Newspace parameters
Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 81.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(12.9910894049\) |
Analytic rank: | \(1\) |
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: | \(\Z[a_1, \ldots, a_{25}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 3) |
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 \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
28.1 |
|
−3.00000 | − | 5.19615i | 0 | −2.00000 | + | 3.46410i | 3.00000 | − | 5.19615i | 0 | 20.0000 | + | 34.6410i | −168.000 | 0 | −36.0000 | ||||||||||||||||
55.1 | −3.00000 | + | 5.19615i | 0 | −2.00000 | − | 3.46410i | 3.00000 | + | 5.19615i | 0 | 20.0000 | − | 34.6410i | −168.000 | 0 | −36.0000 | |||||||||||||||||
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 | 81.6.c.a | 2 | |
3.b | odd | 2 | 1 | 81.6.c.c | 2 | ||
9.c | even | 3 | 1 | 9.6.a.a | 1 | ||
9.c | even | 3 | 1 | inner | 81.6.c.a | 2 | |
9.d | odd | 6 | 1 | 3.6.a.a | ✓ | 1 | |
9.d | odd | 6 | 1 | 81.6.c.c | 2 | ||
36.f | odd | 6 | 1 | 144.6.a.f | 1 | ||
36.h | even | 6 | 1 | 48.6.a.a | 1 | ||
45.h | odd | 6 | 1 | 75.6.a.e | 1 | ||
45.j | even | 6 | 1 | 225.6.a.a | 1 | ||
45.k | odd | 12 | 2 | 225.6.b.b | 2 | ||
45.l | even | 12 | 2 | 75.6.b.b | 2 | ||
63.i | even | 6 | 1 | 147.6.e.k | 2 | ||
63.j | odd | 6 | 1 | 147.6.e.h | 2 | ||
63.l | odd | 6 | 1 | 441.6.a.i | 1 | ||
63.n | odd | 6 | 1 | 147.6.e.h | 2 | ||
63.o | even | 6 | 1 | 147.6.a.a | 1 | ||
63.s | even | 6 | 1 | 147.6.e.k | 2 | ||
72.j | odd | 6 | 1 | 192.6.a.d | 1 | ||
72.l | even | 6 | 1 | 192.6.a.l | 1 | ||
72.n | even | 6 | 1 | 576.6.a.s | 1 | ||
72.p | odd | 6 | 1 | 576.6.a.t | 1 | ||
99.g | even | 6 | 1 | 363.6.a.d | 1 | ||
99.h | odd | 6 | 1 | 1089.6.a.b | 1 | ||
117.n | odd | 6 | 1 | 507.6.a.b | 1 | ||
144.u | even | 12 | 2 | 768.6.d.h | 2 | ||
144.w | odd | 12 | 2 | 768.6.d.k | 2 | ||
153.i | odd | 6 | 1 | 867.6.a.a | 1 | ||
171.l | even | 6 | 1 | 1083.6.a.c | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3.6.a.a | ✓ | 1 | 9.d | odd | 6 | 1 | |
9.6.a.a | 1 | 9.c | even | 3 | 1 | ||
48.6.a.a | 1 | 36.h | even | 6 | 1 | ||
75.6.a.e | 1 | 45.h | odd | 6 | 1 | ||
75.6.b.b | 2 | 45.l | even | 12 | 2 | ||
81.6.c.a | 2 | 1.a | even | 1 | 1 | trivial | |
81.6.c.a | 2 | 9.c | even | 3 | 1 | inner | |
81.6.c.c | 2 | 3.b | odd | 2 | 1 | ||
81.6.c.c | 2 | 9.d | odd | 6 | 1 | ||
144.6.a.f | 1 | 36.f | odd | 6 | 1 | ||
147.6.a.a | 1 | 63.o | even | 6 | 1 | ||
147.6.e.h | 2 | 63.j | odd | 6 | 1 | ||
147.6.e.h | 2 | 63.n | odd | 6 | 1 | ||
147.6.e.k | 2 | 63.i | even | 6 | 1 | ||
147.6.e.k | 2 | 63.s | even | 6 | 1 | ||
192.6.a.d | 1 | 72.j | odd | 6 | 1 | ||
192.6.a.l | 1 | 72.l | even | 6 | 1 | ||
225.6.a.a | 1 | 45.j | even | 6 | 1 | ||
225.6.b.b | 2 | 45.k | odd | 12 | 2 | ||
363.6.a.d | 1 | 99.g | even | 6 | 1 | ||
441.6.a.i | 1 | 63.l | odd | 6 | 1 | ||
507.6.a.b | 1 | 117.n | odd | 6 | 1 | ||
576.6.a.s | 1 | 72.n | even | 6 | 1 | ||
576.6.a.t | 1 | 72.p | odd | 6 | 1 | ||
768.6.d.h | 2 | 144.u | even | 12 | 2 | ||
768.6.d.k | 2 | 144.w | odd | 12 | 2 | ||
867.6.a.a | 1 | 153.i | odd | 6 | 1 | ||
1083.6.a.c | 1 | 171.l | even | 6 | 1 | ||
1089.6.a.b | 1 | 99.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 6T_{2} + 36 \)
acting on \(S_{6}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 6T + 36 \)
$3$
\( T^{2} \)
$5$
\( T^{2} - 6T + 36 \)
$7$
\( T^{2} - 40T + 1600 \)
$11$
\( T^{2} + 564T + 318096 \)
$13$
\( T^{2} + 638T + 407044 \)
$17$
\( (T + 882)^{2} \)
$19$
\( (T + 556)^{2} \)
$23$
\( T^{2} + 840T + 705600 \)
$29$
\( T^{2} - 4638 T + 21511044 \)
$31$
\( T^{2} + 4400 T + 19360000 \)
$37$
\( (T + 2410)^{2} \)
$41$
\( T^{2} + 6870 T + 47196900 \)
$43$
\( T^{2} + 9644 T + 93006736 \)
$47$
\( T^{2} + 18672 T + 348643584 \)
$53$
\( (T + 33750)^{2} \)
$59$
\( T^{2} + 18084 T + 327031056 \)
$61$
\( T^{2} + 39758 T + 1580698564 \)
$67$
\( T^{2} - 23068 T + 532132624 \)
$71$
\( (T - 4248)^{2} \)
$73$
\( (T + 41110)^{2} \)
$79$
\( T^{2} + 21920 T + 480486400 \)
$83$
\( T^{2} - 82452 T + 6798332304 \)
$89$
\( (T - 94086)^{2} \)
$97$
\( T^{2} + 49442 T + 2444511364 \)
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