Newspace parameters
Level: | \( N \) | = | \( 36 = 2^{2} \cdot 3^{2} \) |
Weight: | \( k \) | = | \( 2 \) |
Character orbit: | \([\chi]\) | = | 36.h (of order \(6\) and degree \(2\)) |
Newform invariants
Self dual: | No |
Analytic conductor: | \(0.287461447277\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 8.0.170772624.1 |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
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 \(x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 6 x^{4} - 12 x^{3} + 20 x^{2} - 24 x + 16\):
\(\beta_{0}\) | \(=\) | \( 1 \) |
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \((\)\( \nu^{7} - \nu^{6} - \nu^{5} - 2 \nu^{3} - 4 \nu^{2} + 4 \nu + 8 \)\()/8\) |
\(\beta_{3}\) | \(=\) | \((\)\( -\nu^{6} + \nu^{5} - \nu^{4} + 2 \nu^{3} + 4 \nu - 4 \)\()/4\) |
\(\beta_{4}\) | \(=\) | \((\)\( \nu^{7} - 3 \nu^{6} + 5 \nu^{5} - 2 \nu^{4} + 2 \nu^{3} - 8 \nu^{2} + 20 \nu - 24 \)\()/8\) |
\(\beta_{5}\) | \(=\) | \((\)\( -3 \nu^{7} + 3 \nu^{6} - 5 \nu^{5} + 4 \nu^{4} - 6 \nu^{3} + 16 \nu^{2} - 12 \nu + 16 \)\()/8\) |
\(\beta_{6}\) | \(=\) | \((\)\( \nu^{7} - 3 \nu^{6} + 5 \nu^{5} - 6 \nu^{4} + 6 \nu^{3} - 12 \nu^{2} + 20 \nu - 24 \)\()/8\) |
\(\beta_{7}\) | \(=\) | \((\)\( 3 \nu^{7} - 5 \nu^{6} + 7 \nu^{5} - 6 \nu^{4} + 10 \nu^{3} - 24 \nu^{2} + 28 \nu - 32 \)\()/8\) |
\(1\) | \(=\) | \(\beta_0\) |
\(\nu\) | \(=\) | \(\beta_{1}\) |
\(\nu^{2}\) | \(=\) | \(-\beta_{7} - \beta_{5} + \beta_{3} + \beta_{1} - 1\) |
\(\nu^{3}\) | \(=\) | \(\beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1}\) |
\(\nu^{4}\) | \(=\) | \(2 \beta_{7} - 3 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + 1\) |
\(\nu^{5}\) | \(=\) | \(-\beta_{7} - \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_{1} + 5\) |
\(\nu^{6}\) | \(=\) | \(-\beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4} - 3 \beta_{3} - 3 \beta_{2} + 5 \beta_{1}\) |
\(\nu^{7}\) | \(=\) | \(-4 \beta_{7} - \beta_{6} - 7 \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + 6 \beta_{1} - 7\) |
Character Values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).
\(n\) | \(19\) | \(29\) |
\(\chi(n)\) | \(-1\) | \(-\beta_{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 |
|
−1.41203 | − | 0.0786378i | 0.637910 | + | 1.61030i | 1.98763 | + | 0.222077i | 0.686141 | − | 0.396143i | −0.774115 | − | 2.32395i | −2.35143 | − | 1.35760i | −2.78912 | − | 0.469882i | −2.18614 | + | 2.05446i | −1.00000 | + | 0.505408i | ||||||||||||||||||||||||
11.2 | −0.774115 | − | 1.18353i | −0.637910 | − | 1.61030i | −0.801492 | + | 1.83238i | 0.686141 | − | 0.396143i | −1.41203 | + | 2.00155i | 2.35143 | + | 1.35760i | 2.78912 | − | 0.469882i | −2.18614 | + | 2.05446i | −1.00000 | − | 0.505408i | |||||||||||||||||||||||||
11.3 | −0.335728 | + | 1.37379i | 1.35760 | − | 1.07561i | −1.77457 | − | 0.922437i | −2.18614 | + | 1.26217i | 1.02187 | + | 2.22616i | −1.10489 | − | 0.637910i | 1.86301 | − | 2.12819i | 0.686141 | − | 2.92048i | −1.00000 | − | 3.42703i | |||||||||||||||||||||||||
11.4 | 1.02187 | − | 0.977642i | −1.35760 | + | 1.07561i | 0.0884324 | − | 1.99804i | −2.18614 | + | 1.26217i | −0.335728 | + | 2.42637i | 1.10489 | + | 0.637910i | −1.86301 | − | 2.12819i | 0.686141 | − | 2.92048i | −1.00000 | + | 3.42703i | |||||||||||||||||||||||||
23.1 | −1.41203 | + | 0.0786378i | 0.637910 | − | 1.61030i | 1.98763 | − | 0.222077i | 0.686141 | + | 0.396143i | −0.774115 | + | 2.32395i | −2.35143 | + | 1.35760i | −2.78912 | + | 0.469882i | −2.18614 | − | 2.05446i | −1.00000 | − | 0.505408i | |||||||||||||||||||||||||
23.2 | −0.774115 | + | 1.18353i | −0.637910 | + | 1.61030i | −0.801492 | − | 1.83238i | 0.686141 | + | 0.396143i | −1.41203 | − | 2.00155i | 2.35143 | − | 1.35760i | 2.78912 | + | 0.469882i | −2.18614 | − | 2.05446i | −1.00000 | + | 0.505408i | |||||||||||||||||||||||||
23.3 | −0.335728 | − | 1.37379i | 1.35760 | + | 1.07561i | −1.77457 | + | 0.922437i | −2.18614 | − | 1.26217i | 1.02187 | − | 2.22616i | −1.10489 | + | 0.637910i | 1.86301 | + | 2.12819i | 0.686141 | + | 2.92048i | −1.00000 | + | 3.42703i | |||||||||||||||||||||||||
23.4 | 1.02187 | + | 0.977642i | −1.35760 | − | 1.07561i | 0.0884324 | + | 1.99804i | −2.18614 | − | 1.26217i | −0.335728 | − | 2.42637i | 1.10489 | − | 0.637910i | −1.86301 | + | 2.12819i | 0.686141 | + | 2.92048i | −1.00000 | − | 3.42703i |
Inner twists
Char. orbit | Parity | Mult. | Self Twist | Proved |
---|---|---|---|---|
1.a | Even | 1 | trivial | yes |
4.b | Odd | 1 | yes | |
9.d | Odd | 1 | yes | |
36.h | Even | 1 | yes |
Hecke kernels
There are no other newforms in \(S_{2}^{\mathrm{new}}(36, [\chi])\).