Newspace parameters
| Level: | \( N \) | \(=\) | \( 189 = 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 189.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(5.14987699641\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.1166592.2 |
|
|
|
| Defining polynomial: |
\( x^{4} + 20x^{2} + 93 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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
\( x^{4} + 20x^{2} + 93 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 10 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 11\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 11\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(136\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 134.1 |
|
− | 3.55609i | 0 | −8.64575 | 9.40852i | 0 | −2.64575 | 16.5207i | 0 | 33.4575 | |||||||||||||||||||||||||||||
| 134.2 | − | 2.71187i | 0 | −3.35425 | − | 7.17494i | 0 | 2.64575 | − | 1.75119i | 0 | −19.4575 | ||||||||||||||||||||||||||||
| 134.3 | 2.71187i | 0 | −3.35425 | 7.17494i | 0 | 2.64575 | 1.75119i | 0 | −19.4575 | |||||||||||||||||||||||||||||||
| 134.4 | 3.55609i | 0 | −8.64575 | − | 9.40852i | 0 | −2.64575 | − | 16.5207i | 0 | 33.4575 | |||||||||||||||||||||||||||||
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 | 189.3.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 189.3.b.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 3024.3.d.f | 4 | ||
| 9.c | even | 3 | 2 | 567.3.r.d | 8 | ||
| 9.d | odd | 6 | 2 | 567.3.r.d | 8 | ||
| 12.b | even | 2 | 1 | 3024.3.d.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 189.3.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 189.3.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 567.3.r.d | 8 | 9.c | even | 3 | 2 | ||
| 567.3.r.d | 8 | 9.d | odd | 6 | 2 | ||
| 3024.3.d.f | 4 | 4.b | odd | 2 | 1 | ||
| 3024.3.d.f | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 20T_{2}^{2} + 93 \)
acting on \(S_{3}^{\mathrm{new}}(189, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 20T^{2} + 93 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 140T^{2} + 4557 \)
|
| $7$ |
\( (T^{2} - 7)^{2} \)
|
| $11$ |
\( T^{4} + 20T^{2} + 93 \)
|
| $13$ |
\( (T^{2} - 18 T + 74)^{2} \)
|
| $17$ |
\( T^{4} + 780 T^{2} + 30132 \)
|
| $19$ |
\( (T^{2} + 24 T - 199)^{2} \)
|
| $23$ |
\( T^{4} + 780T^{2} + 837 \)
|
| $29$ |
\( T^{4} + 2252 T^{2} + 1208628 \)
|
| $31$ |
\( (T^{2} - 20 T - 1083)^{2} \)
|
| $37$ |
\( (T^{2} - 26 T - 3219)^{2} \)
|
| $41$ |
\( T^{4} + 1196 T^{2} + 302157 \)
|
| $43$ |
\( (T^{2} - 70 T - 350)^{2} \)
|
| $47$ |
\( T^{4} + 4380 T^{2} + 271188 \)
|
| $53$ |
\( T^{4} + 1392 T^{2} + 120528 \)
|
| $59$ |
\( T^{4} + 10356 T^{2} + 30132 \)
|
| $61$ |
\( (T^{2} - 144 T + 4484)^{2} \)
|
| $67$ |
\( (T^{2} - 66 T - 2614)^{2} \)
|
| $71$ |
\( T^{4} + 10548 T^{2} + 26222373 \)
|
| $73$ |
\( (T^{2} + 94 T + 2034)^{2} \)
|
| $79$ |
\( (T^{2} + 38 T - 1214)^{2} \)
|
| $83$ |
\( T^{4} + 13572 T^{2} + 41250708 \)
|
| $89$ |
\( T^{4} + 3380 T^{2} + 1796853 \)
|
| $97$ |
\( (T^{2} - 80 T - 1788)^{2} \)
|
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