Newspace parameters
| Level: | \( N \) | \(=\) | \( 1053 = 3^{4} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1053.e (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(8.40824733284\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{13})\) |
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|
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| Defining polynomial: |
\( x^{4} - x^{3} + 4x^{2} + 3x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 351) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} + 4x^{2} + 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 7 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 3\beta_{2} + \beta _1 - 1 \)
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| \(\nu^{3}\) | \(=\) |
\( 4\beta_{3} - 7 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times\).
| \(n\) | \(326\) | \(730\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 352.1 |
|
−0.651388 | − | 1.12824i | 0 | 0.151388 | − | 0.262211i | 2.15139 | − | 3.72631i | 0 | 0.500000 | + | 0.866025i | −3.00000 | 0 | −5.60555 | ||||||||||||||||||||||
| 352.2 | 1.15139 | + | 1.99426i | 0 | −1.65139 | + | 2.86029i | 0.348612 | − | 0.603814i | 0 | 0.500000 | + | 0.866025i | −3.00000 | 0 | 1.60555 | |||||||||||||||||||||||
| 703.1 | −0.651388 | + | 1.12824i | 0 | 0.151388 | + | 0.262211i | 2.15139 | + | 3.72631i | 0 | 0.500000 | − | 0.866025i | −3.00000 | 0 | −5.60555 | |||||||||||||||||||||||
| 703.2 | 1.15139 | − | 1.99426i | 0 | −1.65139 | − | 2.86029i | 0.348612 | + | 0.603814i | 0 | 0.500000 | − | 0.866025i | −3.00000 | 0 | 1.60555 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1053.2.e.j | 4 | |
| 3.b | odd | 2 | 1 | 1053.2.e.f | 4 | ||
| 9.c | even | 3 | 1 | 351.2.a.b | ✓ | 2 | |
| 9.c | even | 3 | 1 | inner | 1053.2.e.j | 4 | |
| 9.d | odd | 6 | 1 | 351.2.a.d | yes | 2 | |
| 9.d | odd | 6 | 1 | 1053.2.e.f | 4 | ||
| 36.f | odd | 6 | 1 | 5616.2.a.bi | 2 | ||
| 36.h | even | 6 | 1 | 5616.2.a.bz | 2 | ||
| 45.h | odd | 6 | 1 | 8775.2.a.w | 2 | ||
| 45.j | even | 6 | 1 | 8775.2.a.bd | 2 | ||
| 117.n | odd | 6 | 1 | 4563.2.a.i | 2 | ||
| 117.t | even | 6 | 1 | 4563.2.a.q | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 351.2.a.b | ✓ | 2 | 9.c | even | 3 | 1 | |
| 351.2.a.d | yes | 2 | 9.d | odd | 6 | 1 | |
| 1053.2.e.f | 4 | 3.b | odd | 2 | 1 | ||
| 1053.2.e.f | 4 | 9.d | odd | 6 | 1 | ||
| 1053.2.e.j | 4 | 1.a | even | 1 | 1 | trivial | |
| 1053.2.e.j | 4 | 9.c | even | 3 | 1 | inner | |
| 4563.2.a.i | 2 | 117.n | odd | 6 | 1 | ||
| 4563.2.a.q | 2 | 117.t | even | 6 | 1 | ||
| 5616.2.a.bi | 2 | 36.f | odd | 6 | 1 | ||
| 5616.2.a.bz | 2 | 36.h | even | 6 | 1 | ||
| 8775.2.a.w | 2 | 45.h | odd | 6 | 1 | ||
| 8775.2.a.bd | 2 | 45.j | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1053, [\chi])\):
|
\( T_{2}^{4} - T_{2}^{3} + 4T_{2}^{2} + 3T_{2} + 9 \)
|
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\( T_{5}^{4} - 5T_{5}^{3} + 22T_{5}^{2} - 15T_{5} + 9 \)
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\( T_{7}^{2} - T_{7} + 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} + 4 T^{2} + \cdots + 9 \)
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| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 5 T^{3} + \cdots + 9 \)
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| $7$ |
\( (T^{2} - T + 1)^{2} \)
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| $11$ |
\( T^{4} - T^{3} + 4 T^{2} + \cdots + 9 \)
|
| $13$ |
\( (T^{2} + T + 1)^{2} \)
|
| $17$ |
\( (T^{2} + 5 T + 3)^{2} \)
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| $19$ |
\( (T^{2} + 7 T + 9)^{2} \)
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| $23$ |
\( T^{4} + 3 T^{3} + \cdots + 729 \)
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| $29$ |
\( T^{4} - 8 T^{3} + \cdots + 9 \)
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| $31$ |
\( T^{4} + 52T^{2} + 2704 \)
|
| $37$ |
\( (T^{2} + 6 T - 43)^{2} \)
|
| $41$ |
\( (T^{2} - 9 T + 81)^{2} \)
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| $43$ |
\( T^{4} + 5 T^{3} + \cdots + 9 \)
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| $47$ |
\( T^{4} - 9 T^{3} + \cdots + 81 \)
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| $53$ |
\( (T^{2} - 15 T + 27)^{2} \)
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| $59$ |
\( (T^{2} - 3 T + 9)^{2} \)
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| $61$ |
\( T^{4} + 12 T^{3} + \cdots + 529 \)
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| $67$ |
\( T^{4} - 9 T^{3} + \cdots + 3721 \)
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| $71$ |
\( (T^{2} + 6 T - 108)^{2} \)
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| $73$ |
\( (T^{2} + 11 T + 1)^{2} \)
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| $79$ |
\( T^{4} - 17 T^{3} + \cdots + 1849 \)
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| $83$ |
\( T^{4} + 8 T^{3} + \cdots + 9 \)
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| $89$ |
\( (T^{2} + 10 T + 12)^{2} \)
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| $97$ |
\( T^{4} + 208 T^{2} + 43264 \)
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