Newspace parameters
| Level: | \( N \) | \(=\) | \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1350.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(139.549450163\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-33})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 32x^{2} + 289 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2} \) |
| 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
\( x^{4} - 32x^{2} + 289 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} - 30\nu ) / 17 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -6\nu^{3} + 294\nu ) / 17 \)
|
| \(\beta_{3}\) | \(=\) |
\( 6\nu^{2} - 96 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta_1 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 96 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{2} + 49\beta_1 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).
| \(n\) | \(1001\) | \(1027\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 701.1 |
|
− | 2.82843i | 0 | −8.00000 | 0 | 0 | −47.7442 | 22.6274i | 0 | 0 | |||||||||||||||||||||||||||||
| 701.2 | − | 2.82843i | 0 | −8.00000 | 0 | 0 | 49.7442 | 22.6274i | 0 | 0 | ||||||||||||||||||||||||||||||
| 701.3 | 2.82843i | 0 | −8.00000 | 0 | 0 | −47.7442 | − | 22.6274i | 0 | 0 | ||||||||||||||||||||||||||||||
| 701.4 | 2.82843i | 0 | −8.00000 | 0 | 0 | 49.7442 | − | 22.6274i | 0 | 0 | ||||||||||||||||||||||||||||||
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 | 1350.5.d.e | yes | 4 |
| 3.b | odd | 2 | 1 | inner | 1350.5.d.e | yes | 4 |
| 5.b | even | 2 | 1 | 1350.5.d.d | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 1350.5.b.f | 8 | ||
| 15.d | odd | 2 | 1 | 1350.5.d.d | ✓ | 4 | |
| 15.e | even | 4 | 2 | 1350.5.b.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1350.5.b.f | 8 | 5.c | odd | 4 | 2 | ||
| 1350.5.b.f | 8 | 15.e | even | 4 | 2 | ||
| 1350.5.d.d | ✓ | 4 | 5.b | even | 2 | 1 | |
| 1350.5.d.d | ✓ | 4 | 15.d | odd | 2 | 1 | |
| 1350.5.d.e | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 1350.5.d.e | yes | 4 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} - 2T_{7} - 2375 \)
acting on \(S_{5}^{\mathrm{new}}(1350, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} - 2 T - 2375)^{2} \)
|
| $11$ |
\( T^{4} + 3672 T^{2} + 291600 \)
|
| $13$ |
\( (T^{2} + 148 T - 32540)^{2} \)
|
| $17$ |
\( T^{4} + 18720 T^{2} + 20736 \)
|
| $19$ |
\( (T^{2} + 182 T - 1223)^{2} \)
|
| $23$ |
\( T^{4} + 54360 T^{2} + 615238416 \)
|
| $29$ |
\( T^{4} + \cdots + 643101348096 \)
|
| $31$ |
\( (T^{2} + 98 T + 25)^{2} \)
|
| $37$ |
\( (T^{2} - 230 T - 521375)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 14368799984400 \)
|
| $43$ |
\( (T^{2} + 1642 T - 475943)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 39805752272400 \)
|
| $53$ |
\( T^{4} + \cdots + 64614781342224 \)
|
| $59$ |
\( T^{4} + \cdots + 2514419118864 \)
|
| $61$ |
\( (T^{2} - 1858 T + 670585)^{2} \)
|
| $67$ |
\( (T^{2} - 3500 T - 2411804)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 23324147522064 \)
|
| $73$ |
\( (T^{2} + 4414 T + 4785313)^{2} \)
|
| $79$ |
\( (T^{2} - 8782 T + 9850537)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 110015261081856 \)
|
| $89$ |
\( T^{4} + \cdots + 524452686468624 \)
|
| $97$ |
\( (T^{2} - 11042 T + 30015745)^{2} \)
|
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