Newspace parameters
Level: | \( N \) | \(=\) | \( 16 = 2^{4} \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 16.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(8.24057337862\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 2) |
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 | 156.000 | 0 | 870.000 | 0 | 952.000 | 0 | 4653.00 | 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 | 16.10.a.d | 1 | |
3.b | odd | 2 | 1 | 144.10.a.d | 1 | ||
4.b | odd | 2 | 1 | 2.10.a.a | ✓ | 1 | |
5.b | even | 2 | 1 | 400.10.a.b | 1 | ||
5.c | odd | 4 | 2 | 400.10.c.d | 2 | ||
8.b | even | 2 | 1 | 64.10.a.b | 1 | ||
8.d | odd | 2 | 1 | 64.10.a.h | 1 | ||
12.b | even | 2 | 1 | 18.10.a.a | 1 | ||
16.e | even | 4 | 2 | 256.10.b.e | 2 | ||
16.f | odd | 4 | 2 | 256.10.b.g | 2 | ||
20.d | odd | 2 | 1 | 50.10.a.c | 1 | ||
20.e | even | 4 | 2 | 50.10.b.a | 2 | ||
28.d | even | 2 | 1 | 98.10.a.c | 1 | ||
28.f | even | 6 | 2 | 98.10.c.b | 2 | ||
28.g | odd | 6 | 2 | 98.10.c.c | 2 | ||
36.f | odd | 6 | 2 | 162.10.c.b | 2 | ||
36.h | even | 6 | 2 | 162.10.c.i | 2 | ||
44.c | even | 2 | 1 | 242.10.a.a | 1 | ||
52.b | odd | 2 | 1 | 338.10.a.a | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2.10.a.a | ✓ | 1 | 4.b | odd | 2 | 1 | |
16.10.a.d | 1 | 1.a | even | 1 | 1 | trivial | |
18.10.a.a | 1 | 12.b | even | 2 | 1 | ||
50.10.a.c | 1 | 20.d | odd | 2 | 1 | ||
50.10.b.a | 2 | 20.e | even | 4 | 2 | ||
64.10.a.b | 1 | 8.b | even | 2 | 1 | ||
64.10.a.h | 1 | 8.d | odd | 2 | 1 | ||
98.10.a.c | 1 | 28.d | even | 2 | 1 | ||
98.10.c.b | 2 | 28.f | even | 6 | 2 | ||
98.10.c.c | 2 | 28.g | odd | 6 | 2 | ||
144.10.a.d | 1 | 3.b | odd | 2 | 1 | ||
162.10.c.b | 2 | 36.f | odd | 6 | 2 | ||
162.10.c.i | 2 | 36.h | even | 6 | 2 | ||
242.10.a.a | 1 | 44.c | even | 2 | 1 | ||
256.10.b.e | 2 | 16.e | even | 4 | 2 | ||
256.10.b.g | 2 | 16.f | odd | 4 | 2 | ||
338.10.a.a | 1 | 52.b | odd | 2 | 1 | ||
400.10.a.b | 1 | 5.b | even | 2 | 1 | ||
400.10.c.d | 2 | 5.c | odd | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 156 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(16))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 156 \)
$5$
\( T - 870 \)
$7$
\( T - 952 \)
$11$
\( T - 56148 \)
$13$
\( T - 178094 \)
$17$
\( T + 247662 \)
$19$
\( T + 315380 \)
$23$
\( T + 204504 \)
$29$
\( T + 3840450 \)
$31$
\( T - 1309408 \)
$37$
\( T - 4307078 \)
$41$
\( T - 1512042 \)
$43$
\( T + 33670604 \)
$47$
\( T - 10581072 \)
$53$
\( T - 16616214 \)
$59$
\( T + 112235100 \)
$61$
\( T + 33197218 \)
$67$
\( T - 121372252 \)
$71$
\( T - 387172728 \)
$73$
\( T - 255240074 \)
$79$
\( T + 492101840 \)
$83$
\( T - 457420236 \)
$89$
\( T + 31809510 \)
$97$
\( T + 673532062 \)
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