Newspace parameters
Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 81.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.646788256372\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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|
Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
\(\beta_{1}\) | \(=\) | \( \zeta_{12}^{2} \) |
\(\beta_{2}\) | \(=\) | \( \zeta_{12}^{3} + \zeta_{12} \) |
\(\beta_{3}\) | \(=\) | \( -\zeta_{12}^{3} + 2\zeta_{12} \) |
\(\zeta_{12}\) | \(=\) | \( ( \beta_{3} + \beta_{2} ) / 3 \) |
\(\zeta_{12}^{2}\) | \(=\) | \( \beta_1 \) |
\(\zeta_{12}^{3}\) | \(=\) | \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).
\(n\) | \(2\) |
\(\chi(n)\) | \(-1 + \beta_{1}\) |
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 |
|
−0.866025 | − | 1.50000i | 0 | −0.500000 | + | 0.866025i | 0.866025 | − | 1.50000i | 0 | −1.00000 | − | 1.73205i | −1.73205 | 0 | −3.00000 | ||||||||||||||||||||||
28.2 | 0.866025 | + | 1.50000i | 0 | −0.500000 | + | 0.866025i | −0.866025 | + | 1.50000i | 0 | −1.00000 | − | 1.73205i | 1.73205 | 0 | −3.00000 | |||||||||||||||||||||||
55.1 | −0.866025 | + | 1.50000i | 0 | −0.500000 | − | 0.866025i | 0.866025 | + | 1.50000i | 0 | −1.00000 | + | 1.73205i | −1.73205 | 0 | −3.00000 | |||||||||||||||||||||||
55.2 | 0.866025 | − | 1.50000i | 0 | −0.500000 | − | 0.866025i | −0.866025 | − | 1.50000i | 0 | −1.00000 | + | 1.73205i | 1.73205 | 0 | −3.00000 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
9.c | even | 3 | 1 | inner |
9.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 81.2.c.b | 4 | |
3.b | odd | 2 | 1 | inner | 81.2.c.b | 4 | |
4.b | odd | 2 | 1 | 1296.2.i.s | 4 | ||
9.c | even | 3 | 1 | 81.2.a.a | ✓ | 2 | |
9.c | even | 3 | 1 | inner | 81.2.c.b | 4 | |
9.d | odd | 6 | 1 | 81.2.a.a | ✓ | 2 | |
9.d | odd | 6 | 1 | inner | 81.2.c.b | 4 | |
12.b | even | 2 | 1 | 1296.2.i.s | 4 | ||
27.e | even | 9 | 6 | 729.2.e.o | 12 | ||
27.f | odd | 18 | 6 | 729.2.e.o | 12 | ||
36.f | odd | 6 | 1 | 1296.2.a.o | 2 | ||
36.f | odd | 6 | 1 | 1296.2.i.s | 4 | ||
36.h | even | 6 | 1 | 1296.2.a.o | 2 | ||
36.h | even | 6 | 1 | 1296.2.i.s | 4 | ||
45.h | odd | 6 | 1 | 2025.2.a.j | 2 | ||
45.j | even | 6 | 1 | 2025.2.a.j | 2 | ||
45.k | odd | 12 | 2 | 2025.2.b.k | 4 | ||
45.l | even | 12 | 2 | 2025.2.b.k | 4 | ||
63.l | odd | 6 | 1 | 3969.2.a.i | 2 | ||
63.o | even | 6 | 1 | 3969.2.a.i | 2 | ||
72.j | odd | 6 | 1 | 5184.2.a.br | 2 | ||
72.l | even | 6 | 1 | 5184.2.a.bq | 2 | ||
72.n | even | 6 | 1 | 5184.2.a.br | 2 | ||
72.p | odd | 6 | 1 | 5184.2.a.bq | 2 | ||
99.g | even | 6 | 1 | 9801.2.a.v | 2 | ||
99.h | odd | 6 | 1 | 9801.2.a.v | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
81.2.a.a | ✓ | 2 | 9.c | even | 3 | 1 | |
81.2.a.a | ✓ | 2 | 9.d | odd | 6 | 1 | |
81.2.c.b | 4 | 1.a | even | 1 | 1 | trivial | |
81.2.c.b | 4 | 3.b | odd | 2 | 1 | inner | |
81.2.c.b | 4 | 9.c | even | 3 | 1 | inner | |
81.2.c.b | 4 | 9.d | odd | 6 | 1 | inner | |
729.2.e.o | 12 | 27.e | even | 9 | 6 | ||
729.2.e.o | 12 | 27.f | odd | 18 | 6 | ||
1296.2.a.o | 2 | 36.f | odd | 6 | 1 | ||
1296.2.a.o | 2 | 36.h | even | 6 | 1 | ||
1296.2.i.s | 4 | 4.b | odd | 2 | 1 | ||
1296.2.i.s | 4 | 12.b | even | 2 | 1 | ||
1296.2.i.s | 4 | 36.f | odd | 6 | 1 | ||
1296.2.i.s | 4 | 36.h | even | 6 | 1 | ||
2025.2.a.j | 2 | 45.h | odd | 6 | 1 | ||
2025.2.a.j | 2 | 45.j | even | 6 | 1 | ||
2025.2.b.k | 4 | 45.k | odd | 12 | 2 | ||
2025.2.b.k | 4 | 45.l | even | 12 | 2 | ||
3969.2.a.i | 2 | 63.l | odd | 6 | 1 | ||
3969.2.a.i | 2 | 63.o | even | 6 | 1 | ||
5184.2.a.bq | 2 | 72.l | even | 6 | 1 | ||
5184.2.a.bq | 2 | 72.p | odd | 6 | 1 | ||
5184.2.a.br | 2 | 72.j | odd | 6 | 1 | ||
5184.2.a.br | 2 | 72.n | even | 6 | 1 | ||
9801.2.a.v | 2 | 99.g | even | 6 | 1 | ||
9801.2.a.v | 2 | 99.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 3T_{2}^{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 3T^{2} + 9 \)
$3$
\( T^{4} \)
$5$
\( T^{4} + 3T^{2} + 9 \)
$7$
\( (T^{2} + 2 T + 4)^{2} \)
$11$
\( T^{4} + 12T^{2} + 144 \)
$13$
\( (T^{2} - T + 1)^{2} \)
$17$
\( (T^{2} - 27)^{2} \)
$19$
\( (T - 2)^{4} \)
$23$
\( T^{4} + 12T^{2} + 144 \)
$29$
\( T^{4} + 3T^{2} + 9 \)
$31$
\( (T^{2} + 8 T + 64)^{2} \)
$37$
\( (T + 7)^{4} \)
$41$
\( T^{4} + 48T^{2} + 2304 \)
$43$
\( (T^{2} + 2 T + 4)^{2} \)
$47$
\( T^{4} + 48T^{2} + 2304 \)
$53$
\( T^{4} \)
$59$
\( T^{4} + 192 T^{2} + 36864 \)
$61$
\( (T^{2} - 7 T + 49)^{2} \)
$67$
\( (T^{2} - 10 T + 100)^{2} \)
$71$
\( (T^{2} - 108)^{2} \)
$73$
\( (T + 7)^{4} \)
$79$
\( (T^{2} + 2 T + 4)^{2} \)
$83$
\( T^{4} + 192 T^{2} + 36864 \)
$89$
\( (T^{2} - 27)^{2} \)
$97$
\( (T^{2} + 2 T + 4)^{2} \)
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