Newspace parameters
Level: | \( N \) | \(=\) | \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1764.t (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(14.0856109166\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{48})\) |
Defining polynomial: |
\( x^{16} - x^{8} + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{25}]\) |
Coefficient ring index: | \( 2^{8} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
\(\beta_{1}\) | \(=\) |
\( 2\zeta_{48}^{4} \)
|
\(\beta_{2}\) | \(=\) |
\( \zeta_{48}^{7} + \zeta_{48} \)
|
\(\beta_{3}\) | \(=\) |
\( \zeta_{48}^{8} \)
|
\(\beta_{4}\) | \(=\) |
\( \zeta_{48}^{11} + \zeta_{48}^{5} \)
|
\(\beta_{5}\) | \(=\) |
\( 2\zeta_{48}^{12} \)
|
\(\beta_{6}\) | \(=\) |
\( \zeta_{48}^{14} + \zeta_{48}^{2} \)
|
\(\beta_{7}\) | \(=\) |
\( \zeta_{48}^{15} + \zeta_{48}^{9} \)
|
\(\beta_{8}\) | \(=\) |
\( -\zeta_{48}^{7} + \zeta_{48} \)
|
\(\beta_{9}\) | \(=\) |
\( -\zeta_{48}^{14} + \zeta_{48}^{2} \)
|
\(\beta_{10}\) | \(=\) |
\( -\zeta_{48}^{11} + \zeta_{48}^{5} \)
|
\(\beta_{11}\) | \(=\) |
\( -\zeta_{48}^{15} + \zeta_{48}^{9} \)
|
\(\beta_{12}\) | \(=\) |
\( -\zeta_{48}^{14} + \zeta_{48}^{10} + \zeta_{48}^{6} \)
|
\(\beta_{13}\) | \(=\) |
\( \zeta_{48}^{13} - \zeta_{48}^{11} + \zeta_{48}^{3} \)
|
\(\beta_{14}\) | \(=\) |
\( -\zeta_{48}^{10} + \zeta_{48}^{6} + \zeta_{48}^{2} \)
|
\(\beta_{15}\) | \(=\) |
\( -\zeta_{48}^{13} + \zeta_{48}^{5} + \zeta_{48}^{3} \)
|
\(\zeta_{48}\) | \(=\) |
\( ( \beta_{8} + \beta_{2} ) / 2 \)
|
\(\zeta_{48}^{2}\) | \(=\) |
\( ( \beta_{9} + \beta_{6} ) / 2 \)
|
\(\zeta_{48}^{3}\) | \(=\) |
\( ( \beta_{15} + \beta_{13} - \beta_{10} ) / 2 \)
|
\(\zeta_{48}^{4}\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
\(\zeta_{48}^{5}\) | \(=\) |
\( ( \beta_{10} + \beta_{4} ) / 2 \)
|
\(\zeta_{48}^{6}\) | \(=\) |
\( ( \beta_{14} + \beta_{12} - \beta_{9} ) / 2 \)
|
\(\zeta_{48}^{7}\) | \(=\) |
\( ( -\beta_{8} + \beta_{2} ) / 2 \)
|
\(\zeta_{48}^{8}\) | \(=\) |
\( \beta_{3} \)
|
\(\zeta_{48}^{9}\) | \(=\) |
\( ( \beta_{11} + \beta_{7} ) / 2 \)
|
\(\zeta_{48}^{10}\) | \(=\) |
\( ( -\beta_{14} + \beta_{12} + \beta_{6} ) / 2 \)
|
\(\zeta_{48}^{11}\) | \(=\) |
\( ( -\beta_{10} + \beta_{4} ) / 2 \)
|
\(\zeta_{48}^{12}\) | \(=\) |
\( ( \beta_{5} ) / 2 \)
|
\(\zeta_{48}^{13}\) | \(=\) |
\( ( -\beta_{15} + \beta_{13} + \beta_{4} ) / 2 \)
|
\(\zeta_{48}^{14}\) | \(=\) |
\( ( -\beta_{9} + \beta_{6} ) / 2 \)
|
\(\zeta_{48}^{15}\) | \(=\) |
\( ( -\beta_{11} + \beta_{7} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).
\(n\) | \(785\) | \(883\) | \(1081\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1 - \beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
521.1 |
|
0 | 0 | 0 | −0.923880 | − | 1.60021i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
521.2 | 0 | 0 | 0 | −0.923880 | − | 1.60021i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
521.3 | 0 | 0 | 0 | −0.382683 | − | 0.662827i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
521.4 | 0 | 0 | 0 | −0.382683 | − | 0.662827i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
521.5 | 0 | 0 | 0 | 0.382683 | + | 0.662827i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
521.6 | 0 | 0 | 0 | 0.382683 | + | 0.662827i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
521.7 | 0 | 0 | 0 | 0.923880 | + | 1.60021i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
521.8 | 0 | 0 | 0 | 0.923880 | + | 1.60021i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1097.1 | 0 | 0 | 0 | −0.923880 | + | 1.60021i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1097.2 | 0 | 0 | 0 | −0.923880 | + | 1.60021i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1097.3 | 0 | 0 | 0 | −0.382683 | + | 0.662827i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1097.4 | 0 | 0 | 0 | −0.382683 | + | 0.662827i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1097.5 | 0 | 0 | 0 | 0.382683 | − | 0.662827i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1097.6 | 0 | 0 | 0 | 0.382683 | − | 0.662827i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1097.7 | 0 | 0 | 0 | 0.923880 | − | 1.60021i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1097.8 | 0 | 0 | 0 | 0.923880 | − | 1.60021i | 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 |
7.c | even | 3 | 1 | inner |
7.d | odd | 6 | 1 | inner |
21.c | even | 2 | 1 | inner |
21.g | even | 6 | 1 | inner |
21.h | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1764.2.t.c | 16 | |
3.b | odd | 2 | 1 | inner | 1764.2.t.c | 16 | |
7.b | odd | 2 | 1 | inner | 1764.2.t.c | 16 | |
7.c | even | 3 | 1 | 1764.2.f.b | ✓ | 8 | |
7.c | even | 3 | 1 | inner | 1764.2.t.c | 16 | |
7.d | odd | 6 | 1 | 1764.2.f.b | ✓ | 8 | |
7.d | odd | 6 | 1 | inner | 1764.2.t.c | 16 | |
21.c | even | 2 | 1 | inner | 1764.2.t.c | 16 | |
21.g | even | 6 | 1 | 1764.2.f.b | ✓ | 8 | |
21.g | even | 6 | 1 | inner | 1764.2.t.c | 16 | |
21.h | odd | 6 | 1 | 1764.2.f.b | ✓ | 8 | |
21.h | odd | 6 | 1 | inner | 1764.2.t.c | 16 | |
28.f | even | 6 | 1 | 7056.2.k.e | 8 | ||
28.g | odd | 6 | 1 | 7056.2.k.e | 8 | ||
84.j | odd | 6 | 1 | 7056.2.k.e | 8 | ||
84.n | even | 6 | 1 | 7056.2.k.e | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1764.2.f.b | ✓ | 8 | 7.c | even | 3 | 1 | |
1764.2.f.b | ✓ | 8 | 7.d | odd | 6 | 1 | |
1764.2.f.b | ✓ | 8 | 21.g | even | 6 | 1 | |
1764.2.f.b | ✓ | 8 | 21.h | odd | 6 | 1 | |
1764.2.t.c | 16 | 1.a | even | 1 | 1 | trivial | |
1764.2.t.c | 16 | 3.b | odd | 2 | 1 | inner | |
1764.2.t.c | 16 | 7.b | odd | 2 | 1 | inner | |
1764.2.t.c | 16 | 7.c | even | 3 | 1 | inner | |
1764.2.t.c | 16 | 7.d | odd | 6 | 1 | inner | |
1764.2.t.c | 16 | 21.c | even | 2 | 1 | inner | |
1764.2.t.c | 16 | 21.g | even | 6 | 1 | inner | |
1764.2.t.c | 16 | 21.h | odd | 6 | 1 | inner | |
7056.2.k.e | 8 | 28.f | even | 6 | 1 | ||
7056.2.k.e | 8 | 28.g | odd | 6 | 1 | ||
7056.2.k.e | 8 | 84.j | odd | 6 | 1 | ||
7056.2.k.e | 8 | 84.n | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 4T_{5}^{6} + 14T_{5}^{4} + 8T_{5}^{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} \)
$5$
\( (T^{8} + 4 T^{6} + 14 T^{4} + 8 T^{2} + 4)^{2} \)
$7$
\( T^{16} \)
$11$
\( (T^{4} - 4 T^{2} + 16)^{4} \)
$13$
\( (T^{4} + 20 T^{2} + 2)^{4} \)
$17$
\( (T^{8} + 36 T^{6} + 1134 T^{4} + \cdots + 26244)^{2} \)
$19$
\( (T^{8} - 16 T^{6} + 224 T^{4} - 512 T^{2} + \cdots + 1024)^{2} \)
$23$
\( (T^{8} - 88 T^{6} + 6960 T^{4} + \cdots + 614656)^{2} \)
$29$
\( (T^{4} + 48 T^{2} + 64)^{4} \)
$31$
\( (T^{8} - 80 T^{6} + 4832 T^{4} + \cdots + 2458624)^{2} \)
$37$
\( (T^{4} + 8 T^{3} + 66 T^{2} - 16 T + 4)^{4} \)
$41$
\( (T^{4} - 148 T^{2} + 1058)^{4} \)
$43$
\( (T^{2} - 8 T + 8)^{8} \)
$47$
\( (T^{8} + 208 T^{6} + 34016 T^{4} + \cdots + 85525504)^{2} \)
$53$
\( (T^{8} - 164 T^{6} + 24780 T^{4} + \cdots + 4477456)^{2} \)
$59$
\( (T^{8} + 272 T^{6} + 57056 T^{4} + \cdots + 286557184)^{2} \)
$61$
\( (T^{8} - 164 T^{6} + 24974 T^{4} + \cdots + 3694084)^{2} \)
$67$
\( (T^{4} + 8 T^{3} + 120 T^{2} - 448 T + 3136)^{4} \)
$71$
\( (T^{4} + 264 T^{2} + 15376)^{4} \)
$73$
\( (T^{8} - 148 T^{6} + 20846 T^{4} + \cdots + 1119364)^{2} \)
$79$
\( (T^{4} + 128 T^{2} + 16384)^{4} \)
$83$
\( (T^{4} - 128 T^{2} + 2048)^{4} \)
$89$
\( (T^{8} + 212 T^{6} + 44366 T^{4} + \cdots + 334084)^{2} \)
$97$
\( (T^{4} + 148 T^{2} + 1058)^{4} \)
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