Newspace parameters
| Level: | \( N \) | \(=\) | \( 6174 = 2 \cdot 3^{2} \cdot 7^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6174.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(49.2996382079\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\Q(\zeta_{14})^+\) |
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| Defining polynomial: |
\( x^{3} - x^{2} - 2x + 1 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 686) |
| 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 \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
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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 |
|
1.00000 | 0 | 1.00000 | −3.04892 | 0 | 0 | 1.00000 | 0 | −3.04892 | |||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0 | 1.00000 | 1.35690 | 0 | 0 | 1.00000 | 0 | 1.35690 | ||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0 | 1.00000 | 1.69202 | 0 | 0 | 1.00000 | 0 | 1.69202 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(7\) | \( +1 \) |
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 | 6174.2.a.k | 3 | |
| 3.b | odd | 2 | 1 | 686.2.a.a | ✓ | 3 | |
| 7.b | odd | 2 | 1 | 6174.2.a.l | 3 | ||
| 12.b | even | 2 | 1 | 5488.2.a.e | 3 | ||
| 21.c | even | 2 | 1 | 686.2.a.b | yes | 3 | |
| 21.g | even | 6 | 2 | 686.2.c.c | 6 | ||
| 21.h | odd | 6 | 2 | 686.2.c.d | 6 | ||
| 84.h | odd | 2 | 1 | 5488.2.a.b | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 686.2.a.a | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 686.2.a.b | yes | 3 | 21.c | even | 2 | 1 | |
| 686.2.c.c | 6 | 21.g | even | 6 | 2 | ||
| 686.2.c.d | 6 | 21.h | odd | 6 | 2 | ||
| 5488.2.a.b | 3 | 84.h | odd | 2 | 1 | ||
| 5488.2.a.e | 3 | 12.b | even | 2 | 1 | ||
| 6174.2.a.k | 3 | 1.a | even | 1 | 1 | trivial | |
| 6174.2.a.l | 3 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6174))\):
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\( T_{5}^{3} - 7T_{5} + 7 \)
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\( T_{11}^{3} - 21T_{11} + 7 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{3} \)
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| $3$ |
\( T^{3} \)
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| $5$ |
\( T^{3} - 7T + 7 \)
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| $7$ |
\( T^{3} \)
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| $11$ |
\( T^{3} - 21T + 7 \)
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| $13$ |
\( T^{3} + 9 T^{2} + \cdots - 113 \)
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| $17$ |
\( T^{3} - 4 T^{2} + \cdots - 13 \)
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| $19$ |
\( T^{3} + 13 T^{2} + \cdots + 43 \)
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| $23$ |
\( T^{3} + 7 T^{2} + \cdots - 91 \)
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| $29$ |
\( T^{3} + 3 T^{2} + \cdots + 1 \)
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| $31$ |
\( T^{3} + 16 T^{2} + \cdots + 41 \)
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| $37$ |
\( T^{3} - 6 T^{2} + \cdots + 27 \)
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| $41$ |
\( T^{3} - 11 T^{2} + \cdots - 13 \)
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| $43$ |
\( T^{3} + 11 T^{2} + \cdots - 29 \)
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| $47$ |
\( T^{3} + 8 T^{2} + \cdots - 1 \)
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| $53$ |
\( T^{3} - 3 T^{2} + \cdots + 783 \)
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| $59$ |
\( T^{3} + 4 T^{2} + \cdots - 211 \)
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| $61$ |
\( T^{3} + 31 T^{2} + \cdots + 911 \)
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| $67$ |
\( T^{3} - 23 T^{2} + \cdots - 307 \)
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| $71$ |
\( T^{3} + 7 T^{2} + \cdots - 637 \)
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| $73$ |
\( T^{3} + 27 T^{2} + \cdots + 239 \)
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| $79$ |
\( T^{3} + 32 T^{2} + \cdots + 1063 \)
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| $83$ |
\( T^{3} + 11 T^{2} + \cdots + 181 \)
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| $89$ |
\( T^{3} - 10 T^{2} + \cdots - 29 \)
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| $97$ |
\( T^{3} + 5 T^{2} + \cdots + 1 \)
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