Newspace parameters
| Level: | \( N \) | \(=\) | \( 36 = 2^{2} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 36.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(100.611843943\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 12) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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 | 0 | 0 | 1.12681e7 | 0 | 2.81914e8 | 0 | 0 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
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 | 36.22.a.a | 1 | |
| 3.b | odd | 2 | 1 | 12.22.a.a | ✓ | 1 | |
| 12.b | even | 2 | 1 | 48.22.a.b | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 12.22.a.a | ✓ | 1 | 3.b | odd | 2 | 1 | |
| 36.22.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
| 48.22.a.b | 1 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 11268090 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(36))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T \)
$5$
\( T - 11268090 \)
$7$
\( T - 281914136 \)
$11$
\( T - 36172082484 \)
$13$
\( T + 449098578370 \)
$17$
\( T + 2121858786546 \)
$19$
\( T + 4609406233900 \)
$23$
\( T + 95095276921656 \)
$29$
\( T - 2245742383351266 \)
$31$
\( T + 3155693201792656 \)
$37$
\( T + 18\!\cdots\!82 \)
$41$
\( T - 16\!\cdots\!10 \)
$43$
\( T + 15\!\cdots\!44 \)
$47$
\( T - 13\!\cdots\!36 \)
$53$
\( T - 15\!\cdots\!38 \)
$59$
\( T + 29\!\cdots\!84 \)
$61$
\( T - 36\!\cdots\!02 \)
$67$
\( T - 21\!\cdots\!04 \)
$71$
\( T + 21\!\cdots\!60 \)
$73$
\( T + 17\!\cdots\!22 \)
$79$
\( T + 11\!\cdots\!52 \)
$83$
\( T - 96\!\cdots\!44 \)
$89$
\( T + 60\!\cdots\!50 \)
$97$
\( T + 40\!\cdots\!98 \)
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