Newspace parameters
| Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 546.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(32.2150428631\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 248x^{2} + 790x + 1756 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 248x^{2} + 790x + 1756 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -3\nu^{3} + 11\nu^{2} + 654\nu - 2840 ) / 182 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{3} - 12\nu^{2} + 1454\nu - 881 ) / 91 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} + 23\nu^{2} - 254\nu - 1959 ) / 91 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 2\beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{3} - \beta_{2} + 6\beta _1 + 127 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 131\beta_{3} + 98\beta_{2} - 334\beta _1 - 1443 ) / 3 \)
|
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 |
|
2.00000 | 3.00000 | 4.00000 | −17.8495 | 6.00000 | 7.00000 | 8.00000 | 9.00000 | −35.6990 | ||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | 3.00000 | 4.00000 | 4.19402 | 6.00000 | 7.00000 | 8.00000 | 9.00000 | 8.38804 | |||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 3.00000 | 4.00000 | 4.82722 | 6.00000 | 7.00000 | 8.00000 | 9.00000 | 9.65445 | |||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 3.00000 | 4.00000 | 18.8283 | 6.00000 | 7.00000 | 8.00000 | 9.00000 | 37.6565 | |||||||||||||||||||||||||||||||
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 | 546.4.a.q | ✓ | 4 |
| 3.b | odd | 2 | 1 | 1638.4.a.be | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.4.a.q | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1638.4.a.be | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 10T_{5}^{3} - 307T_{5}^{2} + 3012T_{5} - 6804 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(546))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{4} \)
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| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( T^{4} - 10 T^{3} + \cdots - 6804 \)
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| $7$ |
\( (T - 7)^{4} \)
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| $11$ |
\( T^{4} - 67 T^{3} + \cdots - 1081296 \)
|
| $13$ |
\( (T - 13)^{4} \)
|
| $17$ |
\( T^{4} - 65 T^{3} + \cdots - 14939568 \)
|
| $19$ |
\( T^{4} - 24 T^{3} + \cdots + 6500816 \)
|
| $23$ |
\( T^{4} - 24 T^{3} + \cdots + 153599472 \)
|
| $29$ |
\( T^{4} - 106 T^{3} + \cdots + 575667972 \)
|
| $31$ |
\( T^{4} + 35 T^{3} + \cdots + 151611264 \)
|
| $37$ |
\( T^{4} - 97 T^{3} + \cdots - 21415568 \)
|
| $41$ |
\( T^{4} + \cdots + 3696192864 \)
|
| $43$ |
\( T^{4} + \cdots + 1211942416 \)
|
| $47$ |
\( T^{4} + \cdots - 2517721920 \)
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| $53$ |
\( T^{4} - 49 T^{3} + \cdots - 356040360 \)
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| $59$ |
\( T^{4} + \cdots + 40422154752 \)
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| $61$ |
\( T^{4} + \cdots - 16114854680 \)
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| $67$ |
\( T^{4} + \cdots - 3814230592 \)
|
| $71$ |
\( T^{4} + \cdots + 159889856256 \)
|
| $73$ |
\( T^{4} + 338 T^{3} + \cdots - 191287388 \)
|
| $79$ |
\( T^{4} + \cdots - 64002938048 \)
|
| $83$ |
\( T^{4} + \cdots + 249415738992 \)
|
| $89$ |
\( T^{4} + \cdots - 1294775208 \)
|
| $97$ |
\( T^{4} + \cdots - 8259391336 \)
|
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