Newspace parameters
Level: | \( N \) | \(=\) | \( 36 = 2^{2} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 36.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(18.5412901019\) |
Analytic rank: | \(1\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 4) |
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 | 666.000 | 0 | −6328.00 | 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 | 36.10.a.b | 1 | |
3.b | odd | 2 | 1 | 4.10.a.a | ✓ | 1 | |
4.b | odd | 2 | 1 | 144.10.a.j | 1 | ||
9.c | even | 3 | 2 | 324.10.e.b | 2 | ||
9.d | odd | 6 | 2 | 324.10.e.e | 2 | ||
12.b | even | 2 | 1 | 16.10.a.a | 1 | ||
15.d | odd | 2 | 1 | 100.10.a.a | 1 | ||
15.e | even | 4 | 2 | 100.10.c.a | 2 | ||
21.c | even | 2 | 1 | 196.10.a.a | 1 | ||
21.g | even | 6 | 2 | 196.10.e.b | 2 | ||
21.h | odd | 6 | 2 | 196.10.e.a | 2 | ||
24.f | even | 2 | 1 | 64.10.a.i | 1 | ||
24.h | odd | 2 | 1 | 64.10.a.a | 1 | ||
48.i | odd | 4 | 2 | 256.10.b.j | 2 | ||
48.k | even | 4 | 2 | 256.10.b.b | 2 | ||
60.h | even | 2 | 1 | 400.10.a.k | 1 | ||
60.l | odd | 4 | 2 | 400.10.c.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4.10.a.a | ✓ | 1 | 3.b | odd | 2 | 1 | |
16.10.a.a | 1 | 12.b | even | 2 | 1 | ||
36.10.a.b | 1 | 1.a | even | 1 | 1 | trivial | |
64.10.a.a | 1 | 24.h | odd | 2 | 1 | ||
64.10.a.i | 1 | 24.f | even | 2 | 1 | ||
100.10.a.a | 1 | 15.d | odd | 2 | 1 | ||
100.10.c.a | 2 | 15.e | even | 4 | 2 | ||
144.10.a.j | 1 | 4.b | odd | 2 | 1 | ||
196.10.a.a | 1 | 21.c | even | 2 | 1 | ||
196.10.e.a | 2 | 21.h | odd | 6 | 2 | ||
196.10.e.b | 2 | 21.g | even | 6 | 2 | ||
256.10.b.b | 2 | 48.k | even | 4 | 2 | ||
256.10.b.j | 2 | 48.i | odd | 4 | 2 | ||
324.10.e.b | 2 | 9.c | even | 3 | 2 | ||
324.10.e.e | 2 | 9.d | odd | 6 | 2 | ||
400.10.a.k | 1 | 60.h | even | 2 | 1 | ||
400.10.c.a | 2 | 60.l | odd | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 666 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(36))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T \)
$5$
\( T - 666 \)
$7$
\( T + 6328 \)
$11$
\( T - 30420 \)
$13$
\( T + 32338 \)
$17$
\( T + 590994 \)
$19$
\( T - 34676 \)
$23$
\( T + 1048536 \)
$29$
\( T + 4409406 \)
$31$
\( T + 7401184 \)
$37$
\( T - 10234502 \)
$41$
\( T + 18352746 \)
$43$
\( T + 252340 \)
$47$
\( T - 49517136 \)
$53$
\( T - 66396906 \)
$59$
\( T - 61523748 \)
$61$
\( T - 35638622 \)
$67$
\( T - 181742372 \)
$71$
\( T + 90904968 \)
$73$
\( T + 262978678 \)
$79$
\( T + 116502832 \)
$83$
\( T - 9563724 \)
$89$
\( T + 611826714 \)
$97$
\( T + 259312798 \)
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