Newspace parameters
| Level: | \( N \) | \(=\) | \( 1734 = 2 \cdot 3 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1734.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(13.8460597105\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
|
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| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 102) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1734\mathbb{Z}\right)^\times\).
| \(n\) | \(1157\) | \(1159\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 577.1 |
|
1.00000 | − | 1.00000i | 1.00000 | 0 | − | 1.00000i | 2.00000i | 1.00000 | −1.00000 | 0 | ||||||||||||||||||||||
| 577.2 | 1.00000 | 1.00000i | 1.00000 | 0 | 1.00000i | − | 2.00000i | 1.00000 | −1.00000 | 0 | ||||||||||||||||||||||||
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 | 1734.2.b.f | 2 | |
| 17.b | even | 2 | 1 | inner | 1734.2.b.f | 2 | |
| 17.c | even | 4 | 1 | 102.2.a.b | ✓ | 1 | |
| 17.c | even | 4 | 1 | 1734.2.a.b | 1 | ||
| 17.d | even | 8 | 4 | 1734.2.f.b | 4 | ||
| 51.f | odd | 4 | 1 | 306.2.a.c | 1 | ||
| 51.f | odd | 4 | 1 | 5202.2.a.j | 1 | ||
| 68.f | odd | 4 | 1 | 816.2.a.d | 1 | ||
| 85.f | odd | 4 | 1 | 2550.2.d.g | 2 | ||
| 85.i | odd | 4 | 1 | 2550.2.d.g | 2 | ||
| 85.j | even | 4 | 1 | 2550.2.a.u | 1 | ||
| 119.f | odd | 4 | 1 | 4998.2.a.d | 1 | ||
| 136.i | even | 4 | 1 | 3264.2.a.i | 1 | ||
| 136.j | odd | 4 | 1 | 3264.2.a.w | 1 | ||
| 204.l | even | 4 | 1 | 2448.2.a.i | 1 | ||
| 255.i | odd | 4 | 1 | 7650.2.a.j | 1 | ||
| 408.q | even | 4 | 1 | 9792.2.a.ba | 1 | ||
| 408.t | odd | 4 | 1 | 9792.2.a.bg | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 102.2.a.b | ✓ | 1 | 17.c | even | 4 | 1 | |
| 306.2.a.c | 1 | 51.f | odd | 4 | 1 | ||
| 816.2.a.d | 1 | 68.f | odd | 4 | 1 | ||
| 1734.2.a.b | 1 | 17.c | even | 4 | 1 | ||
| 1734.2.b.f | 2 | 1.a | even | 1 | 1 | trivial | |
| 1734.2.b.f | 2 | 17.b | even | 2 | 1 | inner | |
| 1734.2.f.b | 4 | 17.d | even | 8 | 4 | ||
| 2448.2.a.i | 1 | 204.l | even | 4 | 1 | ||
| 2550.2.a.u | 1 | 85.j | even | 4 | 1 | ||
| 2550.2.d.g | 2 | 85.f | odd | 4 | 1 | ||
| 2550.2.d.g | 2 | 85.i | odd | 4 | 1 | ||
| 3264.2.a.i | 1 | 136.i | even | 4 | 1 | ||
| 3264.2.a.w | 1 | 136.j | odd | 4 | 1 | ||
| 4998.2.a.d | 1 | 119.f | odd | 4 | 1 | ||
| 5202.2.a.j | 1 | 51.f | odd | 4 | 1 | ||
| 7650.2.a.j | 1 | 255.i | odd | 4 | 1 | ||
| 9792.2.a.ba | 1 | 408.q | even | 4 | 1 | ||
| 9792.2.a.bg | 1 | 408.t | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1734, [\chi])\):
|
\( T_{5} \)
|
|
\( T_{7}^{2} + 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{2} \)
|
| $3$ |
\( T^{2} + 1 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} + 4 \)
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| $11$ |
\( T^{2} \)
|
| $13$ |
\( (T - 2)^{2} \)
|
| $17$ |
\( T^{2} \)
|
| $19$ |
\( (T - 4)^{2} \)
|
| $23$ |
\( T^{2} + 36 \)
|
| $29$ |
\( T^{2} \)
|
| $31$ |
\( T^{2} + 100 \)
|
| $37$ |
\( T^{2} + 64 \)
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| $41$ |
\( T^{2} + 36 \)
|
| $43$ |
\( (T - 4)^{2} \)
|
| $47$ |
\( (T - 12)^{2} \)
|
| $53$ |
\( (T + 6)^{2} \)
|
| $59$ |
\( (T - 12)^{2} \)
|
| $61$ |
\( T^{2} + 64 \)
|
| $67$ |
\( (T + 4)^{2} \)
|
| $71$ |
\( T^{2} + 36 \)
|
| $73$ |
\( T^{2} + 4 \)
|
| $79$ |
\( T^{2} + 100 \)
|
| $83$ |
\( (T + 12)^{2} \)
|
| $89$ |
\( (T + 18)^{2} \)
|
| $97$ |
\( T^{2} + 196 \)
|
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