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: | 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 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 | −1.50000 | − | 2.59808i | 0 | −14.5000 | + | 25.1147i | 8.00000 | 0 | 6.00000 | ||||||||||||||||
109.1 | −1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | −1.50000 | + | 2.59808i | 0 | −14.5000 | − | 25.1147i | 8.00000 | 0 | 6.00000 | |||||||||||||||||
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.b | 2 | |
3.b | odd | 2 | 1 | 162.4.c.g | 2 | ||
9.c | even | 3 | 1 | 54.4.a.c | yes | 1 | |
9.c | even | 3 | 1 | inner | 162.4.c.b | 2 | |
9.d | odd | 6 | 1 | 54.4.a.b | ✓ | 1 | |
9.d | odd | 6 | 1 | 162.4.c.g | 2 | ||
36.f | odd | 6 | 1 | 432.4.a.j | 1 | ||
36.h | even | 6 | 1 | 432.4.a.e | 1 | ||
45.h | odd | 6 | 1 | 1350.4.a.o | 1 | ||
45.j | even | 6 | 1 | 1350.4.a.a | 1 | ||
45.k | odd | 12 | 2 | 1350.4.c.b | 2 | ||
45.l | even | 12 | 2 | 1350.4.c.s | 2 | ||
72.j | odd | 6 | 1 | 1728.4.a.v | 1 | ||
72.l | even | 6 | 1 | 1728.4.a.u | 1 | ||
72.n | even | 6 | 1 | 1728.4.a.l | 1 | ||
72.p | odd | 6 | 1 | 1728.4.a.k | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
54.4.a.b | ✓ | 1 | 9.d | odd | 6 | 1 | |
54.4.a.c | yes | 1 | 9.c | even | 3 | 1 | |
162.4.c.b | 2 | 1.a | even | 1 | 1 | trivial | |
162.4.c.b | 2 | 9.c | even | 3 | 1 | inner | |
162.4.c.g | 2 | 3.b | odd | 2 | 1 | ||
162.4.c.g | 2 | 9.d | odd | 6 | 1 | ||
432.4.a.e | 1 | 36.h | even | 6 | 1 | ||
432.4.a.j | 1 | 36.f | odd | 6 | 1 | ||
1350.4.a.a | 1 | 45.j | even | 6 | 1 | ||
1350.4.a.o | 1 | 45.h | odd | 6 | 1 | ||
1350.4.c.b | 2 | 45.k | odd | 12 | 2 | ||
1350.4.c.s | 2 | 45.l | even | 12 | 2 | ||
1728.4.a.k | 1 | 72.p | odd | 6 | 1 | ||
1728.4.a.l | 1 | 72.n | even | 6 | 1 | ||
1728.4.a.u | 1 | 72.l | even | 6 | 1 | ||
1728.4.a.v | 1 | 72.j | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 3T_{5} + 9 \)
acting on \(S_{4}^{\mathrm{new}}(162, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 2T + 4 \)
$3$
\( T^{2} \)
$5$
\( T^{2} + 3T + 9 \)
$7$
\( T^{2} + 29T + 841 \)
$11$
\( T^{2} - 57T + 3249 \)
$13$
\( T^{2} + 20T + 400 \)
$17$
\( (T + 72)^{2} \)
$19$
\( (T + 106)^{2} \)
$23$
\( T^{2} + 174T + 30276 \)
$29$
\( T^{2} - 210T + 44100 \)
$31$
\( T^{2} + 47T + 2209 \)
$37$
\( (T - 2)^{2} \)
$41$
\( T^{2} - 6T + 36 \)
$43$
\( T^{2} + 218T + 47524 \)
$47$
\( T^{2} + 474T + 224676 \)
$53$
\( (T - 81)^{2} \)
$59$
\( T^{2} + 84T + 7056 \)
$61$
\( T^{2} + 56T + 3136 \)
$67$
\( T^{2} - 142T + 20164 \)
$71$
\( (T - 360)^{2} \)
$73$
\( (T + 1159)^{2} \)
$79$
\( T^{2} - 160T + 25600 \)
$83$
\( T^{2} + 735T + 540225 \)
$89$
\( (T + 954)^{2} \)
$97$
\( T^{2} + 191T + 36481 \)
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