Newspace parameters
| Level: | \( N \) | \(=\) | \( 3249 = 3^{2} \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3249.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(25.9433956167\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.27648.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - 6x^{2} + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 171) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{4} - 6x^{2} + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.33441 | 0 | 3.44949 | 1.04930 | 0 | −3.44949 | −3.38371 | 0 | −2.44949 | ||||||||||||||||||||||||||||||
| 1.2 | −0.741964 | 0 | −1.44949 | −3.30136 | 0 | 1.44949 | 2.55940 | 0 | 2.44949 | |||||||||||||||||||||||||||||||
| 1.3 | 0.741964 | 0 | −1.44949 | 3.30136 | 0 | 1.44949 | −2.55940 | 0 | 2.44949 | |||||||||||||||||||||||||||||||
| 1.4 | 2.33441 | 0 | 3.44949 | −1.04930 | 0 | −3.44949 | 3.38371 | 0 | −2.44949 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(19\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3249.2.a.be | 4 | |
| 3.b | odd | 2 | 1 | inner | 3249.2.a.be | 4 | |
| 19.b | odd | 2 | 1 | 3249.2.a.bd | 4 | ||
| 19.d | odd | 6 | 2 | 171.2.f.c | ✓ | 8 | |
| 57.d | even | 2 | 1 | 3249.2.a.bd | 4 | ||
| 57.f | even | 6 | 2 | 171.2.f.c | ✓ | 8 | |
| 76.f | even | 6 | 2 | 2736.2.s.bb | 8 | ||
| 228.n | odd | 6 | 2 | 2736.2.s.bb | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 171.2.f.c | ✓ | 8 | 19.d | odd | 6 | 2 | |
| 171.2.f.c | ✓ | 8 | 57.f | even | 6 | 2 | |
| 2736.2.s.bb | 8 | 76.f | even | 6 | 2 | ||
| 2736.2.s.bb | 8 | 228.n | odd | 6 | 2 | ||
| 3249.2.a.bd | 4 | 19.b | odd | 2 | 1 | ||
| 3249.2.a.bd | 4 | 57.d | even | 2 | 1 | ||
| 3249.2.a.be | 4 | 1.a | even | 1 | 1 | trivial | |
| 3249.2.a.be | 4 | 3.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3249))\):
|
\( T_{2}^{4} - 6T_{2}^{2} + 3 \)
|
|
\( T_{5}^{4} - 12T_{5}^{2} + 12 \)
|
|
\( T_{13} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 6T^{2} + 3 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 12T^{2} + 12 \)
|
| $7$ |
\( (T^{2} + 2 T - 5)^{2} \)
|
| $11$ |
\( T^{4} - 36T^{2} + 108 \)
|
| $13$ |
\( (T - 1)^{4} \)
|
| $17$ |
\( T^{4} - 48T^{2} + 192 \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} - 36T^{2} + 300 \)
|
| $29$ |
\( T^{4} - 144T^{2} + 4800 \)
|
| $31$ |
\( (T^{2} - 14 T + 43)^{2} \)
|
| $37$ |
\( (T^{2} - 2 T - 23)^{2} \)
|
| $41$ |
\( T^{4} - 96T^{2} + 768 \)
|
| $43$ |
\( (T^{2} + 2 T - 53)^{2} \)
|
| $47$ |
\( T^{4} - 96T^{2} + 768 \)
|
| $53$ |
\( T^{4} - 12T^{2} + 12 \)
|
| $59$ |
\( T^{4} - 132T^{2} + 4332 \)
|
| $61$ |
\( (T - 5)^{4} \)
|
| $67$ |
\( (T^{2} - 14 T - 5)^{2} \)
|
| $71$ |
\( T^{4} - 144T^{2} + 4800 \)
|
| $73$ |
\( (T - 5)^{4} \)
|
| $79$ |
\( (T^{2} - 14 T - 5)^{2} \)
|
| $83$ |
\( T^{4} - 144T^{2} + 1728 \)
|
| $89$ |
\( T^{4} - 300T^{2} + 7500 \)
|
| $97$ |
\( (T^{2} + 16 T + 40)^{2} \)
|
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