Newspace parameters
| Level: | \( N \) | \(=\) | \( 1003 = 17 \cdot 59 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1003.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(8.00899532273\) |
| 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: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(\left(\mathbb{Z}/1003\mathbb{Z}\right)^\times\).
| \(n\) | \(120\) | \(768\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 237.1 |
|
−0.618034 | − | 1.23607i | −1.61803 | − | 1.61803i | 0.763932i | − | 4.23607i | 2.23607 | 1.47214 | 1.00000i | |||||||||||||||||||||||||||
| 237.2 | −0.618034 | 1.23607i | −1.61803 | 1.61803i | − | 0.763932i | 4.23607i | 2.23607 | 1.47214 | − | 1.00000i | |||||||||||||||||||||||||||||
| 237.3 | 1.61803 | − | 3.23607i | 0.618034 | − | 0.618034i | − | 5.23607i | − | 0.236068i | −2.23607 | −7.47214 | − | 1.00000i | ||||||||||||||||||||||||||
| 237.4 | 1.61803 | 3.23607i | 0.618034 | 0.618034i | 5.23607i | 0.236068i | −2.23607 | −7.47214 | 1.00000i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1003.2.d.b | ✓ | 4 |
| 17.b | even | 2 | 1 | inner | 1003.2.d.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1003.2.d.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1003.2.d.b | ✓ | 4 | 17.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(1003, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T - 1)^{2} \)
|
| $3$ |
\( T^{4} + 12T^{2} + 16 \)
|
| $5$ |
\( T^{4} + 3T^{2} + 1 \)
|
| $7$ |
\( T^{4} + 18T^{2} + 1 \)
|
| $11$ |
\( T^{4} + 15T^{2} + 25 \)
|
| $13$ |
\( (T^{2} + 7 T + 11)^{2} \)
|
| $17$ |
\( (T^{2} - 8 T + 17)^{2} \)
|
| $19$ |
\( (T^{2} - 5)^{2} \)
|
| $23$ |
\( T^{4} + 127T^{2} + 3481 \)
|
| $29$ |
\( T^{4} + 67T^{2} + 841 \)
|
| $31$ |
\( T^{4} + 90T^{2} + 25 \)
|
| $37$ |
\( T^{4} + 138T^{2} + 3481 \)
|
| $41$ |
\( (T^{2} + 25)^{2} \)
|
| $43$ |
\( (T^{2} - 13 T + 31)^{2} \)
|
| $47$ |
\( (T^{2} + 9 T - 11)^{2} \)
|
| $53$ |
\( (T^{2} + 12 T - 9)^{2} \)
|
| $59$ |
\( (T + 1)^{4} \)
|
| $61$ |
\( T^{4} + 210T^{2} + 9025 \)
|
| $67$ |
\( (T^{2} + 14 T + 29)^{2} \)
|
| $71$ |
\( T^{4} + 115T^{2} + 3025 \)
|
| $73$ |
\( T^{4} + 162T^{2} + 81 \)
|
| $79$ |
\( T^{4} + 87T^{2} + 361 \)
|
| $83$ |
\( (T^{2} + 2 T - 79)^{2} \)
|
| $89$ |
\( (T^{2} - 5 T - 55)^{2} \)
|
| $97$ |
\( T^{4} + 3T^{2} + 1 \)
|
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