Newspace parameters
| Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 162.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(124.471595251\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{109})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 28x^{2} + 27x + 729 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{6} \) |
| Twist minimal: | no (minimal twist has level 54) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} + 28x^{2} + 27x + 729 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 28\nu^{2} - 28\nu + 729 ) / 756 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 28\nu^{2} + 1540\nu - 729 ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 54\nu^{3} + 2214 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 108\beta_1 ) / 216 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 5940\beta _1 - 5940 ) / 216 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{3} - 2214 ) / 54 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).
| \(n\) | \(83\) |
| \(\chi(n)\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 55.1 |
|
−16.0000 | + | 27.7128i | 0 | −512.000 | − | 886.810i | −1888.88 | − | 3271.64i | 0 | 26504.7 | − | 45907.4i | 32768.0 | 0 | 120888. | ||||||||||||||||||||||
| 55.2 | −16.0000 | + | 27.7128i | 0 | −512.000 | − | 886.810i | 3748.88 | + | 6493.26i | 0 | −23107.7 | + | 40023.7i | 32768.0 | 0 | −239928. | |||||||||||||||||||||||
| 109.1 | −16.0000 | − | 27.7128i | 0 | −512.000 | + | 886.810i | −1888.88 | + | 3271.64i | 0 | 26504.7 | + | 45907.4i | 32768.0 | 0 | 120888. | |||||||||||||||||||||||
| 109.2 | −16.0000 | − | 27.7128i | 0 | −512.000 | + | 886.810i | 3748.88 | − | 6493.26i | 0 | −23107.7 | − | 40023.7i | 32768.0 | 0 | −239928. | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 162.12.c.l | 4 | |
| 3.b | odd | 2 | 1 | 162.12.c.q | 4 | ||
| 9.c | even | 3 | 1 | 54.12.a.g | yes | 2 | |
| 9.c | even | 3 | 1 | inner | 162.12.c.l | 4 | |
| 9.d | odd | 6 | 1 | 54.12.a.d | ✓ | 2 | |
| 9.d | odd | 6 | 1 | 162.12.c.q | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 54.12.a.d | ✓ | 2 | 9.d | odd | 6 | 1 | |
| 54.12.a.g | yes | 2 | 9.c | even | 3 | 1 | |
| 162.12.c.l | 4 | 1.a | even | 1 | 1 | trivial | |
| 162.12.c.l | 4 | 9.c | even | 3 | 1 | inner | |
| 162.12.c.q | 4 | 3.b | odd | 2 | 1 | ||
| 162.12.c.q | 4 | 9.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 3720T_{5}^{3} + 42163200T_{5}^{2} + 105368256000T_{5} + 802294295040000 \)
acting on \(S_{12}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 32 T + 1024)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 802294295040000 \)
|
| $7$ |
\( T^{4} + \cdots + 60\!\cdots\!29 \)
|
| $11$ |
\( T^{4} + \cdots + 38\!\cdots\!00 \)
|
| $13$ |
\( T^{4} + \cdots + 93\!\cdots\!25 \)
|
| $17$ |
\( (T^{2} + \cdots - 16856880103872)^{2} \)
|
| $19$ |
\( (T^{2} + \cdots + 102225689174569)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 36\!\cdots\!56 \)
|
| $29$ |
\( T^{4} + \cdots + 31\!\cdots\!44 \)
|
| $31$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
|
| $37$ |
\( (T^{2} + \cdots - 20\!\cdots\!75)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 28\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots + 12\!\cdots\!16 \)
|
| $47$ |
\( T^{4} + \cdots + 23\!\cdots\!84 \)
|
| $53$ |
\( (T^{2} + \cdots + 50\!\cdots\!00)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 29\!\cdots\!64 \)
|
| $61$ |
\( T^{4} + \cdots + 28\!\cdots\!81 \)
|
| $67$ |
\( T^{4} + \cdots + 16\!\cdots\!61 \)
|
| $71$ |
\( (T^{2} + \cdots - 11\!\cdots\!92)^{2} \)
|
| $73$ |
\( (T^{2} + \cdots - 16\!\cdots\!95)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 28\!\cdots\!89 \)
|
| $83$ |
\( T^{4} + \cdots + 15\!\cdots\!76 \)
|
| $89$ |
\( (T^{2} + \cdots - 28\!\cdots\!92)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 43\!\cdots\!25 \)
|
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