Newspace parameters
Level: | \( N \) | \(=\) | \( 20 = 2^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 20.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(8.14757220122\) |
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 \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).
\(n\) | \(11\) | \(17\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 |
|
−16.0000 | 158.000 | 256.000 | 625.000 | −2528.00 | −1922.00 | −4096.00 | 18403.0 | −10000.0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
20.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-5}) \) |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 20.9.d.a | ✓ | 1 |
4.b | odd | 2 | 1 | 20.9.d.b | yes | 1 | |
5.b | even | 2 | 1 | 20.9.d.b | yes | 1 | |
5.c | odd | 4 | 2 | 100.9.b.b | 2 | ||
8.b | even | 2 | 1 | 320.9.h.a | 1 | ||
8.d | odd | 2 | 1 | 320.9.h.b | 1 | ||
20.d | odd | 2 | 1 | CM | 20.9.d.a | ✓ | 1 |
20.e | even | 4 | 2 | 100.9.b.b | 2 | ||
40.e | odd | 2 | 1 | 320.9.h.a | 1 | ||
40.f | even | 2 | 1 | 320.9.h.b | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
20.9.d.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
20.9.d.a | ✓ | 1 | 20.d | odd | 2 | 1 | CM |
20.9.d.b | yes | 1 | 4.b | odd | 2 | 1 | |
20.9.d.b | yes | 1 | 5.b | even | 2 | 1 | |
100.9.b.b | 2 | 5.c | odd | 4 | 2 | ||
100.9.b.b | 2 | 20.e | even | 4 | 2 | ||
320.9.h.a | 1 | 8.b | even | 2 | 1 | ||
320.9.h.a | 1 | 40.e | odd | 2 | 1 | ||
320.9.h.b | 1 | 8.d | odd | 2 | 1 | ||
320.9.h.b | 1 | 40.f | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 158 \)
acting on \(S_{9}^{\mathrm{new}}(20, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 16 \)
$3$
\( T - 158 \)
$5$
\( T - 625 \)
$7$
\( T + 1922 \)
$11$
\( T \)
$13$
\( T \)
$17$
\( T \)
$19$
\( T \)
$23$
\( T + 211202 \)
$29$
\( T - 20642 \)
$31$
\( T \)
$37$
\( T \)
$41$
\( T + 5419198 \)
$43$
\( T - 2519518 \)
$47$
\( T + 9618242 \)
$53$
\( T \)
$59$
\( T \)
$61$
\( T + 11061598 \)
$67$
\( T - 20249758 \)
$71$
\( T \)
$73$
\( T \)
$79$
\( T \)
$83$
\( T - 30884638 \)
$89$
\( T + 106804798 \)
$97$
\( T \)
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