Newspace parameters
Level: | \( N \) | \(=\) | \( 36 = 2^{2} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 13 \) |
Character orbit: | \([\chi]\) | \(=\) | 36.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(32.9037774219\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 4) |
Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).
\(n\) | \(19\) | \(29\) |
\(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 |
|
64.0000 | 0 | 4096.00 | −23506.0 | 0 | 0 | 262144. | 0 | −1.50438e6 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 36.13.d.a | 1 | |
3.b | odd | 2 | 1 | 4.13.b.a | ✓ | 1 | |
4.b | odd | 2 | 1 | CM | 36.13.d.a | 1 | |
12.b | even | 2 | 1 | 4.13.b.a | ✓ | 1 | |
24.f | even | 2 | 1 | 64.13.c.a | 1 | ||
24.h | odd | 2 | 1 | 64.13.c.a | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4.13.b.a | ✓ | 1 | 3.b | odd | 2 | 1 | |
4.13.b.a | ✓ | 1 | 12.b | even | 2 | 1 | |
36.13.d.a | 1 | 1.a | even | 1 | 1 | trivial | |
36.13.d.a | 1 | 4.b | odd | 2 | 1 | CM | |
64.13.c.a | 1 | 24.f | even | 2 | 1 | ||
64.13.c.a | 1 | 24.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 23506 \)
acting on \(S_{13}^{\mathrm{new}}(36, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T - 64 \)
$3$
\( T \)
$5$
\( T + 23506 \)
$7$
\( T \)
$11$
\( T \)
$13$
\( T - 6911282 \)
$17$
\( T - 47295038 \)
$19$
\( T \)
$23$
\( T \)
$29$
\( T - 173439758 \)
$31$
\( T \)
$37$
\( T + 2050092718 \)
$41$
\( T - 2285065118 \)
$43$
\( T \)
$47$
\( T \)
$53$
\( T - 43462597358 \)
$59$
\( T \)
$61$
\( T + 47844884878 \)
$67$
\( T \)
$71$
\( T \)
$73$
\( T + 119852347678 \)
$79$
\( T \)
$83$
\( T \)
$89$
\( T + 907573615522 \)
$97$
\( T - 502341690242 \)
show more
show less