Newspace parameters
| Level: | \( N \) | \(=\) | \( 156 = 2^{2} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 156.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(48.7320639755\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 11078x - 379248 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 11078x - 379248 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} - 29\nu - 7371 ) / 15 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{2} + 89\nu + 7356 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 3 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 29\beta_{2} + 267\beta _1 + 88539 ) / 12 \)
|
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 | −27.0000 | 0 | −59.3952 | 0 | −23.1416 | 0 | 729.000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −27.0000 | 0 | 103.038 | 0 | 1545.57 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | −27.0000 | 0 | 480.358 | 0 | −800.424 | 0 | 729.000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 156.8.a.c | ✓ | 3 |
| 3.b | odd | 2 | 1 | 468.8.a.d | 3 | ||
| 4.b | odd | 2 | 1 | 624.8.a.m | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 156.8.a.c | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 468.8.a.d | 3 | 3.b | odd | 2 | 1 | ||
| 624.8.a.m | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 524T_{5}^{2} + 14844T_{5} + 2939760 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(156))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T + 27)^{3} \)
|
| $5$ |
\( T^{3} - 524 T^{2} + \cdots + 2939760 \)
|
| $7$ |
\( T^{3} - 722 T^{2} + \cdots - 28628704 \)
|
| $11$ |
\( T^{3} + \cdots + 45830002176 \)
|
| $13$ |
\( (T - 2197)^{3} \)
|
| $17$ |
\( T^{3} + \cdots - 3211140134664 \)
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| $19$ |
\( T^{3} + \cdots + 311539185920 \)
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| $23$ |
\( T^{3} + \cdots - 87018827710464 \)
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| $29$ |
\( T^{3} + \cdots - 12\!\cdots\!48 \)
|
| $31$ |
\( T^{3} + \cdots + 516613206809440 \)
|
| $37$ |
\( T^{3} + \cdots + 36\!\cdots\!96 \)
|
| $41$ |
\( T^{3} + \cdots + 46\!\cdots\!80 \)
|
| $43$ |
\( T^{3} + \cdots + 38\!\cdots\!16 \)
|
| $47$ |
\( T^{3} + \cdots + 12\!\cdots\!40 \)
|
| $53$ |
\( T^{3} + \cdots + 28\!\cdots\!20 \)
|
| $59$ |
\( T^{3} + \cdots + 11\!\cdots\!12 \)
|
| $61$ |
\( T^{3} + \cdots - 36\!\cdots\!44 \)
|
| $67$ |
\( T^{3} + \cdots - 96\!\cdots\!44 \)
|
| $71$ |
\( T^{3} + \cdots - 70\!\cdots\!48 \)
|
| $73$ |
\( T^{3} + \cdots + 95\!\cdots\!84 \)
|
| $79$ |
\( T^{3} + \cdots + 18\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 41\!\cdots\!20 \)
|
| $89$ |
\( T^{3} + \cdots + 49\!\cdots\!52 \)
|
| $97$ |
\( T^{3} + \cdots + 87\!\cdots\!32 \)
|
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