Newspace parameters
Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 81.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(4.77915471046\) |
Analytic rank: | \(1\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{33}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
\( x^{2} - x - 8 \)
|
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 9) |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
|
−4.37228 | 0 | 11.1168 | −4.62772 | 0 | 12.1168 | −13.6277 | 0 | 20.2337 | ||||||||||||||||||||||||
1.2 | 1.37228 | 0 | −6.11684 | −10.3723 | 0 | −5.11684 | −19.3723 | 0 | −14.2337 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 81.4.a.a | 2 | |
3.b | odd | 2 | 1 | 81.4.a.d | 2 | ||
4.b | odd | 2 | 1 | 1296.4.a.i | 2 | ||
5.b | even | 2 | 1 | 2025.4.a.n | 2 | ||
9.c | even | 3 | 2 | 27.4.c.a | 4 | ||
9.d | odd | 6 | 2 | 9.4.c.a | ✓ | 4 | |
12.b | even | 2 | 1 | 1296.4.a.u | 2 | ||
15.d | odd | 2 | 1 | 2025.4.a.g | 2 | ||
36.f | odd | 6 | 2 | 432.4.i.c | 4 | ||
36.h | even | 6 | 2 | 144.4.i.c | 4 | ||
45.h | odd | 6 | 2 | 225.4.e.b | 4 | ||
45.l | even | 12 | 4 | 225.4.k.b | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
9.4.c.a | ✓ | 4 | 9.d | odd | 6 | 2 | |
27.4.c.a | 4 | 9.c | even | 3 | 2 | ||
81.4.a.a | 2 | 1.a | even | 1 | 1 | trivial | |
81.4.a.d | 2 | 3.b | odd | 2 | 1 | ||
144.4.i.c | 4 | 36.h | even | 6 | 2 | ||
225.4.e.b | 4 | 45.h | odd | 6 | 2 | ||
225.4.k.b | 8 | 45.l | even | 12 | 4 | ||
432.4.i.c | 4 | 36.f | odd | 6 | 2 | ||
1296.4.a.i | 2 | 4.b | odd | 2 | 1 | ||
1296.4.a.u | 2 | 12.b | even | 2 | 1 | ||
2025.4.a.g | 2 | 15.d | odd | 2 | 1 | ||
2025.4.a.n | 2 | 5.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 3T_{2} - 6 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(81))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 3T - 6 \)
$3$
\( T^{2} \)
$5$
\( T^{2} + 15T + 48 \)
$7$
\( T^{2} - 7T - 62 \)
$11$
\( T^{2} + 66T + 561 \)
$13$
\( T^{2} + 11T - 1826 \)
$17$
\( T^{2} + 99T + 1782 \)
$19$
\( T^{2} + 77T - 4532 \)
$23$
\( T^{2} + 33T - 2706 \)
$29$
\( T^{2} - 51T + 642 \)
$31$
\( T^{2} - 43T + 388 \)
$37$
\( T^{2} + 50T - 23432 \)
$41$
\( T^{2} + 132T - 74877 \)
$43$
\( T^{2} - 88T + 1639 \)
$47$
\( T^{2} + 399T - 28518 \)
$53$
\( T^{2} + 54T - 215784 \)
$59$
\( T^{2} + 798T + 6609 \)
$61$
\( T^{2} - 439T - 42776 \)
$67$
\( T^{2} - 988T + 208099 \)
$71$
\( T^{2} + 1368 T + 296784 \)
$73$
\( T^{2} + 455T - 435398 \)
$79$
\( T^{2} + 803T - 626516 \)
$83$
\( T^{2} + 813T + 70788 \)
$89$
\( T^{2} - 396T - 3564 \)
$97$
\( T^{2} - 736T + 49591 \)
show more
show less