Newspace parameters
| Level: | \( N \) | \(=\) | \( 1456 = 2^{4} \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1456.r (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(11.6262185343\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{7})\) |
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|
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| Defining polynomial: |
\( x^{4} + 7x^{2} + 49 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 728) |
| 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
\( x^{4} + 7x^{2} + 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 7 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( 7\beta_{2} \)
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| \(\nu^{3}\) | \(=\) |
\( 7\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).
| \(n\) | \(561\) | \(911\) | \(1093\) | \(1249\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 417.1 |
|
0 | 1.00000 | − | 1.73205i | 0 | 0 | 0 | −1.32288 | − | 2.29129i | 0 | −0.500000 | − | 0.866025i | 0 | ||||||||||||||||||||||||
| 417.2 | 0 | 1.00000 | − | 1.73205i | 0 | 0 | 0 | 1.32288 | + | 2.29129i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||
| 625.1 | 0 | 1.00000 | + | 1.73205i | 0 | 0 | 0 | −1.32288 | + | 2.29129i | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||
| 625.2 | 0 | 1.00000 | + | 1.73205i | 0 | 0 | 0 | 1.32288 | − | 2.29129i | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1456.2.r.k | 4 | |
| 4.b | odd | 2 | 1 | 728.2.r.d | ✓ | 4 | |
| 7.c | even | 3 | 1 | inner | 1456.2.r.k | 4 | |
| 28.f | even | 6 | 1 | 5096.2.a.l | 2 | ||
| 28.g | odd | 6 | 1 | 728.2.r.d | ✓ | 4 | |
| 28.g | odd | 6 | 1 | 5096.2.a.p | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 728.2.r.d | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 728.2.r.d | ✓ | 4 | 28.g | odd | 6 | 1 | |
| 1456.2.r.k | 4 | 1.a | even | 1 | 1 | trivial | |
| 1456.2.r.k | 4 | 7.c | even | 3 | 1 | inner | |
| 5096.2.a.l | 2 | 28.f | even | 6 | 1 | ||
| 5096.2.a.p | 2 | 28.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1456, [\chi])\):
|
\( T_{3}^{2} - 2T_{3} + 4 \)
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\( T_{5} \)
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\( T_{11}^{4} + 4T_{11}^{3} + 19T_{11}^{2} - 12T_{11} + 9 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( (T^{2} - 2 T + 4)^{2} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} + 7T^{2} + 49 \)
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| $11$ |
\( T^{4} + 4 T^{3} + \cdots + 9 \)
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| $13$ |
\( (T + 1)^{4} \)
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| $17$ |
\( (T^{2} + T + 1)^{2} \)
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| $19$ |
\( T^{4} + 4 T^{3} + \cdots + 9 \)
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| $23$ |
\( T^{4} - 4 T^{3} + \cdots + 576 \)
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| $29$ |
\( (T^{2} + 2 T - 27)^{2} \)
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| $31$ |
\( T^{4} + 8 T^{3} + \cdots + 144 \)
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| $37$ |
\( T^{4} - 8 T^{3} + \cdots + 144 \)
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| $41$ |
\( (T^{2} + 8 T - 12)^{2} \)
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| $43$ |
\( (T^{2} + 4 T - 24)^{2} \)
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| $47$ |
\( T^{4} + 16 T^{3} + \cdots + 3249 \)
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| $53$ |
\( T^{4} - 10 T^{3} + \cdots + 9 \)
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| $59$ |
\( T^{4} - 4 T^{3} + \cdots + 9 \)
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| $61$ |
\( T^{4} + 6 T^{3} + \cdots + 361 \)
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| $67$ |
\( T^{4} + 4 T^{3} + \cdots + 9 \)
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| $71$ |
\( (T^{2} - 8 T - 47)^{2} \)
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| $73$ |
\( (T^{2} + 2 T + 4)^{2} \)
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| $79$ |
\( T^{4} - 16 T^{3} + \cdots + 1296 \)
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| $83$ |
\( (T^{2} - 28)^{2} \)
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| $89$ |
\( T^{4} + 28T^{2} + 784 \)
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| $97$ |
\( (T^{2} - 4 T - 248)^{2} \)
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