Newspace parameters
| Level: | \( N \) | \(=\) | \( 538 = 2 \cdot 269 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 538.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(4.29595162874\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 3x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 3x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 2\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 2\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/538\mathbb{Z}\right)^\times\).
| \(n\) | \(271\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 537.1 |
|
− | 1.00000i | − | 0.618034i | −1.00000 | 0.381966 | −0.618034 | 3.00000i | 1.00000i | 2.61803 | − | 0.381966i | |||||||||||||||||||||||||||
| 537.2 | − | 1.00000i | 1.61803i | −1.00000 | 2.61803 | 1.61803 | 3.00000i | 1.00000i | 0.381966 | − | 2.61803i | |||||||||||||||||||||||||||||
| 537.3 | 1.00000i | − | 1.61803i | −1.00000 | 2.61803 | 1.61803 | − | 3.00000i | − | 1.00000i | 0.381966 | 2.61803i | ||||||||||||||||||||||||||||
| 537.4 | 1.00000i | 0.618034i | −1.00000 | 0.381966 | −0.618034 | − | 3.00000i | − | 1.00000i | 2.61803 | 0.381966i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 269.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 538.2.b.a | ✓ | 4 |
| 269.b | even | 2 | 1 | inner | 538.2.b.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 538.2.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 538.2.b.a | ✓ | 4 | 269.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 3T_{3}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(538, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{2} \)
|
| $3$ |
\( T^{4} + 3T^{2} + 1 \)
|
| $5$ |
\( (T^{2} - 3 T + 1)^{2} \)
|
| $7$ |
\( (T^{2} + 9)^{2} \)
|
| $11$ |
\( (T^{2} - 5)^{2} \)
|
| $13$ |
\( (T^{2} - 45)^{2} \)
|
| $17$ |
\( (T^{2} + 9)^{2} \)
|
| $19$ |
\( T^{4} + 28T^{2} + 16 \)
|
| $23$ |
\( (T^{2} - 12 T + 31)^{2} \)
|
| $29$ |
\( T^{4} + 83T^{2} + 361 \)
|
| $31$ |
\( T^{4} + 82T^{2} + 961 \)
|
| $37$ |
\( (T - 1)^{4} \)
|
| $41$ |
\( (T^{2} + 6 T + 4)^{2} \)
|
| $43$ |
\( (T^{2} - 4 T - 41)^{2} \)
|
| $47$ |
\( (T^{2} - 9 T - 41)^{2} \)
|
| $53$ |
\( (T^{2} - 12 T + 16)^{2} \)
|
| $59$ |
\( T^{4} + 27T^{2} + 81 \)
|
| $61$ |
\( (T + 1)^{4} \)
|
| $67$ |
\( (T^{2} - 4 T - 41)^{2} \)
|
| $71$ |
\( T^{4} + 368 T^{2} + 30976 \)
|
| $73$ |
\( (T + 9)^{4} \)
|
| $79$ |
\( (T^{2} + 15 T + 45)^{2} \)
|
| $83$ |
\( T^{4} + 108T^{2} + 1296 \)
|
| $89$ |
\( (T^{2} - 9 T - 11)^{2} \)
|
| $97$ |
\( (T^{2} + 4 T - 176)^{2} \)
|
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