Newspace parameters
Level: | \( N \) | \(=\) | \( 16 = 2^{4} \) |
Weight: | \( k \) | \(=\) | \( 22 \) |
Character orbit: | \([\chi]\) | \(=\) | 16.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(44.7163750859\) |
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 | −71604.0 | 0 | −2.86938e7 | 0 | 8.53202e8 | 0 | −5.33322e9 | 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 | 16.22.a.a | 1 | |
4.b | odd | 2 | 1 | 2.22.a.a | ✓ | 1 | |
8.b | even | 2 | 1 | 64.22.a.f | 1 | ||
8.d | odd | 2 | 1 | 64.22.a.b | 1 | ||
12.b | even | 2 | 1 | 18.22.a.e | 1 | ||
20.d | odd | 2 | 1 | 50.22.a.c | 1 | ||
20.e | even | 4 | 2 | 50.22.b.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2.22.a.a | ✓ | 1 | 4.b | odd | 2 | 1 | |
16.22.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
18.22.a.e | 1 | 12.b | even | 2 | 1 | ||
50.22.a.c | 1 | 20.d | odd | 2 | 1 | ||
50.22.b.a | 2 | 20.e | even | 4 | 2 | ||
64.22.a.b | 1 | 8.d | odd | 2 | 1 | ||
64.22.a.f | 1 | 8.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} + 71604 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(16))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T + 71604 \)
$5$
\( T + 28693770 \)
$7$
\( T - 853202392 \)
$11$
\( T + 86731179612 \)
$13$
\( T + 895323442786 \)
$17$
\( T - 3257566804818 \)
$19$
\( T + 23032467644420 \)
$23$
\( T + 146495714575224 \)
$29$
\( T + 734051633521170 \)
$31$
\( T - 3146664162057568 \)
$37$
\( T + 12\!\cdots\!62 \)
$41$
\( T - 45\!\cdots\!42 \)
$43$
\( T - 24\!\cdots\!56 \)
$47$
\( T - 44\!\cdots\!52 \)
$53$
\( T - 20\!\cdots\!54 \)
$59$
\( T - 37\!\cdots\!40 \)
$61$
\( T + 76\!\cdots\!38 \)
$67$
\( T - 18\!\cdots\!32 \)
$71$
\( T - 45\!\cdots\!28 \)
$73$
\( T + 25\!\cdots\!26 \)
$79$
\( T + 99\!\cdots\!80 \)
$83$
\( T + 29\!\cdots\!84 \)
$89$
\( T - 11\!\cdots\!90 \)
$97$
\( T + 56\!\cdots\!02 \)
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