Newspace parameters
Level: | \( N \) | \(=\) | \( 16 = 2^{4} \) |
Weight: | \( k \) | \(=\) | \( 16 \) |
Character orbit: | \([\chi]\) | \(=\) | 16.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(22.8309608160\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 8) |
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 | −2700.00 | 0 | −251890. | 0 | −1.37407e6 | 0 | −7.05891e6 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
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 | 16.16.a.b | 1 | |
3.b | odd | 2 | 1 | 144.16.a.m | 1 | ||
4.b | odd | 2 | 1 | 8.16.a.b | ✓ | 1 | |
8.b | even | 2 | 1 | 64.16.a.h | 1 | ||
8.d | odd | 2 | 1 | 64.16.a.d | 1 | ||
12.b | even | 2 | 1 | 72.16.a.c | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8.16.a.b | ✓ | 1 | 4.b | odd | 2 | 1 | |
16.16.a.b | 1 | 1.a | even | 1 | 1 | trivial | |
64.16.a.d | 1 | 8.d | odd | 2 | 1 | ||
64.16.a.h | 1 | 8.b | even | 2 | 1 | ||
72.16.a.c | 1 | 12.b | even | 2 | 1 | ||
144.16.a.m | 1 | 3.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} + 2700 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(16))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T + 2700 \)
$5$
\( T + 251890 \)
$7$
\( T + 1374072 \)
$11$
\( T - 43286716 \)
$13$
\( T + 323161466 \)
$17$
\( T + 191653646 \)
$19$
\( T - 6515456644 \)
$23$
\( T + 23880801512 \)
$29$
\( T - 176820596982 \)
$31$
\( T - 152007193888 \)
$37$
\( T - 21581233902 \)
$41$
\( T + 245334499686 \)
$43$
\( T + 2769961534756 \)
$47$
\( T + 2811771943248 \)
$53$
\( T + 3491413730722 \)
$59$
\( T - 15827800893676 \)
$61$
\( T + 24609047974442 \)
$67$
\( T - 20706233653684 \)
$71$
\( T - 719982528200 \)
$73$
\( T - 29883036220282 \)
$79$
\( T - 148100908648400 \)
$83$
\( T - 302806756982468 \)
$89$
\( T + 496150966996374 \)
$97$
\( T - 309183128990882 \)
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