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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{2161}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 540 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 4) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2688\sqrt{2161}\). We also show the integral \(q\)-expansion of the trace form.
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 | −157776. | 0 | 3.23357e7 | 0 | 8.65744e8 | 0 | 1.44329e10 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 92135.9 | 0 | −1.86463e7 | 0 | −6.05236e8 | 0 | −1.97134e9 | 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.d | 2 | |
| 4.b | odd | 2 | 1 | 4.22.a.a | ✓ | 2 | |
| 8.b | even | 2 | 1 | 64.22.a.j | 2 | ||
| 8.d | odd | 2 | 1 | 64.22.a.i | 2 | ||
| 12.b | even | 2 | 1 | 36.22.a.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4.22.a.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 16.22.a.d | 2 | 1.a | even | 1 | 1 | trivial | |
| 36.22.a.c | 2 | 12.b | even | 2 | 1 | ||
| 64.22.a.i | 2 | 8.d | odd | 2 | 1 | ||
| 64.22.a.j | 2 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 65640T_{3} - 14536815984 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(16))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 65640 T - 14536815984 \)
$5$
\( T^{2} + \cdots - 602941510374300 \)
$7$
\( T^{2} - 260508080 T - 52\!\cdots\!64 \)
$11$
\( T^{2} + 145435963320 T + 37\!\cdots\!00 \)
$13$
\( T^{2} - 1428900417340 T + 50\!\cdots\!76 \)
$17$
\( T^{2} - 1840620576420 T - 11\!\cdots\!56 \)
$19$
\( T^{2} - 16780743928568 T - 10\!\cdots\!44 \)
$23$
\( T^{2} - 319691925426960 T + 19\!\cdots\!04 \)
$29$
\( T^{2} + \cdots + 31\!\cdots\!16 \)
$31$
\( T^{2} + 112042353462592 T - 32\!\cdots\!84 \)
$37$
\( T^{2} + \cdots - 66\!\cdots\!44 \)
$41$
\( T^{2} + \cdots + 65\!\cdots\!56 \)
$43$
\( T^{2} + \cdots - 25\!\cdots\!00 \)
$47$
\( T^{2} + \cdots + 11\!\cdots\!64 \)
$53$
\( T^{2} + \cdots - 70\!\cdots\!16 \)
$59$
\( T^{2} + \cdots - 56\!\cdots\!96 \)
$61$
\( T^{2} + \cdots + 45\!\cdots\!24 \)
$67$
\( T^{2} + \cdots + 22\!\cdots\!56 \)
$71$
\( T^{2} + \cdots - 19\!\cdots\!76 \)
$73$
\( T^{2} + \cdots - 63\!\cdots\!24 \)
$79$
\( T^{2} + \cdots + 35\!\cdots\!04 \)
$83$
\( T^{2} + \cdots + 22\!\cdots\!64 \)
$89$
\( T^{2} + \cdots - 88\!\cdots\!04 \)
$97$
\( T^{2} + \cdots - 28\!\cdots\!24 \)
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