Newspace parameters
| Level: | \( N \) | \(=\) | \( 6728 = 2^{3} \cdot 29^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6728.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(53.7233504799\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.8468.1 |
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| Defining polynomial: |
\( x^{4} - x^{3} - 5x^{2} + 3x + 4 \)
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| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 232) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{4} - x^{3} - 5x^{2} + 3x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 3 \)
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| \(\beta_{2}\) | \(=\) |
\( -\nu^{3} + 5\nu \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 2 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 3 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{3} + 3\beta_{2} + 5\beta _1 + 5 ) / 2 \)
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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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −3.19117 | 0 | −1.80122 | 0 | 4.55683 | 0 | 7.18356 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.57671 | 0 | 2.66740 | 0 | −3.78244 | 0 | −0.513978 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.264377 | 0 | 1.40135 | 0 | 2.63486 | 0 | −2.93011 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.50350 | 0 | −3.26753 | 0 | −1.40925 | 0 | −0.739474 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(29\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 6728.2.a.s | 4 | |
| 29.b | even | 2 | 1 | 6728.2.a.t | 4 | ||
| 29.c | odd | 4 | 2 | 232.2.e.a | ✓ | 8 | |
| 87.f | even | 4 | 2 | 2088.2.o.e | 8 | ||
| 116.e | even | 4 | 2 | 464.2.e.d | 8 | ||
| 232.k | even | 4 | 2 | 1856.2.e.i | 8 | ||
| 232.l | odd | 4 | 2 | 1856.2.e.j | 8 | ||
| 348.k | odd | 4 | 2 | 4176.2.o.r | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 232.2.e.a | ✓ | 8 | 29.c | odd | 4 | 2 | |
| 464.2.e.d | 8 | 116.e | even | 4 | 2 | ||
| 1856.2.e.i | 8 | 232.k | even | 4 | 2 | ||
| 1856.2.e.j | 8 | 232.l | odd | 4 | 2 | ||
| 2088.2.o.e | 8 | 87.f | even | 4 | 2 | ||
| 4176.2.o.r | 8 | 348.k | odd | 4 | 2 | ||
| 6728.2.a.s | 4 | 1.a | even | 1 | 1 | trivial | |
| 6728.2.a.t | 4 | 29.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6728))\):
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\( T_{3}^{4} + 3T_{3}^{3} - 3T_{3}^{2} - 7T_{3} + 2 \)
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\( T_{5}^{4} + T_{5}^{3} - 11T_{5}^{2} - 5T_{5} + 22 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} + 3 T^{3} + \cdots + 2 \)
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| $5$ |
\( T^{4} + T^{3} + \cdots + 22 \)
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| $7$ |
\( T^{4} - 2 T^{3} + \cdots + 64 \)
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| $11$ |
\( T^{4} + T^{3} - 19 T^{2} + \cdots - 2 \)
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| $13$ |
\( T^{4} - 5 T^{3} + \cdots - 118 \)
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| $17$ |
\( T^{4} - 6 T^{3} + \cdots - 256 \)
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| $19$ |
\( T^{4} - 2 T^{3} + \cdots + 32 \)
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| $23$ |
\( T^{4} + 2 T^{3} + \cdots + 64 \)
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| $29$ |
\( T^{4} \)
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| $31$ |
\( T^{4} + 11 T^{3} + \cdots + 158 \)
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| $37$ |
\( T^{4} - 12 T^{3} + \cdots + 512 \)
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| $41$ |
\( T^{4} + 6 T^{3} + \cdots - 1856 \)
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| $43$ |
\( T^{4} + 5 T^{3} + \cdots + 118 \)
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| $47$ |
\( T^{4} - 13 T^{3} + \cdots + 2 \)
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| $53$ |
\( T^{4} - 3 T^{3} + \cdots + 2 \)
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| $59$ |
\( T^{4} + 6 T^{3} + \cdots + 1696 \)
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| $61$ |
\( T^{4} + 22 T^{3} + \cdots - 6112 \)
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| $67$ |
\( T^{4} - 12 T^{3} + \cdots + 4096 \)
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| $71$ |
\( T^{4} - 192 T^{2} + \cdots - 1024 \)
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| $73$ |
\( (T + 8)^{4} \)
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| $79$ |
\( T^{4} + 17 T^{3} + \cdots - 2614 \)
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| $83$ |
\( T^{4} + 26 T^{3} + \cdots - 18272 \)
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| $89$ |
\( T^{4} + 26 T^{3} + \cdots + 704 \)
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| $97$ |
\( T^{4} + 14 T^{3} + \cdots - 128 \)
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