Newspace parameters
Level: | \( N \) | \(=\) | \( 1728 = 2^{6} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1728.r (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(13.7981494693\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{24})\) |
Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{25}]\) |
Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
Twist minimal: | no (minimal twist has level 576) |
Sato-Tate group: | $\mathrm{U}(1)[D_{6}]$ |
$q$-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
\(\beta_{1}\) | \(=\) |
\( 2\zeta_{24}^{2} \)
|
\(\beta_{2}\) | \(=\) |
\( \zeta_{24}^{4} \)
|
\(\beta_{3}\) | \(=\) |
\( \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}^{2} + 2\zeta_{24} \)
|
\(\beta_{4}\) | \(=\) |
\( -2\zeta_{24}^{6} \)
|
\(\beta_{5}\) | \(=\) |
\( -2\zeta_{24}^{7} - 2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 4\zeta_{24} \)
|
\(\beta_{6}\) | \(=\) |
\( -2\zeta_{24}^{7} + \zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24} \)
|
\(\beta_{7}\) | \(=\) |
\( -2\zeta_{24}^{7} + 4\zeta_{24}^{5} + 4\zeta_{24}^{3} - 2\zeta_{24} \)
|
\(\zeta_{24}\) | \(=\) |
\( ( \beta_{7} - 2\beta_{6} + 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_1 ) / 12 \)
|
\(\zeta_{24}^{2}\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
\(\zeta_{24}^{3}\) | \(=\) |
\( ( \beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} + 4\beta_{3} - 2\beta_1 ) / 12 \)
|
\(\zeta_{24}^{4}\) | \(=\) |
\( \beta_{2} \)
|
\(\zeta_{24}^{5}\) | \(=\) |
\( ( 2\beta_{7} - 4\beta_{6} + \beta_{5} - 2\beta_{4} - 2\beta_{3} + \beta_1 ) / 12 \)
|
\(\zeta_{24}^{6}\) | \(=\) |
\( ( -\beta_{4} ) / 2 \)
|
\(\zeta_{24}^{7}\) | \(=\) |
\( ( -\beta_{7} - 2\beta_{6} - 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_1 ) / 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).
\(n\) | \(325\) | \(703\) | \(1217\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
289.1 |
|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
289.2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
289.3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
289.4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
1441.1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
1441.2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
1441.3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
1441.4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
4.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
9.c | even | 3 | 1 | inner |
36.f | odd | 6 | 1 | inner |
72.n | even | 6 | 1 | inner |
72.p | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1728.2.r.c | 8 | |
3.b | odd | 2 | 1 | 576.2.r.c | ✓ | 8 | |
4.b | odd | 2 | 1 | inner | 1728.2.r.c | 8 | |
8.b | even | 2 | 1 | inner | 1728.2.r.c | 8 | |
8.d | odd | 2 | 1 | CM | 1728.2.r.c | 8 | |
9.c | even | 3 | 1 | inner | 1728.2.r.c | 8 | |
9.c | even | 3 | 1 | 5184.2.d.e | 4 | ||
9.d | odd | 6 | 1 | 576.2.r.c | ✓ | 8 | |
9.d | odd | 6 | 1 | 5184.2.d.l | 4 | ||
12.b | even | 2 | 1 | 576.2.r.c | ✓ | 8 | |
24.f | even | 2 | 1 | 576.2.r.c | ✓ | 8 | |
24.h | odd | 2 | 1 | 576.2.r.c | ✓ | 8 | |
36.f | odd | 6 | 1 | inner | 1728.2.r.c | 8 | |
36.f | odd | 6 | 1 | 5184.2.d.e | 4 | ||
36.h | even | 6 | 1 | 576.2.r.c | ✓ | 8 | |
36.h | even | 6 | 1 | 5184.2.d.l | 4 | ||
72.j | odd | 6 | 1 | 576.2.r.c | ✓ | 8 | |
72.j | odd | 6 | 1 | 5184.2.d.l | 4 | ||
72.l | even | 6 | 1 | 576.2.r.c | ✓ | 8 | |
72.l | even | 6 | 1 | 5184.2.d.l | 4 | ||
72.n | even | 6 | 1 | inner | 1728.2.r.c | 8 | |
72.n | even | 6 | 1 | 5184.2.d.e | 4 | ||
72.p | odd | 6 | 1 | inner | 1728.2.r.c | 8 | |
72.p | odd | 6 | 1 | 5184.2.d.e | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
576.2.r.c | ✓ | 8 | 3.b | odd | 2 | 1 | |
576.2.r.c | ✓ | 8 | 9.d | odd | 6 | 1 | |
576.2.r.c | ✓ | 8 | 12.b | even | 2 | 1 | |
576.2.r.c | ✓ | 8 | 24.f | even | 2 | 1 | |
576.2.r.c | ✓ | 8 | 24.h | odd | 2 | 1 | |
576.2.r.c | ✓ | 8 | 36.h | even | 6 | 1 | |
576.2.r.c | ✓ | 8 | 72.j | odd | 6 | 1 | |
576.2.r.c | ✓ | 8 | 72.l | even | 6 | 1 | |
1728.2.r.c | 8 | 1.a | even | 1 | 1 | trivial | |
1728.2.r.c | 8 | 4.b | odd | 2 | 1 | inner | |
1728.2.r.c | 8 | 8.b | even | 2 | 1 | inner | |
1728.2.r.c | 8 | 8.d | odd | 2 | 1 | CM | |
1728.2.r.c | 8 | 9.c | even | 3 | 1 | inner | |
1728.2.r.c | 8 | 36.f | odd | 6 | 1 | inner | |
1728.2.r.c | 8 | 72.n | even | 6 | 1 | inner | |
1728.2.r.c | 8 | 72.p | odd | 6 | 1 | inner | |
5184.2.d.e | 4 | 9.c | even | 3 | 1 | ||
5184.2.d.e | 4 | 36.f | odd | 6 | 1 | ||
5184.2.d.e | 4 | 72.n | even | 6 | 1 | ||
5184.2.d.e | 4 | 72.p | odd | 6 | 1 | ||
5184.2.d.l | 4 | 9.d | odd | 6 | 1 | ||
5184.2.d.l | 4 | 36.h | even | 6 | 1 | ||
5184.2.d.l | 4 | 72.j | odd | 6 | 1 | ||
5184.2.d.l | 4 | 72.l | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} \)
acting on \(S_{2}^{\mathrm{new}}(1728, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} \)
$5$
\( T^{8} \)
$7$
\( T^{8} \)
$11$
\( T^{8} - 30 T^{6} + 891 T^{4} + \cdots + 81 \)
$13$
\( T^{8} \)
$17$
\( (T^{2} + 6 T - 15)^{4} \)
$19$
\( (T^{4} + 110 T^{2} + 2809)^{2} \)
$23$
\( T^{8} \)
$29$
\( T^{8} \)
$31$
\( T^{8} \)
$37$
\( T^{8} \)
$41$
\( (T^{4} - 6 T^{3} + 123 T^{2} + 522 T + 7569)^{2} \)
$43$
\( T^{8} - 158 T^{6} + 24123 T^{4} + \cdots + 707281 \)
$47$
\( T^{8} \)
$53$
\( T^{8} \)
$59$
\( T^{8} - 318 T^{6} + \cdots + 395254161 \)
$61$
\( T^{8} \)
$67$
\( T^{8} - 206 T^{6} + 42411 T^{4} + \cdots + 625 \)
$71$
\( T^{8} \)
$73$
\( (T^{2} + 2 T - 215)^{4} \)
$79$
\( T^{8} \)
$83$
\( (T^{4} - 324 T^{2} + 104976)^{2} \)
$89$
\( (T + 18)^{8} \)
$97$
\( (T^{4} - 10 T^{3} + 291 T^{2} + \cdots + 36481)^{2} \)
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