Newspace parameters
| Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 162.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(25.9821788097\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{25}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 6) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).
| \(n\) | \(83\) |
| \(\chi(n)\) | \(-\zeta_{6}\) |
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 |
|
2.00000 | − | 3.46410i | 0 | −8.00000 | − | 13.8564i | −33.0000 | − | 57.1577i | 0 | −88.0000 | + | 152.420i | −64.0000 | 0 | −264.000 | ||||||||||||||||
| 109.1 | 2.00000 | + | 3.46410i | 0 | −8.00000 | + | 13.8564i | −33.0000 | + | 57.1577i | 0 | −88.0000 | − | 152.420i | −64.0000 | 0 | −264.000 | |||||||||||||||||
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 | 162.6.c.h | 2 | |
| 3.b | odd | 2 | 1 | 162.6.c.e | 2 | ||
| 9.c | even | 3 | 1 | 18.6.a.b | 1 | ||
| 9.c | even | 3 | 1 | inner | 162.6.c.h | 2 | |
| 9.d | odd | 6 | 1 | 6.6.a.a | ✓ | 1 | |
| 9.d | odd | 6 | 1 | 162.6.c.e | 2 | ||
| 36.f | odd | 6 | 1 | 144.6.a.j | 1 | ||
| 36.h | even | 6 | 1 | 48.6.a.c | 1 | ||
| 45.h | odd | 6 | 1 | 150.6.a.d | 1 | ||
| 45.j | even | 6 | 1 | 450.6.a.m | 1 | ||
| 45.k | odd | 12 | 2 | 450.6.c.j | 2 | ||
| 45.l | even | 12 | 2 | 150.6.c.b | 2 | ||
| 63.i | even | 6 | 1 | 294.6.e.a | 2 | ||
| 63.j | odd | 6 | 1 | 294.6.e.g | 2 | ||
| 63.l | odd | 6 | 1 | 882.6.a.a | 1 | ||
| 63.n | odd | 6 | 1 | 294.6.e.g | 2 | ||
| 63.o | even | 6 | 1 | 294.6.a.m | 1 | ||
| 63.s | even | 6 | 1 | 294.6.e.a | 2 | ||
| 72.j | odd | 6 | 1 | 192.6.a.o | 1 | ||
| 72.l | even | 6 | 1 | 192.6.a.g | 1 | ||
| 72.n | even | 6 | 1 | 576.6.a.j | 1 | ||
| 72.p | odd | 6 | 1 | 576.6.a.i | 1 | ||
| 99.g | even | 6 | 1 | 726.6.a.a | 1 | ||
| 117.n | odd | 6 | 1 | 1014.6.a.c | 1 | ||
| 144.u | even | 12 | 2 | 768.6.d.p | 2 | ||
| 144.w | odd | 12 | 2 | 768.6.d.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6.6.a.a | ✓ | 1 | 9.d | odd | 6 | 1 | |
| 18.6.a.b | 1 | 9.c | even | 3 | 1 | ||
| 48.6.a.c | 1 | 36.h | even | 6 | 1 | ||
| 144.6.a.j | 1 | 36.f | odd | 6 | 1 | ||
| 150.6.a.d | 1 | 45.h | odd | 6 | 1 | ||
| 150.6.c.b | 2 | 45.l | even | 12 | 2 | ||
| 162.6.c.e | 2 | 3.b | odd | 2 | 1 | ||
| 162.6.c.e | 2 | 9.d | odd | 6 | 1 | ||
| 162.6.c.h | 2 | 1.a | even | 1 | 1 | trivial | |
| 162.6.c.h | 2 | 9.c | even | 3 | 1 | inner | |
| 192.6.a.g | 1 | 72.l | even | 6 | 1 | ||
| 192.6.a.o | 1 | 72.j | odd | 6 | 1 | ||
| 294.6.a.m | 1 | 63.o | even | 6 | 1 | ||
| 294.6.e.a | 2 | 63.i | even | 6 | 1 | ||
| 294.6.e.a | 2 | 63.s | even | 6 | 1 | ||
| 294.6.e.g | 2 | 63.j | odd | 6 | 1 | ||
| 294.6.e.g | 2 | 63.n | odd | 6 | 1 | ||
| 450.6.a.m | 1 | 45.j | even | 6 | 1 | ||
| 450.6.c.j | 2 | 45.k | odd | 12 | 2 | ||
| 576.6.a.i | 1 | 72.p | odd | 6 | 1 | ||
| 576.6.a.j | 1 | 72.n | even | 6 | 1 | ||
| 726.6.a.a | 1 | 99.g | even | 6 | 1 | ||
| 768.6.d.c | 2 | 144.w | odd | 12 | 2 | ||
| 768.6.d.p | 2 | 144.u | even | 12 | 2 | ||
| 882.6.a.a | 1 | 63.l | odd | 6 | 1 | ||
| 1014.6.a.c | 1 | 117.n | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 66T_{5} + 4356 \)
acting on \(S_{6}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 4T + 16 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 66T + 4356 \)
|
| $7$ |
\( T^{2} + 176T + 30976 \)
|
| $11$ |
\( T^{2} + 60T + 3600 \)
|
| $13$ |
\( T^{2} - 658T + 432964 \)
|
| $17$ |
\( (T - 414)^{2} \)
|
| $19$ |
\( (T - 956)^{2} \)
|
| $23$ |
\( T^{2} - 600T + 360000 \)
|
| $29$ |
\( T^{2} - 5574 T + 31069476 \)
|
| $31$ |
\( T^{2} - 3592 T + 12902464 \)
|
| $37$ |
\( (T + 8458)^{2} \)
|
| $41$ |
\( T^{2} - 19194 T + 368409636 \)
|
| $43$ |
\( T^{2} + 13316 T + 177315856 \)
|
| $47$ |
\( T^{2} + 19680 T + 387302400 \)
|
| $53$ |
\( (T - 31266)^{2} \)
|
| $59$ |
\( T^{2} - 26340 T + 693795600 \)
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| $61$ |
\( T^{2} - 31090 T + 966588100 \)
|
| $67$ |
\( T^{2} - 16804 T + 282374416 \)
|
| $71$ |
\( (T + 6120)^{2} \)
|
| $73$ |
\( (T + 25558)^{2} \)
|
| $79$ |
\( T^{2} + \cdots + 5536550464 \)
|
| $83$ |
\( T^{2} + 6468 T + 41835024 \)
|
| $89$ |
\( (T - 32742)^{2} \)
|
| $97$ |
\( T^{2} + \cdots + 27583230724 \)
|
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