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: | \(2\) |
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: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 27) |
Sato-Tate group: | $\mathrm{U}(1)[D_{3}]$ |
$q$-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 |
|
0 | 0 | 1.00000 | − | 1.73205i | 0 | 0 | 0.500000 | + | 0.866025i | 0 | 0 | 0 | ||||||||||||||||||||
55.1 | 0 | 0 | 1.00000 | + | 1.73205i | 0 | 0 | 0.500000 | − | 0.866025i | 0 | 0 | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
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.a | 2 | |
3.b | odd | 2 | 1 | CM | 81.2.c.a | 2 | |
4.b | odd | 2 | 1 | 1296.2.i.i | 2 | ||
9.c | even | 3 | 1 | 27.2.a.a | ✓ | 1 | |
9.c | even | 3 | 1 | inner | 81.2.c.a | 2 | |
9.d | odd | 6 | 1 | 27.2.a.a | ✓ | 1 | |
9.d | odd | 6 | 1 | inner | 81.2.c.a | 2 | |
12.b | even | 2 | 1 | 1296.2.i.i | 2 | ||
27.e | even | 9 | 6 | 729.2.e.f | 6 | ||
27.f | odd | 18 | 6 | 729.2.e.f | 6 | ||
36.f | odd | 6 | 1 | 432.2.a.e | 1 | ||
36.f | odd | 6 | 1 | 1296.2.i.i | 2 | ||
36.h | even | 6 | 1 | 432.2.a.e | 1 | ||
36.h | even | 6 | 1 | 1296.2.i.i | 2 | ||
45.h | odd | 6 | 1 | 675.2.a.e | 1 | ||
45.j | even | 6 | 1 | 675.2.a.e | 1 | ||
45.k | odd | 12 | 2 | 675.2.b.f | 2 | ||
45.l | even | 12 | 2 | 675.2.b.f | 2 | ||
63.l | odd | 6 | 1 | 1323.2.a.i | 1 | ||
63.o | even | 6 | 1 | 1323.2.a.i | 1 | ||
72.j | odd | 6 | 1 | 1728.2.a.n | 1 | ||
72.l | even | 6 | 1 | 1728.2.a.o | 1 | ||
72.n | even | 6 | 1 | 1728.2.a.n | 1 | ||
72.p | odd | 6 | 1 | 1728.2.a.o | 1 | ||
99.g | even | 6 | 1 | 3267.2.a.f | 1 | ||
99.h | odd | 6 | 1 | 3267.2.a.f | 1 | ||
117.n | odd | 6 | 1 | 4563.2.a.e | 1 | ||
117.t | even | 6 | 1 | 4563.2.a.e | 1 | ||
153.h | even | 6 | 1 | 7803.2.a.k | 1 | ||
153.i | odd | 6 | 1 | 7803.2.a.k | 1 | ||
171.l | even | 6 | 1 | 9747.2.a.f | 1 | ||
171.o | odd | 6 | 1 | 9747.2.a.f | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
27.2.a.a | ✓ | 1 | 9.c | even | 3 | 1 | |
27.2.a.a | ✓ | 1 | 9.d | odd | 6 | 1 | |
81.2.c.a | 2 | 1.a | even | 1 | 1 | trivial | |
81.2.c.a | 2 | 3.b | odd | 2 | 1 | CM | |
81.2.c.a | 2 | 9.c | even | 3 | 1 | inner | |
81.2.c.a | 2 | 9.d | odd | 6 | 1 | inner | |
432.2.a.e | 1 | 36.f | odd | 6 | 1 | ||
432.2.a.e | 1 | 36.h | even | 6 | 1 | ||
675.2.a.e | 1 | 45.h | odd | 6 | 1 | ||
675.2.a.e | 1 | 45.j | even | 6 | 1 | ||
675.2.b.f | 2 | 45.k | odd | 12 | 2 | ||
675.2.b.f | 2 | 45.l | even | 12 | 2 | ||
729.2.e.f | 6 | 27.e | even | 9 | 6 | ||
729.2.e.f | 6 | 27.f | odd | 18 | 6 | ||
1296.2.i.i | 2 | 4.b | odd | 2 | 1 | ||
1296.2.i.i | 2 | 12.b | even | 2 | 1 | ||
1296.2.i.i | 2 | 36.f | odd | 6 | 1 | ||
1296.2.i.i | 2 | 36.h | even | 6 | 1 | ||
1323.2.a.i | 1 | 63.l | odd | 6 | 1 | ||
1323.2.a.i | 1 | 63.o | even | 6 | 1 | ||
1728.2.a.n | 1 | 72.j | odd | 6 | 1 | ||
1728.2.a.n | 1 | 72.n | even | 6 | 1 | ||
1728.2.a.o | 1 | 72.l | even | 6 | 1 | ||
1728.2.a.o | 1 | 72.p | odd | 6 | 1 | ||
3267.2.a.f | 1 | 99.g | even | 6 | 1 | ||
3267.2.a.f | 1 | 99.h | odd | 6 | 1 | ||
4563.2.a.e | 1 | 117.n | odd | 6 | 1 | ||
4563.2.a.e | 1 | 117.t | even | 6 | 1 | ||
7803.2.a.k | 1 | 153.h | even | 6 | 1 | ||
7803.2.a.k | 1 | 153.i | odd | 6 | 1 | ||
9747.2.a.f | 1 | 171.l | even | 6 | 1 | ||
9747.2.a.f | 1 | 171.o | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{2}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} \)
$5$
\( T^{2} \)
$7$
\( T^{2} - T + 1 \)
$11$
\( T^{2} \)
$13$
\( T^{2} + 5T + 25 \)
$17$
\( T^{2} \)
$19$
\( (T + 7)^{2} \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} - 4T + 16 \)
$37$
\( (T - 11)^{2} \)
$41$
\( T^{2} \)
$43$
\( T^{2} + 8T + 64 \)
$47$
\( T^{2} \)
$53$
\( T^{2} \)
$59$
\( T^{2} \)
$61$
\( T^{2} - T + 1 \)
$67$
\( T^{2} + 5T + 25 \)
$71$
\( T^{2} \)
$73$
\( (T + 7)^{2} \)
$79$
\( T^{2} + 17T + 289 \)
$83$
\( T^{2} \)
$89$
\( T^{2} \)
$97$
\( T^{2} - 19T + 361 \)
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