Newspace parameters
| Level: | \( N \) | \(=\) | \( 6321 = 3 \cdot 7^{2} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6321.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(50.4734391177\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.568.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 6x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 129) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 6x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 4 \)
|
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 |
|
−2.76156 | −1.00000 | 5.62620 | −0.761557 | 2.76156 | 0 | −10.0140 | 1.00000 | 2.10308 | |||||||||||||||||||||||||||
| 1.2 | −1.36333 | −1.00000 | −0.141336 | 0.636672 | 1.36333 | 0 | 2.91934 | 1.00000 | −0.867993 | ||||||||||||||||||||||||||||
| 1.3 | 2.12489 | −1.00000 | 2.51514 | 4.12489 | −2.12489 | 0 | 1.09461 | 1.00000 | 8.76491 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(7\) | \( -1 \) |
| \(43\) | \( +1 \) |
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 | 6321.2.a.p | 3 | |
| 7.b | odd | 2 | 1 | 129.2.a.d | ✓ | 3 | |
| 21.c | even | 2 | 1 | 387.2.a.i | 3 | ||
| 28.d | even | 2 | 1 | 2064.2.a.x | 3 | ||
| 35.c | odd | 2 | 1 | 3225.2.a.t | 3 | ||
| 56.e | even | 2 | 1 | 8256.2.a.cu | 3 | ||
| 56.h | odd | 2 | 1 | 8256.2.a.cr | 3 | ||
| 84.h | odd | 2 | 1 | 6192.2.a.bw | 3 | ||
| 105.g | even | 2 | 1 | 9675.2.a.bq | 3 | ||
| 301.c | even | 2 | 1 | 5547.2.a.p | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 129.2.a.d | ✓ | 3 | 7.b | odd | 2 | 1 | |
| 387.2.a.i | 3 | 21.c | even | 2 | 1 | ||
| 2064.2.a.x | 3 | 28.d | even | 2 | 1 | ||
| 3225.2.a.t | 3 | 35.c | odd | 2 | 1 | ||
| 5547.2.a.p | 3 | 301.c | even | 2 | 1 | ||
| 6192.2.a.bw | 3 | 84.h | odd | 2 | 1 | ||
| 6321.2.a.p | 3 | 1.a | even | 1 | 1 | trivial | |
| 8256.2.a.cr | 3 | 56.h | odd | 2 | 1 | ||
| 8256.2.a.cu | 3 | 56.e | even | 2 | 1 | ||
| 9675.2.a.bq | 3 | 105.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6321))\):
|
\( T_{2}^{3} + 2T_{2}^{2} - 5T_{2} - 8 \)
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\( T_{5}^{3} - 4T_{5}^{2} - T_{5} + 2 \)
|
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\( T_{13} + 3 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 2 T^{2} + \cdots - 8 \)
|
| $3$ |
\( (T + 1)^{3} \)
|
| $5$ |
\( T^{3} - 4T^{2} - T + 2 \)
|
| $7$ |
\( T^{3} \)
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| $11$ |
\( T^{3} - T^{2} + \cdots - 25 \)
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| $13$ |
\( (T + 3)^{3} \)
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| $17$ |
\( T^{3} + T^{2} - 8T - 4 \)
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| $19$ |
\( T^{3} - 4 T^{2} + \cdots + 2 \)
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| $23$ |
\( T^{3} - 11 T^{2} + \cdots + 452 \)
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| $29$ |
\( T^{3} - 2 T^{2} + \cdots + 8 \)
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| $31$ |
\( T^{3} - 5 T^{2} + \cdots + 64 \)
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| $37$ |
\( T^{3} - 40T + 64 \)
|
| $41$ |
\( T^{3} - 15 T^{2} + \cdots + 32 \)
|
| $43$ |
\( (T + 1)^{3} \)
|
| $47$ |
\( T^{3} - 2 T^{2} + \cdots + 664 \)
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| $53$ |
\( T^{3} + 5 T^{2} + \cdots - 64 \)
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| $59$ |
\( T^{3} + 8 T^{2} + \cdots - 80 \)
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| $61$ |
\( T^{3} - 16 T^{2} + \cdots + 512 \)
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| $67$ |
\( T^{3} + 11 T^{2} + \cdots - 332 \)
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| $71$ |
\( T^{3} - 22 T^{2} + \cdots + 424 \)
|
| $73$ |
\( T^{3} - 16 T^{2} + \cdots + 16 \)
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| $79$ |
\( T^{3} - 24 T^{2} + \cdots - 256 \)
|
| $83$ |
\( T^{3} - 7 T^{2} + \cdots + 485 \)
|
| $89$ |
\( T^{3} - 38 T^{2} + \cdots - 1744 \)
|
| $97$ |
\( T^{3} + T^{2} + \cdots - 277 \)
|
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