Newspace parameters
Level: | \( N \) | \(=\) | \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1512.q (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(12.0733807856\) |
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 504) |
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 \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).
\(n\) | \(757\) | \(785\) | \(1081\) | \(1135\) |
\(\chi(n)\) | \(1\) | \(-\zeta_{6}\) | \(-\zeta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
793.1 |
|
0 | 0 | 0 | 0.500000 | + | 0.866025i | 0 | 2.00000 | − | 1.73205i | 0 | 0 | 0 | ||||||||||||||||||||
1369.1 | 0 | 0 | 0 | 0.500000 | − | 0.866025i | 0 | 2.00000 | + | 1.73205i | 0 | 0 | 0 | |||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
63.h | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1512.2.q.b | 2 | |
3.b | odd | 2 | 1 | 504.2.q.a | ✓ | 2 | |
4.b | odd | 2 | 1 | 3024.2.q.d | 2 | ||
7.c | even | 3 | 1 | 1512.2.t.a | 2 | ||
9.c | even | 3 | 1 | 1512.2.t.a | 2 | ||
9.d | odd | 6 | 1 | 504.2.t.a | yes | 2 | |
12.b | even | 2 | 1 | 1008.2.q.b | 2 | ||
21.h | odd | 6 | 1 | 504.2.t.a | yes | 2 | |
28.g | odd | 6 | 1 | 3024.2.t.c | 2 | ||
36.f | odd | 6 | 1 | 3024.2.t.c | 2 | ||
36.h | even | 6 | 1 | 1008.2.t.e | 2 | ||
63.h | even | 3 | 1 | inner | 1512.2.q.b | 2 | |
63.j | odd | 6 | 1 | 504.2.q.a | ✓ | 2 | |
84.n | even | 6 | 1 | 1008.2.t.e | 2 | ||
252.u | odd | 6 | 1 | 3024.2.q.d | 2 | ||
252.bb | even | 6 | 1 | 1008.2.q.b | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
504.2.q.a | ✓ | 2 | 3.b | odd | 2 | 1 | |
504.2.q.a | ✓ | 2 | 63.j | odd | 6 | 1 | |
504.2.t.a | yes | 2 | 9.d | odd | 6 | 1 | |
504.2.t.a | yes | 2 | 21.h | odd | 6 | 1 | |
1008.2.q.b | 2 | 12.b | even | 2 | 1 | ||
1008.2.q.b | 2 | 252.bb | even | 6 | 1 | ||
1008.2.t.e | 2 | 36.h | even | 6 | 1 | ||
1008.2.t.e | 2 | 84.n | even | 6 | 1 | ||
1512.2.q.b | 2 | 1.a | even | 1 | 1 | trivial | |
1512.2.q.b | 2 | 63.h | even | 3 | 1 | inner | |
1512.2.t.a | 2 | 7.c | even | 3 | 1 | ||
1512.2.t.a | 2 | 9.c | even | 3 | 1 | ||
3024.2.q.d | 2 | 4.b | odd | 2 | 1 | ||
3024.2.q.d | 2 | 252.u | odd | 6 | 1 | ||
3024.2.t.c | 2 | 28.g | odd | 6 | 1 | ||
3024.2.t.c | 2 | 36.f | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - T_{5} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(1512, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} \)
$5$
\( T^{2} - T + 1 \)
$7$
\( T^{2} - 4T + 7 \)
$11$
\( T^{2} + 3T + 9 \)
$13$
\( T^{2} + T + 1 \)
$17$
\( T^{2} - 3T + 9 \)
$19$
\( T^{2} + 5T + 25 \)
$23$
\( T^{2} - T + 1 \)
$29$
\( T^{2} - 9T + 81 \)
$31$
\( (T - 4)^{2} \)
$37$
\( T^{2} + 5T + 25 \)
$41$
\( T^{2} - 7T + 49 \)
$43$
\( T^{2} + 3T + 9 \)
$47$
\( (T + 8)^{2} \)
$53$
\( T^{2} - 9T + 81 \)
$59$
\( (T - 4)^{2} \)
$61$
\( (T - 2)^{2} \)
$67$
\( (T - 12)^{2} \)
$71$
\( (T + 8)^{2} \)
$73$
\( T^{2} - 13T + 169 \)
$79$
\( (T - 8)^{2} \)
$83$
\( T^{2} + 13T + 169 \)
$89$
\( T^{2} + 9T + 81 \)
$97$
\( T^{2} - 17T + 289 \)
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