Newspace parameters
| Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 162.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(83.4358054585\) |
| 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 2) |
| 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 |
|
8.00000 | − | 13.8564i | 0 | −128.000 | − | 221.703i | 435.000 | + | 753.442i | 0 | 476.000 | − | 824.456i | −4096.00 | 0 | 13920.0 | ||||||||||||||||
| 109.1 | 8.00000 | + | 13.8564i | 0 | −128.000 | + | 221.703i | 435.000 | − | 753.442i | 0 | 476.000 | + | 824.456i | −4096.00 | 0 | 13920.0 | |||||||||||||||||
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 | 162.10.c.i | 2 | |
| 3.b | odd | 2 | 1 | 162.10.c.b | 2 | ||
| 9.c | even | 3 | 1 | 18.10.a.a | 1 | ||
| 9.c | even | 3 | 1 | inner | 162.10.c.i | 2 | |
| 9.d | odd | 6 | 1 | 2.10.a.a | ✓ | 1 | |
| 9.d | odd | 6 | 1 | 162.10.c.b | 2 | ||
| 36.f | odd | 6 | 1 | 144.10.a.d | 1 | ||
| 36.h | even | 6 | 1 | 16.10.a.d | 1 | ||
| 45.h | odd | 6 | 1 | 50.10.a.c | 1 | ||
| 45.l | even | 12 | 2 | 50.10.b.a | 2 | ||
| 63.i | even | 6 | 1 | 98.10.c.b | 2 | ||
| 63.j | odd | 6 | 1 | 98.10.c.c | 2 | ||
| 63.n | odd | 6 | 1 | 98.10.c.c | 2 | ||
| 63.o | even | 6 | 1 | 98.10.a.c | 1 | ||
| 63.s | even | 6 | 1 | 98.10.c.b | 2 | ||
| 72.j | odd | 6 | 1 | 64.10.a.h | 1 | ||
| 72.l | even | 6 | 1 | 64.10.a.b | 1 | ||
| 99.g | even | 6 | 1 | 242.10.a.a | 1 | ||
| 117.n | odd | 6 | 1 | 338.10.a.a | 1 | ||
| 144.u | even | 12 | 2 | 256.10.b.e | 2 | ||
| 144.w | odd | 12 | 2 | 256.10.b.g | 2 | ||
| 180.n | even | 6 | 1 | 400.10.a.b | 1 | ||
| 180.v | odd | 12 | 2 | 400.10.c.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.10.a.a | ✓ | 1 | 9.d | odd | 6 | 1 | |
| 16.10.a.d | 1 | 36.h | even | 6 | 1 | ||
| 18.10.a.a | 1 | 9.c | even | 3 | 1 | ||
| 50.10.a.c | 1 | 45.h | odd | 6 | 1 | ||
| 50.10.b.a | 2 | 45.l | even | 12 | 2 | ||
| 64.10.a.b | 1 | 72.l | even | 6 | 1 | ||
| 64.10.a.h | 1 | 72.j | odd | 6 | 1 | ||
| 98.10.a.c | 1 | 63.o | even | 6 | 1 | ||
| 98.10.c.b | 2 | 63.i | even | 6 | 1 | ||
| 98.10.c.b | 2 | 63.s | even | 6 | 1 | ||
| 98.10.c.c | 2 | 63.j | odd | 6 | 1 | ||
| 98.10.c.c | 2 | 63.n | odd | 6 | 1 | ||
| 144.10.a.d | 1 | 36.f | odd | 6 | 1 | ||
| 162.10.c.b | 2 | 3.b | odd | 2 | 1 | ||
| 162.10.c.b | 2 | 9.d | odd | 6 | 1 | ||
| 162.10.c.i | 2 | 1.a | even | 1 | 1 | trivial | |
| 162.10.c.i | 2 | 9.c | even | 3 | 1 | inner | |
| 242.10.a.a | 1 | 99.g | even | 6 | 1 | ||
| 256.10.b.e | 2 | 144.u | even | 12 | 2 | ||
| 256.10.b.g | 2 | 144.w | odd | 12 | 2 | ||
| 338.10.a.a | 1 | 117.n | odd | 6 | 1 | ||
| 400.10.a.b | 1 | 180.n | even | 6 | 1 | ||
| 400.10.c.d | 2 | 180.v | odd | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 870T_{5} + 756900 \)
acting on \(S_{10}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 16T + 256 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 870T + 756900 \)
|
| $7$ |
\( T^{2} - 952T + 906304 \)
|
| $11$ |
\( T^{2} + \cdots + 3152597904 \)
|
| $13$ |
\( T^{2} + \cdots + 31717472836 \)
|
| $17$ |
\( (T - 247662)^{2} \)
|
| $19$ |
\( (T - 315380)^{2} \)
|
| $23$ |
\( T^{2} + \cdots + 41821886016 \)
|
| $29$ |
\( T^{2} + \cdots + 14749056202500 \)
|
| $31$ |
\( T^{2} + \cdots + 1714549310464 \)
|
| $37$ |
\( (T - 4307078)^{2} \)
|
| $41$ |
\( T^{2} + \cdots + 2286271009764 \)
|
| $43$ |
\( T^{2} + \cdots + 11\!\cdots\!16 \)
|
| $47$ |
\( T^{2} + \cdots + 111959084669184 \)
|
| $53$ |
\( (T + 16616214)^{2} \)
|
| $59$ |
\( T^{2} + \cdots + 12\!\cdots\!00 \)
|
| $61$ |
\( T^{2} + \cdots + 11\!\cdots\!24 \)
|
| $67$ |
\( T^{2} + \cdots + 14\!\cdots\!04 \)
|
| $71$ |
\( (T - 387172728)^{2} \)
|
| $73$ |
\( (T - 255240074)^{2} \)
|
| $79$ |
\( T^{2} + \cdots + 24\!\cdots\!00 \)
|
| $83$ |
\( T^{2} + \cdots + 20\!\cdots\!96 \)
|
| $89$ |
\( (T - 31809510)^{2} \)
|
| $97$ |
\( T^{2} + \cdots + 45\!\cdots\!44 \)
|
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