | + | 0.228425i | −0.866025 | + | 1.50000i | 1.73205 | −1.00000 | + | 1.73205i | −0.680750 | − | 0.761669i | |||||||||||||||||||||||||||
49.4 | 1.09445 | + | 1.89564i | −0.866025 | + | 0.500000i | −1.39564 | + | 2.41733i | −0.456850 | − | 2.18890i | −1.89564 | − | 1.09445i | 0.866025 | − | 1.50000i | −1.73205 | −1.00000 | + | 1.73205i | 3.64938 | − | 3.26167i |
Inner twists
Char. orbit | Parity | Mult. | Self Twist | Proved |
---|---|---|---|---|
1.a | Even | 1 | trivial | yes |
5.b | Even | 1 | yes | |
13.e | Even | 1 | yes | |
65.l | Even | 1 | yes |
Hecke kernels
There are no other newforms in \(S_{2}^{\mathrm{new}}(65, [\chi])\).