Newspace parameters
| Level: | \( N \) | \(=\) | \( 69 = 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 69.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(4.07113179040\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-23})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 5x^{2} - 6x + 36 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{4} - x^{3} - 5x^{2} - 6x + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 5\nu^{2} - 5\nu - 26 ) / 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 6 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -5\beta_{3} + 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).
| \(n\) | \(28\) | \(47\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 68.1 |
|
− | 4.99599i | −5.15331 | − | 0.665865i | −16.9599 | 0 | −3.32666 | + | 25.7459i | 0 | 44.7638i | 26.1132 | + | 6.86282i | 0 | |||||||||||||||||||||||
| 68.2 | − | 0.200160i | 3.15331 | + | 4.12997i | 7.95994 | 0 | 0.826656 | − | 0.631168i | 0 | − | 3.19455i | −7.11325 | + | 26.0461i | 0 | |||||||||||||||||||||||
| 68.3 | 0.200160i | 3.15331 | − | 4.12997i | 7.95994 | 0 | 0.826656 | + | 0.631168i | 0 | 3.19455i | −7.11325 | − | 26.0461i | 0 | |||||||||||||||||||||||||
| 68.4 | 4.99599i | −5.15331 | + | 0.665865i | −16.9599 | 0 | −3.32666 | − | 25.7459i | 0 | − | 44.7638i | 26.1132 | − | 6.86282i | 0 | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-23}) \) |
| 3.b | odd | 2 | 1 | inner |
| 69.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 69.4.c.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 69.4.c.b | ✓ | 4 |
| 23.b | odd | 2 | 1 | CM | 69.4.c.b | ✓ | 4 |
| 69.c | even | 2 | 1 | inner | 69.4.c.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.4.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 69.4.c.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 69.4.c.b | ✓ | 4 | 23.b | odd | 2 | 1 | CM |
| 69.4.c.b | ✓ | 4 | 69.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 25T_{2}^{2} + 1 \)
acting on \(S_{4}^{\mathrm{new}}(69, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 25T^{2} + 1 \)
|
| $3$ |
\( T^{4} + 4 T^{3} + \cdots + 729 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( (T^{2} + 74 T - 1115)^{2} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( (T^{2} + 12167)^{2} \)
|
| $29$ |
\( T^{4} + \cdots + 3039868225 \)
|
| $31$ |
\( (T^{2} + 344 T + 28963)^{2} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} + \cdots + 12668628025 \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} + \cdots + 10306107361 \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( (T^{2} + 664700)^{2} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} + \cdots + 1050758254225 \)
|
| $73$ |
\( (T^{2} + 1226 T + 336025)^{2} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
| show more | |
| show less |