Newspace parameters
Level: | \( N \) | \(=\) | \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2646.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(21.1284163748\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | \(\Q(\zeta_{24})\) |
Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
Twist minimal: | no (minimal twist has level 378) |
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
\(\beta_{1}\) | \(=\) |
\( \zeta_{24}^{6} \)
|
\(\beta_{2}\) | \(=\) |
\( 2\zeta_{24}^{4} - 1 \)
|
\(\beta_{3}\) | \(=\) |
\( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \)
|
\(\beta_{4}\) | \(=\) |
\( -2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \)
|
\(\beta_{5}\) | \(=\) |
\( -2\zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24} \)
|
\(\beta_{6}\) | \(=\) |
\( 3\zeta_{24}^{5} + 3\zeta_{24}^{3} - 3\zeta_{24} \)
|
\(\beta_{7}\) | \(=\) |
\( -3\zeta_{24}^{5} + 3\zeta_{24}^{3} + 3\zeta_{24} \)
|
\(\zeta_{24}\) | \(=\) |
\( ( \beta_{7} - \beta_{6} - 3\beta_{5} + 3\beta_{4} ) / 12 \)
|
\(\zeta_{24}^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
\(\zeta_{24}^{3}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} ) / 6 \)
|
\(\zeta_{24}^{4}\) | \(=\) |
\( ( \beta_{2} + 1 ) / 2 \)
|
\(\zeta_{24}^{5}\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} - 3\beta_{5} + 3\beta_{4} ) / 12 \)
|
\(\zeta_{24}^{6}\) | \(=\) |
\( \beta_1 \)
|
\(\zeta_{24}^{7}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} - 3\beta_{5} - 3\beta_{4} ) / 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).
\(n\) | \(785\) | \(1081\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2645.1 |
|
− | 1.00000i | 0 | −1.00000 | −2.44949 | 0 | 0 | 1.00000i | 0 | 2.44949i | |||||||||||||||||||||||||||||||||||||||||
2645.2 | − | 1.00000i | 0 | −1.00000 | −2.44949 | 0 | 0 | 1.00000i | 0 | 2.44949i | ||||||||||||||||||||||||||||||||||||||||||
2645.3 | − | 1.00000i | 0 | −1.00000 | 2.44949 | 0 | 0 | 1.00000i | 0 | − | 2.44949i | |||||||||||||||||||||||||||||||||||||||||
2645.4 | − | 1.00000i | 0 | −1.00000 | 2.44949 | 0 | 0 | 1.00000i | 0 | − | 2.44949i | |||||||||||||||||||||||||||||||||||||||||
2645.5 | 1.00000i | 0 | −1.00000 | −2.44949 | 0 | 0 | − | 1.00000i | 0 | − | 2.44949i | |||||||||||||||||||||||||||||||||||||||||
2645.6 | 1.00000i | 0 | −1.00000 | −2.44949 | 0 | 0 | − | 1.00000i | 0 | − | 2.44949i | |||||||||||||||||||||||||||||||||||||||||
2645.7 | 1.00000i | 0 | −1.00000 | 2.44949 | 0 | 0 | − | 1.00000i | 0 | 2.44949i | ||||||||||||||||||||||||||||||||||||||||||
2645.8 | 1.00000i | 0 | −1.00000 | 2.44949 | 0 | 0 | − | 1.00000i | 0 | 2.44949i | ||||||||||||||||||||||||||||||||||||||||||
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 | 2646.2.d.d | 8 | |
3.b | odd | 2 | 1 | inner | 2646.2.d.d | 8 | |
7.b | odd | 2 | 1 | inner | 2646.2.d.d | 8 | |
7.c | even | 3 | 1 | 378.2.k.d | ✓ | 8 | |
7.d | odd | 6 | 1 | 378.2.k.d | ✓ | 8 | |
21.c | even | 2 | 1 | inner | 2646.2.d.d | 8 | |
21.g | even | 6 | 1 | 378.2.k.d | ✓ | 8 | |
21.h | odd | 6 | 1 | 378.2.k.d | ✓ | 8 | |
63.g | even | 3 | 1 | 1134.2.t.f | 8 | ||
63.h | even | 3 | 1 | 1134.2.l.e | 8 | ||
63.i | even | 6 | 1 | 1134.2.l.e | 8 | ||
63.j | odd | 6 | 1 | 1134.2.l.e | 8 | ||
63.k | odd | 6 | 1 | 1134.2.t.f | 8 | ||
63.n | odd | 6 | 1 | 1134.2.t.f | 8 | ||
63.s | even | 6 | 1 | 1134.2.t.f | 8 | ||
63.t | odd | 6 | 1 | 1134.2.l.e | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
378.2.k.d | ✓ | 8 | 7.c | even | 3 | 1 | |
378.2.k.d | ✓ | 8 | 7.d | odd | 6 | 1 | |
378.2.k.d | ✓ | 8 | 21.g | even | 6 | 1 | |
378.2.k.d | ✓ | 8 | 21.h | odd | 6 | 1 | |
1134.2.l.e | 8 | 63.h | even | 3 | 1 | ||
1134.2.l.e | 8 | 63.i | even | 6 | 1 | ||
1134.2.l.e | 8 | 63.j | odd | 6 | 1 | ||
1134.2.l.e | 8 | 63.t | odd | 6 | 1 | ||
1134.2.t.f | 8 | 63.g | even | 3 | 1 | ||
1134.2.t.f | 8 | 63.k | odd | 6 | 1 | ||
1134.2.t.f | 8 | 63.n | odd | 6 | 1 | ||
1134.2.t.f | 8 | 63.s | even | 6 | 1 | ||
2646.2.d.d | 8 | 1.a | even | 1 | 1 | trivial | |
2646.2.d.d | 8 | 3.b | odd | 2 | 1 | inner | |
2646.2.d.d | 8 | 7.b | odd | 2 | 1 | inner | |
2646.2.d.d | 8 | 21.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 6 \)
acting on \(S_{2}^{\mathrm{new}}(2646, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} + 1)^{4} \)
$3$
\( T^{8} \)
$5$
\( (T^{2} - 6)^{4} \)
$7$
\( T^{8} \)
$11$
\( (T^{2} + 18)^{4} \)
$13$
\( (T^{4} + 18 T^{2} + 9)^{2} \)
$17$
\( (T^{2} - 6)^{4} \)
$19$
\( (T^{2} + 24)^{4} \)
$23$
\( (T^{2} + 36)^{4} \)
$29$
\( (T^{4} + 108 T^{2} + 324)^{2} \)
$31$
\( (T^{4} + 114 T^{2} + 2601)^{2} \)
$37$
\( (T^{2} - 2 T - 17)^{4} \)
$41$
\( (T^{2} - 6)^{4} \)
$43$
\( (T - 7)^{8} \)
$47$
\( (T^{4} - 228 T^{2} + 10404)^{2} \)
$53$
\( (T^{4} + 216 T^{2} + 1296)^{2} \)
$59$
\( (T^{2} - 6)^{4} \)
$61$
\( (T^{4} + 18 T^{2} + 9)^{2} \)
$67$
\( (T^{2} - 10 T - 47)^{4} \)
$71$
\( (T^{2} + 162)^{4} \)
$73$
\( (T^{4} + 264 T^{2} + 7056)^{2} \)
$79$
\( (T^{2} - 10 T + 7)^{4} \)
$83$
\( (T^{4} - 264 T^{2} + 7056)^{2} \)
$89$
\( (T^{4} - 324 T^{2} + 2916)^{2} \)
$97$
\( (T^{4} + 54 T^{2} + 441)^{2} \)
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