Newspace parameters
| Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 378.s (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(10.2997539928\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(325\) |
| \(\chi(n)\) | \(-1\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 53.1 |
|
−1.22474 | + | 0.707107i | 0 | 1.00000 | − | 1.73205i | −3.67423 | + | 2.12132i | 0 | 5.50000 | + | 4.33013i | 2.82843i | 0 | 3.00000 | − | 5.19615i | ||||||||||||||||||||
| 53.2 | 1.22474 | − | 0.707107i | 0 | 1.00000 | − | 1.73205i | 3.67423 | − | 2.12132i | 0 | 5.50000 | + | 4.33013i | − | 2.82843i | 0 | 3.00000 | − | 5.19615i | ||||||||||||||||||||
| 107.1 | −1.22474 | − | 0.707107i | 0 | 1.00000 | + | 1.73205i | −3.67423 | − | 2.12132i | 0 | 5.50000 | − | 4.33013i | − | 2.82843i | 0 | 3.00000 | + | 5.19615i | ||||||||||||||||||||
| 107.2 | 1.22474 | + | 0.707107i | 0 | 1.00000 | + | 1.73205i | 3.67423 | + | 2.12132i | 0 | 5.50000 | − | 4.33013i | 2.82843i | 0 | 3.00000 | + | 5.19615i | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 21.h | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 378.3.s.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 378.3.s.b | ✓ | 4 |
| 7.c | even | 3 | 1 | inner | 378.3.s.b | ✓ | 4 |
| 21.h | odd | 6 | 1 | inner | 378.3.s.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 378.3.s.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 378.3.s.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 378.3.s.b | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 378.3.s.b | ✓ | 4 | 21.h | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 18T_{5}^{2} + 324 \)
acting on \(S_{3}^{\mathrm{new}}(378, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2T^{2} + 4 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 18T^{2} + 324 \)
|
| $7$ |
\( (T^{2} - 11 T + 49)^{2} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( (T - 8)^{4} \)
|
| $17$ |
\( T^{4} - 18T^{2} + 324 \)
|
| $19$ |
\( (T^{2} + 5 T + 25)^{2} \)
|
| $23$ |
\( T^{4} - 450 T^{2} + 202500 \)
|
| $29$ |
\( (T^{2} + 882)^{2} \)
|
| $31$ |
\( (T^{2} + 29 T + 841)^{2} \)
|
| $37$ |
\( (T^{2} + 8 T + 64)^{2} \)
|
| $41$ |
\( (T^{2} + 5202)^{2} \)
|
| $43$ |
\( (T + 13)^{4} \)
|
| $47$ |
\( T^{4} - 450 T^{2} + 202500 \)
|
| $53$ |
\( T^{4} - 4050 T^{2} + 16402500 \)
|
| $59$ |
\( T^{4} - 1152 T^{2} + 1327104 \)
|
| $61$ |
\( (T^{2} + 41 T + 1681)^{2} \)
|
| $67$ |
\( (T^{2} + 128 T + 16384)^{2} \)
|
| $71$ |
\( (T^{2} + 18432)^{2} \)
|
| $73$ |
\( (T^{2} + 65 T + 4225)^{2} \)
|
| $79$ |
\( (T^{2} + 14 T + 196)^{2} \)
|
| $83$ |
\( (T^{2} + 4608)^{2} \)
|
| $89$ |
\( T^{4} - 11250 T^{2} + 126562500 \)
|
| $97$ |
\( (T + 103)^{4} \)
|
| show more | |
| show less |