Newspace parameters
| Level: | \( N \) | \(=\) | \( 111 = 3 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 111.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.02453093440\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.65712.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 7x^{2} + 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 7x^{2} + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 6\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 7 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 3\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/111\mathbb{Z}\right)^\times\).
| \(n\) | \(38\) | \(76\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 110.1 |
|
−3.75270 | −3.00000 | 10.0828 | −6.15097 | 11.2581 | −12.1655 | −22.8268 | 9.00000 | 23.0828 | ||||||||||||||||||||||||||||||
| 110.2 | −1.38464 | −3.00000 | −2.08276 | −7.88451 | 4.15393 | 12.1655 | 8.42246 | 9.00000 | 10.9172 | |||||||||||||||||||||||||||||||
| 110.3 | 1.38464 | −3.00000 | −2.08276 | 7.88451 | −4.15393 | 12.1655 | −8.42246 | 9.00000 | 10.9172 | |||||||||||||||||||||||||||||||
| 110.4 | 3.75270 | −3.00000 | 10.0828 | 6.15097 | −11.2581 | −12.1655 | 22.8268 | 9.00000 | 23.0828 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 111.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-111}) \) |
| 3.b | odd | 2 | 1 | inner |
| 37.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 111.3.d.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 111.3.d.a | ✓ | 4 |
| 37.b | even | 2 | 1 | inner | 111.3.d.a | ✓ | 4 |
| 111.d | odd | 2 | 1 | CM | 111.3.d.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 111.3.d.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 111.3.d.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 111.3.d.a | ✓ | 4 | 37.b | even | 2 | 1 | inner |
| 111.3.d.a | ✓ | 4 | 111.d | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 16T_{2}^{2} + 27 \)
acting on \(S_{3}^{\mathrm{new}}(111, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 16T^{2} + 27 \)
|
| $3$ |
\( (T + 3)^{4} \)
|
| $5$ |
\( T^{4} - 100T^{2} + 2352 \)
|
| $7$ |
\( (T^{2} - 148)^{2} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} - 1156 T^{2} + 255792 \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} - 2116 T^{2} + 330672 \)
|
| $29$ |
\( T^{4} - 3364 T^{2} + 1436592 \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( (T - 37)^{4} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( T^{4} - 13924 T^{2} + 45442992 \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( (T^{2} - 17908)^{2} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( (T^{2} - 148)^{2} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} - 31684 T^{2} + 250180272 \)
|
| $97$ |
\( T^{4} \)
|
| show more | |
| show less |