Newspace parameters
Level: | \( N \) | \(=\) | \( 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 19.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.517712502285\) |
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 \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
18.1 |
|
0 | 0 | 4.00000 | −9.00000 | 0 | −5.00000 | 0 | 9.00000 | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-19}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 19.3.b.a | ✓ | 1 |
3.b | odd | 2 | 1 | 171.3.c.a | 1 | ||
4.b | odd | 2 | 1 | 304.3.e.a | 1 | ||
5.b | even | 2 | 1 | 475.3.c.a | 1 | ||
5.c | odd | 4 | 2 | 475.3.d.a | 2 | ||
8.b | even | 2 | 1 | 1216.3.e.a | 1 | ||
8.d | odd | 2 | 1 | 1216.3.e.b | 1 | ||
12.b | even | 2 | 1 | 2736.3.o.a | 1 | ||
19.b | odd | 2 | 1 | CM | 19.3.b.a | ✓ | 1 |
19.c | even | 3 | 2 | 361.3.d.a | 2 | ||
19.d | odd | 6 | 2 | 361.3.d.a | 2 | ||
19.e | even | 9 | 6 | 361.3.f.a | 6 | ||
19.f | odd | 18 | 6 | 361.3.f.a | 6 | ||
57.d | even | 2 | 1 | 171.3.c.a | 1 | ||
76.d | even | 2 | 1 | 304.3.e.a | 1 | ||
95.d | odd | 2 | 1 | 475.3.c.a | 1 | ||
95.g | even | 4 | 2 | 475.3.d.a | 2 | ||
152.b | even | 2 | 1 | 1216.3.e.b | 1 | ||
152.g | odd | 2 | 1 | 1216.3.e.a | 1 | ||
228.b | odd | 2 | 1 | 2736.3.o.a | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
19.3.b.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
19.3.b.a | ✓ | 1 | 19.b | odd | 2 | 1 | CM |
171.3.c.a | 1 | 3.b | odd | 2 | 1 | ||
171.3.c.a | 1 | 57.d | even | 2 | 1 | ||
304.3.e.a | 1 | 4.b | odd | 2 | 1 | ||
304.3.e.a | 1 | 76.d | even | 2 | 1 | ||
361.3.d.a | 2 | 19.c | even | 3 | 2 | ||
361.3.d.a | 2 | 19.d | odd | 6 | 2 | ||
361.3.f.a | 6 | 19.e | even | 9 | 6 | ||
361.3.f.a | 6 | 19.f | odd | 18 | 6 | ||
475.3.c.a | 1 | 5.b | even | 2 | 1 | ||
475.3.c.a | 1 | 95.d | odd | 2 | 1 | ||
475.3.d.a | 2 | 5.c | odd | 4 | 2 | ||
475.3.d.a | 2 | 95.g | even | 4 | 2 | ||
1216.3.e.a | 1 | 8.b | even | 2 | 1 | ||
1216.3.e.a | 1 | 152.g | odd | 2 | 1 | ||
1216.3.e.b | 1 | 8.d | odd | 2 | 1 | ||
1216.3.e.b | 1 | 152.b | even | 2 | 1 | ||
2736.3.o.a | 1 | 12.b | even | 2 | 1 | ||
2736.3.o.a | 1 | 228.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{3}^{\mathrm{new}}(19, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T \)
$5$
\( T + 9 \)
$7$
\( T + 5 \)
$11$
\( T - 3 \)
$13$
\( T \)
$17$
\( T - 15 \)
$19$
\( T + 19 \)
$23$
\( T + 30 \)
$29$
\( T \)
$31$
\( T \)
$37$
\( T \)
$41$
\( T \)
$43$
\( T + 85 \)
$47$
\( T - 75 \)
$53$
\( T \)
$59$
\( T \)
$61$
\( T - 103 \)
$67$
\( T \)
$71$
\( T \)
$73$
\( T + 25 \)
$79$
\( T \)
$83$
\( T - 90 \)
$89$
\( T \)
$97$
\( T \)
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