Newspace parameters
Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 16 \) |
Character orbit: | \([\chi]\) | \(=\) | 50.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(71.3467525500\) |
Analytic rank: | \(1\) |
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 |
|
128.000 | −6252.00 | 16384.0 | 0 | −800256. | −56.0000 | 2.09715e6 | 2.47386e7 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(5\) | \(1\) |
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 | 50.16.a.b | 1 | |
5.b | even | 2 | 1 | 2.16.a.a | ✓ | 1 | |
5.c | odd | 4 | 2 | 50.16.b.a | 2 | ||
15.d | odd | 2 | 1 | 18.16.a.e | 1 | ||
20.d | odd | 2 | 1 | 16.16.a.a | 1 | ||
35.c | odd | 2 | 1 | 98.16.a.a | 1 | ||
35.i | odd | 6 | 2 | 98.16.c.d | 2 | ||
35.j | even | 6 | 2 | 98.16.c.a | 2 | ||
40.e | odd | 2 | 1 | 64.16.a.k | 1 | ||
40.f | even | 2 | 1 | 64.16.a.a | 1 | ||
60.h | even | 2 | 1 | 144.16.a.d | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2.16.a.a | ✓ | 1 | 5.b | even | 2 | 1 | |
16.16.a.a | 1 | 20.d | odd | 2 | 1 | ||
18.16.a.e | 1 | 15.d | odd | 2 | 1 | ||
50.16.a.b | 1 | 1.a | even | 1 | 1 | trivial | |
50.16.b.a | 2 | 5.c | odd | 4 | 2 | ||
64.16.a.a | 1 | 40.f | even | 2 | 1 | ||
64.16.a.k | 1 | 40.e | odd | 2 | 1 | ||
98.16.a.a | 1 | 35.c | odd | 2 | 1 | ||
98.16.c.a | 2 | 35.j | even | 6 | 2 | ||
98.16.c.d | 2 | 35.i | odd | 6 | 2 | ||
144.16.a.d | 1 | 60.h | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} + 6252 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(50))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T - 128 \)
$3$
\( T + 6252 \)
$5$
\( T \)
$7$
\( T + 56 \)
$11$
\( T + 95889948 \)
$13$
\( T - 59782138 \)
$17$
\( T - 1355814414 \)
$19$
\( T - 3783593180 \)
$23$
\( T - 11608845528 \)
$29$
\( T + 28959105930 \)
$31$
\( T - 253685353952 \)
$37$
\( T + 817641294446 \)
$41$
\( T + 682333284198 \)
$43$
\( T + 366945604292 \)
$47$
\( T + 695741581776 \)
$53$
\( T + 12993372468702 \)
$59$
\( T - 9209035340340 \)
$61$
\( T + 42338641200298 \)
$67$
\( T + 30029787950636 \)
$71$
\( T - 115328696975352 \)
$73$
\( T + 43787346432122 \)
$79$
\( T - 79603813043120 \)
$83$
\( T - 3417068864868 \)
$89$
\( T + 377306179184790 \)
$97$
\( T - 166982186657374 \)
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