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: | \(3\) |
| Coefficient field: | 3.3.1600113.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - 249x - 392 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{3} - 249x - 392 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 164 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 6\beta_{2} + \beta _1 + 164 \)
|
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 | −14.9243 | 6.00000 | −7.00000 | −8.00000 | 9.00000 | 29.8486 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | −3.00000 | 4.00000 | −1.59045 | 6.00000 | −7.00000 | −8.00000 | 9.00000 | 3.18091 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −3.00000 | 4.00000 | 16.5147 | 6.00000 | −7.00000 | −8.00000 | 9.00000 | −33.0295 | ||||||||||||||||||||||||||||
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.m | ✓ | 3 |
| 3.b | odd | 2 | 1 | 1638.4.a.bb | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.4.a.m | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1638.4.a.bb | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 249T_{5} - 392 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(546))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( (T + 3)^{3} \)
|
| $5$ |
\( T^{3} - 249T - 392 \)
|
| $7$ |
\( (T + 7)^{3} \)
|
| $11$ |
\( T^{3} - 11 T^{2} + \cdots + 90276 \)
|
| $13$ |
\( (T + 13)^{3} \)
|
| $17$ |
\( T^{3} + 23 T^{2} + \cdots - 401136 \)
|
| $19$ |
\( T^{3} + 38 T^{2} + \cdots - 54960 \)
|
| $23$ |
\( T^{3} + 44 T^{2} + \cdots - 693984 \)
|
| $29$ |
\( T^{3} + 328 T^{2} + \cdots - 9415330 \)
|
| $31$ |
\( T^{3} - 159 T^{2} + \cdots + 3244544 \)
|
| $37$ |
\( T^{3} + 227 T^{2} + \cdots - 10740408 \)
|
| $41$ |
\( T^{3} + 10 T^{2} + \cdots - 380352 \)
|
| $43$ |
\( T^{3} - 10 T^{2} + \cdots - 7537124 \)
|
| $47$ |
\( T^{3} + 309 T^{2} + \cdots - 12122992 \)
|
| $53$ |
\( T^{3} + 849 T^{2} + \cdots + 12064716 \)
|
| $59$ |
\( T^{3} - 34 T^{2} + \cdots - 136093440 \)
|
| $61$ |
\( T^{3} - 201 T^{2} + \cdots - 33416452 \)
|
| $67$ |
\( T^{3} + 308 T^{2} + \cdots + 5195248 \)
|
| $71$ |
\( T^{3} + 66 T^{2} + \cdots - 213395392 \)
|
| $73$ |
\( T^{3} - 1270 T^{2} + \cdots - 47910282 \)
|
| $79$ |
\( T^{3} - 2947 T^{2} + \cdots - 887673120 \)
|
| $83$ |
\( T^{3} - 1505 T^{2} + \cdots + 663722676 \)
|
| $89$ |
\( T^{3} + \cdots + 2003051400 \)
|
| $97$ |
\( T^{3} + \cdots - 4407218284 \)
|
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