Newspace parameters
| Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 546.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(87.5695656179\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{3} - x^{2} - 1004x + 12000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2\cdot 5 \) |
| Twist minimal: | yes |
| 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
\( x^{3} - x^{2} - 1004x + 12000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 19\nu - 676 ) / 4 \)
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| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{2} + \nu + 670 ) / 2 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 + 3 ) / 10 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -19\beta_{2} + 2\beta _1 + 6703 ) / 10 \)
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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 |
|
−4.00000 | 9.00000 | 16.0000 | −92.4513 | −36.0000 | −49.0000 | −64.0000 | 81.0000 | 369.805 | |||||||||||||||||||||||||||
| 1.2 | −4.00000 | 9.00000 | 16.0000 | −23.8596 | −36.0000 | −49.0000 | −64.0000 | 81.0000 | 95.4385 | ||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 9.00000 | 16.0000 | −0.689076 | −36.0000 | −49.0000 | −64.0000 | 81.0000 | 2.75630 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(7\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 546.6.a.o | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.6.a.o | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(546))\):
|
\( T_{5}^{3} + 117T_{5}^{2} + 2286T_{5} + 1520 \)
|
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\( T_{11}^{3} + 632T_{11}^{2} + 41356T_{11} - 9902208 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{3} \)
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| $3$ |
\( (T - 9)^{3} \)
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| $5$ |
\( T^{3} + 117 T^{2} + \cdots + 1520 \)
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| $7$ |
\( (T + 49)^{3} \)
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| $11$ |
\( T^{3} + 632 T^{2} + \cdots - 9902208 \)
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| $13$ |
\( (T + 169)^{3} \)
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| $17$ |
\( T^{3} + \cdots + 2067277440 \)
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| $19$ |
\( T^{3} + \cdots - 1089896448 \)
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| $23$ |
\( T^{3} + \cdots + 1732557408 \)
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| $29$ |
\( T^{3} + \cdots + 200638595140 \)
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| $31$ |
\( T^{3} + \cdots - 77101252000 \)
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| $37$ |
\( T^{3} + \cdots - 119229184272 \)
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| $41$ |
\( T^{3} + \cdots - 1279232070528 \)
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| $43$ |
\( T^{3} + \cdots - 51443137328 \)
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| $47$ |
\( T^{3} + \cdots - 124495349348 \)
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| $53$ |
\( T^{3} + \cdots + 7087134449628 \)
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| $59$ |
\( T^{3} + \cdots + 7926436627200 \)
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| $61$ |
\( T^{3} + \cdots + 10585536564200 \)
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| $67$ |
\( T^{3} + \cdots + 1006647344800 \)
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| $71$ |
\( T^{3} + \cdots - 10188778270208 \)
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| $73$ |
\( T^{3} + \cdots - 14795972282100 \)
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| $79$ |
\( T^{3} + \cdots + 7571487500544 \)
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| $83$ |
\( T^{3} + \cdots + 279849178080468 \)
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| $89$ |
\( T^{3} + \cdots + 153649191648408 \)
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| $97$ |
\( T^{3} + \cdots + 56787100422068 \)
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