Newspace parameters
| Level: | \( N \) | \(=\) | \( 22 = 2 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 22.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(23.5908043694\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 211996x + 12319228 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{3} - x^{2} - 211996x + 12319228 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + 36\nu - 141364 ) / 63 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 63\beta_{2} - 18\beta _1 + 141346 \)
|
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 |
|
64.0000 | −1872.75 | 4096.00 | −29034.6 | −119856. | −320879. | 262144. | 1.91286e6 | −1.85822e6 | |||||||||||||||||||||||||||
| 1.2 | 64.0000 | −362.863 | 4096.00 | 41298.3 | −23223.2 | 284256. | 262144. | −1.46265e6 | 2.64309e6 | ||||||||||||||||||||||||||||
| 1.3 | 64.0000 | 1937.61 | 4096.00 | 28168.4 | 124007. | 77074.7 | 262144. | 2.16001e6 | 1.80278e6 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(1\) |
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 | 22.14.a.d | ✓ | 3 |
| 3.b | odd | 2 | 1 | 198.14.a.f | 3 | ||
| 4.b | odd | 2 | 1 | 176.14.a.d | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.14.a.d | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 176.14.a.d | 3 | 4.b | odd | 2 | 1 | ||
| 198.14.a.f | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 298T_{3}^{2} - 3652191T_{3} - 1316703276 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(22))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 64)^{3} \)
|
| $3$ |
\( T^{3} + \cdots - 1316703276 \)
|
| $5$ |
\( T^{3} + \cdots + 33776158711050 \)
|
| $7$ |
\( T^{3} + \cdots + 70\!\cdots\!08 \)
|
| $11$ |
\( (T + 1771561)^{3} \)
|
| $13$ |
\( T^{3} + \cdots - 73\!\cdots\!80 \)
|
| $17$ |
\( T^{3} + \cdots + 13\!\cdots\!40 \)
|
| $19$ |
\( T^{3} + \cdots + 20\!\cdots\!28 \)
|
| $23$ |
\( T^{3} + \cdots + 23\!\cdots\!72 \)
|
| $29$ |
\( T^{3} + \cdots - 19\!\cdots\!12 \)
|
| $31$ |
\( T^{3} + \cdots + 43\!\cdots\!60 \)
|
| $37$ |
\( T^{3} + \cdots - 21\!\cdots\!54 \)
|
| $41$ |
\( T^{3} + \cdots + 48\!\cdots\!24 \)
|
| $43$ |
\( T^{3} + \cdots - 63\!\cdots\!00 \)
|
| $47$ |
\( T^{3} + \cdots - 23\!\cdots\!80 \)
|
| $53$ |
\( T^{3} + \cdots - 14\!\cdots\!56 \)
|
| $59$ |
\( T^{3} + \cdots - 18\!\cdots\!52 \)
|
| $61$ |
\( T^{3} + \cdots + 56\!\cdots\!00 \)
|
| $67$ |
\( T^{3} + \cdots - 27\!\cdots\!76 \)
|
| $71$ |
\( T^{3} + \cdots - 95\!\cdots\!24 \)
|
| $73$ |
\( T^{3} + \cdots + 11\!\cdots\!68 \)
|
| $79$ |
\( T^{3} + \cdots - 11\!\cdots\!16 \)
|
| $83$ |
\( T^{3} + \cdots - 18\!\cdots\!60 \)
|
| $89$ |
\( T^{3} + \cdots + 36\!\cdots\!38 \)
|
| $97$ |
\( T^{3} + \cdots + 11\!\cdots\!78 \)
|
| show more | |
| show less |