Newspace parameters
| Level: | \( N \) | \(=\) | \( 23 \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 23.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(9.36970803141\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.621.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 6x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 7 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
\( x^{3} - 6x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + 2\nu - 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( -2\nu^{2} + 3\nu + 8 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta_1 ) / 7 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{2} + 3\beta _1 + 28 ) / 7 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 22.1 |
|
−28.9050 | 80.1002 | 579.497 | 0 | −2315.29 | 0 | −9350.67 | −144.953 | 0 | |||||||||||||||||||||||||||
| 22.2 | 2.56227 | −161.997 | −249.435 | 0 | −415.079 | 0 | −1295.06 | 19681.9 | 0 | ||||||||||||||||||||||||||||
| 22.3 | 26.3427 | 81.8964 | 437.938 | 0 | 2157.37 | 0 | 4792.73 | 146.028 | 0 | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-23}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 23.9.b.a | ✓ | 3 |
| 3.b | odd | 2 | 1 | 207.9.d.a | 3 | ||
| 4.b | odd | 2 | 1 | 368.9.f.a | 3 | ||
| 23.b | odd | 2 | 1 | CM | 23.9.b.a | ✓ | 3 |
| 69.c | even | 2 | 1 | 207.9.d.a | 3 | ||
| 92.b | even | 2 | 1 | 368.9.f.a | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 23.9.b.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 23.9.b.a | ✓ | 3 | 23.b | odd | 2 | 1 | CM |
| 207.9.d.a | 3 | 3.b | odd | 2 | 1 | ||
| 207.9.d.a | 3 | 69.c | even | 2 | 1 | ||
| 368.9.f.a | 3 | 4.b | odd | 2 | 1 | ||
| 368.9.f.a | 3 | 92.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 768T_{2} + 1951 \)
acting on \(S_{9}^{\mathrm{new}}(23, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 768T + 1951 \)
|
| $3$ |
\( T^{3} - 19683 T + 1062686 \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} \)
|
| $13$ |
\( T^{3} + \cdots - 25363320370274 \)
|
| $17$ |
\( T^{3} \)
|
| $19$ |
\( T^{3} \)
|
| $23$ |
\( (T - 279841)^{3} \)
|
| $29$ |
\( T^{3} + \cdots + 64\!\cdots\!06 \)
|
| $31$ |
\( T^{3} + \cdots - 12\!\cdots\!94 \)
|
| $37$ |
\( T^{3} \)
|
| $41$ |
\( T^{3} + \cdots - 12\!\cdots\!14 \)
|
| $43$ |
\( T^{3} \)
|
| $47$ |
\( T^{3} + \cdots - 19\!\cdots\!54 \)
|
| $53$ |
\( T^{3} \)
|
| $59$ |
\( (T - 15279074)^{3} \)
|
| $61$ |
\( T^{3} \)
|
| $67$ |
\( T^{3} \)
|
| $71$ |
\( T^{3} + \cdots - 28\!\cdots\!94 \)
|
| $73$ |
\( T^{3} + \cdots - 39\!\cdots\!54 \)
|
| $79$ |
\( T^{3} \)
|
| $83$ |
\( T^{3} \)
|
| $89$ |
\( T^{3} \)
|
| $97$ |
\( T^{3} \)
|
| show more | |
| show less |