Newspace parameters
Level: | \( N \) | \(=\) | \( 36 = 2^{2} \cdot 3^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 36.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.980928951697\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
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 |
|
−2.00000 | 0 | 4.00000 | 8.00000 | 0 | 0 | −8.00000 | 0 | −16.0000 | |||||||||||||||||||||
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 | 36.3.d.a | ✓ | 1 |
3.b | odd | 2 | 1 | 36.3.d.b | yes | 1 | |
4.b | odd | 2 | 1 | CM | 36.3.d.a | ✓ | 1 |
5.b | even | 2 | 1 | 900.3.c.c | 1 | ||
5.c | odd | 4 | 2 | 900.3.f.b | 2 | ||
8.b | even | 2 | 1 | 576.3.g.a | 1 | ||
8.d | odd | 2 | 1 | 576.3.g.a | 1 | ||
9.c | even | 3 | 2 | 324.3.f.f | 2 | ||
9.d | odd | 6 | 2 | 324.3.f.e | 2 | ||
12.b | even | 2 | 1 | 36.3.d.b | yes | 1 | |
15.d | odd | 2 | 1 | 900.3.c.b | 1 | ||
15.e | even | 4 | 2 | 900.3.f.a | 2 | ||
16.e | even | 4 | 2 | 2304.3.b.d | 2 | ||
16.f | odd | 4 | 2 | 2304.3.b.d | 2 | ||
20.d | odd | 2 | 1 | 900.3.c.c | 1 | ||
20.e | even | 4 | 2 | 900.3.f.b | 2 | ||
24.f | even | 2 | 1 | 576.3.g.c | 1 | ||
24.h | odd | 2 | 1 | 576.3.g.c | 1 | ||
36.f | odd | 6 | 2 | 324.3.f.f | 2 | ||
36.h | even | 6 | 2 | 324.3.f.e | 2 | ||
48.i | odd | 4 | 2 | 2304.3.b.e | 2 | ||
48.k | even | 4 | 2 | 2304.3.b.e | 2 | ||
60.h | even | 2 | 1 | 900.3.c.b | 1 | ||
60.l | odd | 4 | 2 | 900.3.f.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
36.3.d.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
36.3.d.a | ✓ | 1 | 4.b | odd | 2 | 1 | CM |
36.3.d.b | yes | 1 | 3.b | odd | 2 | 1 | |
36.3.d.b | yes | 1 | 12.b | even | 2 | 1 | |
324.3.f.e | 2 | 9.d | odd | 6 | 2 | ||
324.3.f.e | 2 | 36.h | even | 6 | 2 | ||
324.3.f.f | 2 | 9.c | even | 3 | 2 | ||
324.3.f.f | 2 | 36.f | odd | 6 | 2 | ||
576.3.g.a | 1 | 8.b | even | 2 | 1 | ||
576.3.g.a | 1 | 8.d | odd | 2 | 1 | ||
576.3.g.c | 1 | 24.f | even | 2 | 1 | ||
576.3.g.c | 1 | 24.h | odd | 2 | 1 | ||
900.3.c.b | 1 | 15.d | odd | 2 | 1 | ||
900.3.c.b | 1 | 60.h | even | 2 | 1 | ||
900.3.c.c | 1 | 5.b | even | 2 | 1 | ||
900.3.c.c | 1 | 20.d | odd | 2 | 1 | ||
900.3.f.a | 2 | 15.e | even | 4 | 2 | ||
900.3.f.a | 2 | 60.l | odd | 4 | 2 | ||
900.3.f.b | 2 | 5.c | odd | 4 | 2 | ||
900.3.f.b | 2 | 20.e | even | 4 | 2 | ||
2304.3.b.d | 2 | 16.e | even | 4 | 2 | ||
2304.3.b.d | 2 | 16.f | odd | 4 | 2 | ||
2304.3.b.e | 2 | 48.i | odd | 4 | 2 | ||
2304.3.b.e | 2 | 48.k | even | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 8 \)
acting on \(S_{3}^{\mathrm{new}}(36, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 2 \)
$3$
\( T \)
$5$
\( T - 8 \)
$7$
\( T \)
$11$
\( T \)
$13$
\( T + 10 \)
$17$
\( T + 16 \)
$19$
\( T \)
$23$
\( T \)
$29$
\( T + 40 \)
$31$
\( T \)
$37$
\( T + 70 \)
$41$
\( T - 80 \)
$43$
\( T \)
$47$
\( T \)
$53$
\( T - 56 \)
$59$
\( T \)
$61$
\( T + 22 \)
$67$
\( T \)
$71$
\( T \)
$73$
\( T - 110 \)
$79$
\( T \)
$83$
\( T \)
$89$
\( T + 160 \)
$97$
\( T + 130 \)
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