Newspace parameters
| Level: | \( N \) | \(=\) | \( 289 = 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 289.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(17.0515519917\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 17) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/289\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 288.1 |
|
3.00000 | − | 8.00000i | 1.00000 | 6.00000i | − | 24.0000i | 28.0000i | −21.0000 | −37.0000 | 18.0000i | ||||||||||||||||||||||
| 288.2 | 3.00000 | 8.00000i | 1.00000 | − | 6.00000i | 24.0000i | − | 28.0000i | −21.0000 | −37.0000 | − | 18.0000i | ||||||||||||||||||||||
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 | 289.4.b.a | 2 | |
| 17.b | even | 2 | 1 | inner | 289.4.b.a | 2 | |
| 17.c | even | 4 | 1 | 17.4.a.a | ✓ | 1 | |
| 17.c | even | 4 | 1 | 289.4.a.a | 1 | ||
| 51.f | odd | 4 | 1 | 153.4.a.d | 1 | ||
| 68.f | odd | 4 | 1 | 272.4.a.d | 1 | ||
| 85.f | odd | 4 | 1 | 425.4.b.c | 2 | ||
| 85.i | odd | 4 | 1 | 425.4.b.c | 2 | ||
| 85.j | even | 4 | 1 | 425.4.a.d | 1 | ||
| 119.f | odd | 4 | 1 | 833.4.a.a | 1 | ||
| 136.i | even | 4 | 1 | 1088.4.a.l | 1 | ||
| 136.j | odd | 4 | 1 | 1088.4.a.a | 1 | ||
| 187.f | odd | 4 | 1 | 2057.4.a.d | 1 | ||
| 204.l | even | 4 | 1 | 2448.4.a.f | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.4.a.a | ✓ | 1 | 17.c | even | 4 | 1 | |
| 153.4.a.d | 1 | 51.f | odd | 4 | 1 | ||
| 272.4.a.d | 1 | 68.f | odd | 4 | 1 | ||
| 289.4.a.a | 1 | 17.c | even | 4 | 1 | ||
| 289.4.b.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 289.4.b.a | 2 | 17.b | even | 2 | 1 | inner | |
| 425.4.a.d | 1 | 85.j | even | 4 | 1 | ||
| 425.4.b.c | 2 | 85.f | odd | 4 | 1 | ||
| 425.4.b.c | 2 | 85.i | odd | 4 | 1 | ||
| 833.4.a.a | 1 | 119.f | odd | 4 | 1 | ||
| 1088.4.a.a | 1 | 136.j | odd | 4 | 1 | ||
| 1088.4.a.l | 1 | 136.i | even | 4 | 1 | ||
| 2057.4.a.d | 1 | 187.f | odd | 4 | 1 | ||
| 2448.4.a.f | 1 | 204.l | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} - 3 \)
acting on \(S_{4}^{\mathrm{new}}(289, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 3)^{2} \)
|
| $3$ |
\( T^{2} + 64 \)
|
| $5$ |
\( T^{2} + 36 \)
|
| $7$ |
\( T^{2} + 784 \)
|
| $11$ |
\( T^{2} + 576 \)
|
| $13$ |
\( (T + 58)^{2} \)
|
| $17$ |
\( T^{2} \)
|
| $19$ |
\( (T + 116)^{2} \)
|
| $23$ |
\( T^{2} + 3600 \)
|
| $29$ |
\( T^{2} + 900 \)
|
| $31$ |
\( T^{2} + 29584 \)
|
| $37$ |
\( T^{2} + 3364 \)
|
| $41$ |
\( T^{2} + 116964 \)
|
| $43$ |
\( (T - 148)^{2} \)
|
| $47$ |
\( (T - 288)^{2} \)
|
| $53$ |
\( (T + 318)^{2} \)
|
| $59$ |
\( (T + 252)^{2} \)
|
| $61$ |
\( T^{2} + 12100 \)
|
| $67$ |
\( (T + 484)^{2} \)
|
| $71$ |
\( T^{2} + 501264 \)
|
| $73$ |
\( T^{2} + 131044 \)
|
| $79$ |
\( T^{2} + 234256 \)
|
| $83$ |
\( (T + 756)^{2} \)
|
| $89$ |
\( (T + 774)^{2} \)
|
| $97$ |
\( T^{2} + 145924 \)
|
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