Newspace parameters
| Level: | \( N \) | \(=\) | \( 578 = 2 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 578.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(92.7018478519\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 533x^{2} - 726x + 27729 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} - 2x^{3} - 533x^{2} - 726x + 27729 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 4\nu - 267 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 8\nu^{2} - 395\nu + 984 ) / 15 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} + 4\beta _1 + 267 \)
|
| \(\nu^{3}\) | \(=\) |
\( 15\beta_{3} + 24\beta_{2} + 427\beta _1 + 1152 \)
|
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 | −28.6994 | 16.0000 | −94.6218 | −114.798 | 63.2900 | 64.0000 | 580.657 | −378.487 | ||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | −11.7507 | 16.0000 | 54.8104 | −47.0026 | −214.524 | 64.0000 | −104.922 | 219.242 | |||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | 3.83700 | 16.0000 | 23.1865 | 15.3480 | 159.302 | 64.0000 | −228.277 | 92.7460 | |||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | 14.6131 | 16.0000 | −93.3752 | 58.4523 | −50.0680 | 64.0000 | −29.4577 | −373.501 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(17\) | \( -1 \) |
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 | 578.6.a.h | ✓ | 4 |
| 17.b | even | 2 | 1 | 578.6.a.j | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 578.6.a.h | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 578.6.a.j | yes | 4 | 17.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 22T_{3}^{3} - 353T_{3}^{2} - 3954T_{3} + 18909 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(578))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{4} \)
|
| $3$ |
\( T^{4} + 22 T^{3} + \cdots + 18909 \)
|
| $5$ |
\( T^{4} + 110 T^{3} + \cdots + 11228481 \)
|
| $7$ |
\( T^{4} + 42 T^{3} + \cdots + 108290901 \)
|
| $11$ |
\( T^{4} + 148 T^{3} + \cdots + 97456293 \)
|
| $13$ |
\( T^{4} + \cdots + 358222012577 \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} + \cdots + 1061107513685 \)
|
| $23$ |
\( T^{4} + \cdots + 130423589497413 \)
|
| $29$ |
\( T^{4} + \cdots + 190176270474825 \)
|
| $31$ |
\( T^{4} + \cdots + 11\!\cdots\!25 \)
|
| $37$ |
\( T^{4} + \cdots - 546405510933040 \)
|
| $41$ |
\( T^{4} + \cdots - 565348465859760 \)
|
| $43$ |
\( T^{4} + \cdots - 20\!\cdots\!75 \)
|
| $47$ |
\( T^{4} + \cdots - 692867021075883 \)
|
| $53$ |
\( T^{4} + \cdots + 13\!\cdots\!77 \)
|
| $59$ |
\( T^{4} + \cdots - 20\!\cdots\!44 \)
|
| $61$ |
\( T^{4} + \cdots + 28\!\cdots\!25 \)
|
| $67$ |
\( T^{4} + \cdots - 17\!\cdots\!96 \)
|
| $71$ |
\( T^{4} + \cdots - 10\!\cdots\!04 \)
|
| $73$ |
\( T^{4} + \cdots + 50\!\cdots\!05 \)
|
| $79$ |
\( T^{4} + \cdots - 27\!\cdots\!60 \)
|
| $83$ |
\( T^{4} + \cdots - 50\!\cdots\!27 \)
|
| $89$ |
\( T^{4} + \cdots - 20\!\cdots\!76 \)
|
| $97$ |
\( T^{4} + \cdots - 11\!\cdots\!15 \)
|
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