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: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{358549}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 89637 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 8) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 192\sqrt{358549}\). 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 | −62251.6 | 0 | −1.24528e6 | 0 | 2.63007e8 | 0 | −6.58509e9 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 167684. | 0 | 3.35342e6 | 0 | −7.07779e8 | 0 | 1.76574e10 | 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.e | 2 | |
| 4.b | odd | 2 | 1 | 8.22.a.a | ✓ | 2 | |
| 8.b | even | 2 | 1 | 64.22.a.h | 2 | ||
| 8.d | odd | 2 | 1 | 64.22.a.k | 2 | ||
| 12.b | even | 2 | 1 | 72.22.a.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.22.a.a | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 16.22.a.e | 2 | 1.a | even | 1 | 1 | trivial | |
| 64.22.a.h | 2 | 8.b | even | 2 | 1 | ||
| 64.22.a.k | 2 | 8.d | odd | 2 | 1 | ||
| 72.22.a.b | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 105432T_{3} - 10438573680 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(16))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} - 105432 T - 10438573680 \)
$5$
\( T^{2} - 2108140 T - 4175956569500 \)
$7$
\( T^{2} + 444771792 T - 18\!\cdots\!08 \)
$11$
\( T^{2} + 53806403320 T - 29\!\cdots\!96 \)
$13$
\( T^{2} + 490366676932 T - 18\!\cdots\!48 \)
$17$
\( T^{2} + 6593864672092 T - 16\!\cdots\!00 \)
$19$
\( T^{2} + 19302397925320 T - 16\!\cdots\!36 \)
$23$
\( T^{2} - 409737865776272 T + 30\!\cdots\!72 \)
$29$
\( T^{2} + \cdots - 56\!\cdots\!84 \)
$31$
\( T^{2} + \cdots + 14\!\cdots\!20 \)
$37$
\( T^{2} + \cdots - 32\!\cdots\!16 \)
$41$
\( T^{2} + \cdots + 11\!\cdots\!08 \)
$43$
\( T^{2} + \cdots + 96\!\cdots\!48 \)
$47$
\( T^{2} + \cdots - 25\!\cdots\!44 \)
$53$
\( T^{2} + \cdots + 22\!\cdots\!56 \)
$59$
\( T^{2} + \cdots - 55\!\cdots\!12 \)
$61$
\( T^{2} + \cdots + 56\!\cdots\!40 \)
$67$
\( T^{2} + \cdots - 10\!\cdots\!28 \)
$71$
\( T^{2} + \cdots - 68\!\cdots\!40 \)
$73$
\( T^{2} + \cdots - 36\!\cdots\!48 \)
$79$
\( T^{2} + \cdots + 15\!\cdots\!80 \)
$83$
\( T^{2} + \cdots + 31\!\cdots\!48 \)
$89$
\( T^{2} + \cdots - 21\!\cdots\!56 \)
$97$
\( T^{2} + \cdots + 15\!\cdots\!36 \)
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