Newspace parameters
| Level: | \( N \) | \(=\) | \( 160 = 2^{5} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 160.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(49.9816040775\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.77052.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 63x + 183 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 5 \) |
| 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
\( x^{3} - x^{2} - 63x + 183 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{2} - 85 \)
|
| \(\beta_{2}\) | \(=\) |
\( 16\nu^{2} + 80\nu - 704 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 8\beta _1 + 24 ) / 80 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta _1 + 85 ) / 2 \)
|
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 | −42.7859 | 0 | −125.000 | 0 | −911.015 | 0 | −356.364 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 26.6186 | 0 | −125.000 | 0 | −380.850 | 0 | −1478.45 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 86.1674 | 0 | −125.000 | 0 | 1353.87 | 0 | 5237.82 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
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 | 160.8.a.g | yes | 3 |
| 4.b | odd | 2 | 1 | 160.8.a.f | ✓ | 3 | |
| 8.b | even | 2 | 1 | 320.8.a.x | 3 | ||
| 8.d | odd | 2 | 1 | 320.8.a.y | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 160.8.a.f | ✓ | 3 | 4.b | odd | 2 | 1 | |
| 160.8.a.g | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 320.8.a.x | 3 | 8.b | even | 2 | 1 | ||
| 320.8.a.y | 3 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 70T_{3}^{2} - 2532T_{3} + 98136 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(160))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 70 T^{2} + \cdots + 98136 \)
|
| $5$ |
\( (T + 125)^{3} \)
|
| $7$ |
\( T^{3} - 62 T^{2} + \cdots - 469737864 \)
|
| $11$ |
\( T^{3} + \cdots + 19547409600 \)
|
| $13$ |
\( T^{3} + \cdots + 109863915000 \)
|
| $17$ |
\( T^{3} + \cdots - 12865848318600 \)
|
| $19$ |
\( T^{3} + \cdots - 56420678540800 \)
|
| $23$ |
\( T^{3} + \cdots + 244764482088 \)
|
| $29$ |
\( T^{3} + \cdots + 574148661362408 \)
|
| $31$ |
\( T^{3} + \cdots - 26\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots - 19\!\cdots\!00 \)
|
| $41$ |
\( T^{3} + \cdots + 11\!\cdots\!00 \)
|
| $43$ |
\( T^{3} + \cdots + 67\!\cdots\!16 \)
|
| $47$ |
\( T^{3} + \cdots + 45\!\cdots\!76 \)
|
| $53$ |
\( T^{3} + \cdots - 24\!\cdots\!00 \)
|
| $59$ |
\( T^{3} + \cdots - 44\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 17\!\cdots\!00 \)
|
| $67$ |
\( T^{3} + \cdots - 42\!\cdots\!32 \)
|
| $71$ |
\( T^{3} + \cdots - 21\!\cdots\!00 \)
|
| $73$ |
\( T^{3} + \cdots + 46\!\cdots\!00 \)
|
| $79$ |
\( T^{3} + \cdots + 74\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 60\!\cdots\!60 \)
|
| $89$ |
\( T^{3} + \cdots - 30\!\cdots\!96 \)
|
| $97$ |
\( T^{3} + \cdots - 38\!\cdots\!00 \)
|
| show more | |
| show less |