Newspace parameters
| Level: | \( N \) | \(=\) | \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1350.g (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(36.7848356886\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).
| \(n\) | \(1001\) | \(1027\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 757.1 |
|
1.00000 | + | 1.00000i | 0 | 2.00000i | 0 | 0 | −3.67423 | − | 3.67423i | −2.00000 | + | 2.00000i | 0 | 0 | ||||||||||||||||||||||||
| 757.2 | 1.00000 | + | 1.00000i | 0 | 2.00000i | 0 | 0 | 3.67423 | + | 3.67423i | −2.00000 | + | 2.00000i | 0 | 0 | |||||||||||||||||||||||||
| 1243.1 | 1.00000 | − | 1.00000i | 0 | − | 2.00000i | 0 | 0 | −3.67423 | + | 3.67423i | −2.00000 | − | 2.00000i | 0 | 0 | ||||||||||||||||||||||||
| 1243.2 | 1.00000 | − | 1.00000i | 0 | − | 2.00000i | 0 | 0 | 3.67423 | − | 3.67423i | −2.00000 | − | 2.00000i | 0 | 0 | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
| 15.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1350.3.g.j | yes | 4 |
| 3.b | odd | 2 | 1 | 1350.3.g.c | ✓ | 4 | |
| 5.b | even | 2 | 1 | 1350.3.g.c | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 1350.3.g.c | ✓ | 4 | |
| 5.c | odd | 4 | 1 | inner | 1350.3.g.j | yes | 4 |
| 15.d | odd | 2 | 1 | inner | 1350.3.g.j | yes | 4 |
| 15.e | even | 4 | 1 | 1350.3.g.c | ✓ | 4 | |
| 15.e | even | 4 | 1 | inner | 1350.3.g.j | yes | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1350.3.g.c | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 1350.3.g.c | ✓ | 4 | 5.b | even | 2 | 1 | |
| 1350.3.g.c | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 1350.3.g.c | ✓ | 4 | 15.e | even | 4 | 1 | |
| 1350.3.g.j | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 1350.3.g.j | yes | 4 | 5.c | odd | 4 | 1 | inner |
| 1350.3.g.j | yes | 4 | 15.d | odd | 2 | 1 | inner |
| 1350.3.g.j | yes | 4 | 15.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1350, [\chi])\):
|
\( T_{7}^{4} + 729 \)
|
|
\( T_{11} \)
|
|
\( T_{17}^{2} - 12T_{17} + 72 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 2)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 729 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + 729 \)
|
| $17$ |
\( (T^{2} - 12 T + 72)^{2} \)
|
| $19$ |
\( (T^{2} + 49)^{2} \)
|
| $23$ |
\( (T^{2} + 24 T + 288)^{2} \)
|
| $29$ |
\( (T^{2} + 1944)^{2} \)
|
| $31$ |
\( (T - 10)^{4} \)
|
| $37$ |
\( T^{4} + 1750329 \)
|
| $41$ |
\( (T^{2} - 1944)^{2} \)
|
| $43$ |
\( T^{4} + 2985984 \)
|
| $47$ |
\( (T^{2} + 36 T + 648)^{2} \)
|
| $53$ |
\( (T^{2} - 60 T + 1800)^{2} \)
|
| $59$ |
\( (T^{2} + 1944)^{2} \)
|
| $61$ |
\( (T - 23)^{4} \)
|
| $67$ |
\( T^{4} + 4782969 \)
|
| $71$ |
\( (T^{2} - 17496)^{2} \)
|
| $73$ |
\( T^{4} + 4782969 \)
|
| $79$ |
\( (T^{2} + 10609)^{2} \)
|
| $83$ |
\( (T^{2} - 84 T + 3528)^{2} \)
|
| $89$ |
\( (T^{2} + 1944)^{2} \)
|
| $97$ |
\( T^{4} + 284765625 \)
|
| show more | |
| show less |