Newspace parameters
Level: | \( N \) | \(=\) | \( 162 = 2 \cdot 3^{4} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 162.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(9.55830942093\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
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 |
|
−1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 10.5000 | + | 18.1865i | 0 | −4.00000 | + | 6.92820i | 8.00000 | 0 | −42.0000 | ||||||||||||||||
109.1 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 10.5000 | − | 18.1865i | 0 | −4.00000 | − | 6.92820i | 8.00000 | 0 | −42.0000 | |||||||||||||||||
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 | 162.4.c.d | 2 | |
3.b | odd | 2 | 1 | 162.4.c.e | 2 | ||
9.c | even | 3 | 1 | 162.4.a.c | yes | 1 | |
9.c | even | 3 | 1 | inner | 162.4.c.d | 2 | |
9.d | odd | 6 | 1 | 162.4.a.b | ✓ | 1 | |
9.d | odd | 6 | 1 | 162.4.c.e | 2 | ||
36.f | odd | 6 | 1 | 1296.4.a.a | 1 | ||
36.h | even | 6 | 1 | 1296.4.a.h | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
162.4.a.b | ✓ | 1 | 9.d | odd | 6 | 1 | |
162.4.a.c | yes | 1 | 9.c | even | 3 | 1 | |
162.4.c.d | 2 | 1.a | even | 1 | 1 | trivial | |
162.4.c.d | 2 | 9.c | even | 3 | 1 | inner | |
162.4.c.e | 2 | 3.b | odd | 2 | 1 | ||
162.4.c.e | 2 | 9.d | odd | 6 | 1 | ||
1296.4.a.a | 1 | 36.f | odd | 6 | 1 | ||
1296.4.a.h | 1 | 36.h | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 21T_{5} + 441 \)
acting on \(S_{4}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 2T + 4 \)
$3$
\( T^{2} \)
$5$
\( T^{2} - 21T + 441 \)
$7$
\( T^{2} + 8T + 64 \)
$11$
\( T^{2} - 36T + 1296 \)
$13$
\( T^{2} - 49T + 2401 \)
$17$
\( (T + 21)^{2} \)
$19$
\( (T + 112)^{2} \)
$23$
\( T^{2} - 180T + 32400 \)
$29$
\( T^{2} + 135T + 18225 \)
$31$
\( T^{2} + 308T + 94864 \)
$37$
\( (T + 1)^{2} \)
$41$
\( T^{2} + 42T + 1764 \)
$43$
\( T^{2} + 20T + 400 \)
$47$
\( T^{2} - 84T + 7056 \)
$53$
\( (T - 174)^{2} \)
$59$
\( T^{2} - 504T + 254016 \)
$61$
\( T^{2} - 385T + 148225 \)
$67$
\( T^{2} + 272T + 73984 \)
$71$
\( (T - 888)^{2} \)
$73$
\( (T - 371)^{2} \)
$79$
\( T^{2} - 652T + 425104 \)
$83$
\( T^{2} - 84T + 7056 \)
$89$
\( (T + 21)^{2} \)
$97$
\( T^{2} - 1246 T + 1552516 \)
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