Newspace parameters
| Level: | \( N \) | \(=\) | \( 52 = 2^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 52.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(16.2440213252\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{3} - 1777x - 25896 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 1777x - 25896 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} - 7\nu - 1188 ) / 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 27\nu - 1188 ) / 10 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -7\beta_{2} + 27\beta _1 + 2376 ) / 2 \)
|
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 |
|
0 | −70.0379 | 0 | 89.0248 | 0 | −1705.69 | 0 | 2718.31 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 13.8979 | 0 | 454.815 | 0 | 723.875 | 0 | −1993.85 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 84.1400 | 0 | −23.8394 | 0 | 245.814 | 0 | 4892.54 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( -1 \) |
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 | 52.8.a.c | ✓ | 3 |
| 3.b | odd | 2 | 1 | 468.8.a.e | 3 | ||
| 4.b | odd | 2 | 1 | 208.8.a.i | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 52.8.a.c | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 208.8.a.i | 3 | 4.b | odd | 2 | 1 | ||
| 468.8.a.e | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 28T_{3}^{2} - 5697T_{3} + 81900 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(52))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 28 T^{2} + \cdots + 81900 \)
|
| $5$ |
\( T^{3} - 520 T^{2} + \cdots + 965250 \)
|
| $7$ |
\( T^{3} + 736 T^{2} + \cdots + 303508232 \)
|
| $11$ |
\( T^{3} + \cdots - 10225397232 \)
|
| $13$ |
\( (T - 2197)^{3} \)
|
| $17$ |
\( T^{3} + \cdots + 10492423160958 \)
|
| $19$ |
\( T^{3} + \cdots - 2718238288336 \)
|
| $23$ |
\( T^{3} + \cdots - 6944593065984 \)
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| $29$ |
\( T^{3} + \cdots + 241960594897224 \)
|
| $31$ |
\( T^{3} + \cdots - 785478510713600 \)
|
| $37$ |
\( T^{3} + \cdots + 16\!\cdots\!50 \)
|
| $41$ |
\( T^{3} + \cdots - 26\!\cdots\!84 \)
|
| $43$ |
\( T^{3} + \cdots - 16\!\cdots\!52 \)
|
| $47$ |
\( T^{3} + \cdots + 75\!\cdots\!28 \)
|
| $53$ |
\( T^{3} + \cdots - 35\!\cdots\!96 \)
|
| $59$ |
\( T^{3} + \cdots + 37\!\cdots\!80 \)
|
| $61$ |
\( T^{3} + \cdots - 20\!\cdots\!24 \)
|
| $67$ |
\( T^{3} + \cdots + 11\!\cdots\!00 \)
|
| $71$ |
\( T^{3} + \cdots - 40\!\cdots\!20 \)
|
| $73$ |
\( T^{3} + \cdots - 30\!\cdots\!12 \)
|
| $79$ |
\( T^{3} + \cdots + 89\!\cdots\!32 \)
|
| $83$ |
\( T^{3} + \cdots + 67\!\cdots\!24 \)
|
| $89$ |
\( T^{3} + \cdots - 13\!\cdots\!24 \)
|
| $97$ |
\( T^{3} + \cdots - 36\!\cdots\!32 \)
|
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