Newspace parameters
Level: | \( N \) | \(=\) | \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1764.f (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(14.0856109166\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | \(\Q(\zeta_{16})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{8} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{4} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
\(\beta_{1}\) | \(=\) | \( 2\zeta_{16}^{4} \) |
\(\beta_{2}\) | \(=\) | \( \zeta_{16}^{5} + \zeta_{16}^{3} \) |
\(\beta_{3}\) | \(=\) | \( \zeta_{16}^{6} + \zeta_{16}^{2} \) |
\(\beta_{4}\) | \(=\) | \( \zeta_{16}^{7} + \zeta_{16} \) |
\(\beta_{5}\) | \(=\) | \( -\zeta_{16}^{7} + \zeta_{16} \) |
\(\beta_{6}\) | \(=\) | \( -\zeta_{16}^{6} + \zeta_{16}^{2} \) |
\(\beta_{7}\) | \(=\) | \( -\zeta_{16}^{5} + \zeta_{16}^{3} \) |
\(\zeta_{16}\) | \(=\) | \( ( \beta_{5} + \beta_{4} ) / 2 \) |
\(\zeta_{16}^{2}\) | \(=\) | \( ( \beta_{6} + \beta_{3} ) / 2 \) |
\(\zeta_{16}^{3}\) | \(=\) | \( ( \beta_{7} + \beta_{2} ) / 2 \) |
\(\zeta_{16}^{4}\) | \(=\) | \( ( \beta_1 ) / 2 \) |
\(\zeta_{16}^{5}\) | \(=\) | \( ( -\beta_{7} + \beta_{2} ) / 2 \) |
\(\zeta_{16}^{6}\) | \(=\) | \( ( -\beta_{6} + \beta_{3} ) / 2 \) |
\(\zeta_{16}^{7}\) | \(=\) | \( ( -\beta_{5} + \beta_{4} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).
\(n\) | \(785\) | \(883\) | \(1081\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
881.1 |
|
0 | 0 | 0 | −1.84776 | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
881.2 | 0 | 0 | 0 | −1.84776 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
881.3 | 0 | 0 | 0 | −0.765367 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
881.4 | 0 | 0 | 0 | −0.765367 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
881.5 | 0 | 0 | 0 | 0.765367 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
881.6 | 0 | 0 | 0 | 0.765367 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
881.7 | 0 | 0 | 0 | 1.84776 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
881.8 | 0 | 0 | 0 | 1.84776 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.b | odd | 2 | 1 | inner |
21.c | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1764.2.f.b | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 1764.2.f.b | ✓ | 8 |
4.b | odd | 2 | 1 | 7056.2.k.e | 8 | ||
7.b | odd | 2 | 1 | inner | 1764.2.f.b | ✓ | 8 |
7.c | even | 3 | 2 | 1764.2.t.c | 16 | ||
7.d | odd | 6 | 2 | 1764.2.t.c | 16 | ||
12.b | even | 2 | 1 | 7056.2.k.e | 8 | ||
21.c | even | 2 | 1 | inner | 1764.2.f.b | ✓ | 8 |
21.g | even | 6 | 2 | 1764.2.t.c | 16 | ||
21.h | odd | 6 | 2 | 1764.2.t.c | 16 | ||
28.d | even | 2 | 1 | 7056.2.k.e | 8 | ||
84.h | odd | 2 | 1 | 7056.2.k.e | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1764.2.f.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
1764.2.f.b | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
1764.2.f.b | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
1764.2.f.b | ✓ | 8 | 21.c | even | 2 | 1 | inner |
1764.2.t.c | 16 | 7.c | even | 3 | 2 | ||
1764.2.t.c | 16 | 7.d | odd | 6 | 2 | ||
1764.2.t.c | 16 | 21.g | even | 6 | 2 | ||
1764.2.t.c | 16 | 21.h | odd | 6 | 2 | ||
7056.2.k.e | 8 | 4.b | odd | 2 | 1 | ||
7056.2.k.e | 8 | 12.b | even | 2 | 1 | ||
7056.2.k.e | 8 | 28.d | even | 2 | 1 | ||
7056.2.k.e | 8 | 84.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 4T_{5}^{2} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} \)
$5$
\( (T^{4} - 4 T^{2} + 2)^{2} \)
$7$
\( T^{8} \)
$11$
\( (T^{2} + 4)^{4} \)
$13$
\( (T^{4} + 20 T^{2} + 2)^{2} \)
$17$
\( (T^{4} - 36 T^{2} + 162)^{2} \)
$19$
\( (T^{4} + 16 T^{2} + 32)^{2} \)
$23$
\( (T^{4} + 88 T^{2} + 784)^{2} \)
$29$
\( (T^{4} + 48 T^{2} + 64)^{2} \)
$31$
\( (T^{4} + 80 T^{2} + 1568)^{2} \)
$37$
\( (T^{2} - 8 T - 2)^{4} \)
$41$
\( (T^{4} - 148 T^{2} + 1058)^{2} \)
$43$
\( (T^{2} - 8 T + 8)^{4} \)
$47$
\( (T^{4} - 208 T^{2} + 9248)^{2} \)
$53$
\( (T^{4} + 164 T^{2} + 2116)^{2} \)
$59$
\( (T^{4} - 272 T^{2} + 16928)^{2} \)
$61$
\( (T^{4} + 164 T^{2} + 1922)^{2} \)
$67$
\( (T^{2} - 8 T - 56)^{4} \)
$71$
\( (T^{4} + 264 T^{2} + 15376)^{2} \)
$73$
\( (T^{4} + 148 T^{2} + 1058)^{2} \)
$79$
\( (T^{2} - 128)^{4} \)
$83$
\( (T^{4} - 128 T^{2} + 2048)^{2} \)
$89$
\( (T^{4} - 212 T^{2} + 578)^{2} \)
$97$
\( (T^{4} + 148 T^{2} + 1058)^{2} \)
show more
show less