Newspace parameters
Level: | \( N \) | \(=\) | \( 864 = 2^{5} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 864.h (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(23.5422948407\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.242095489024.11 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 2x^{6} + 2x^{4} - 32x^{2} + 256 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
Twist minimal: | no (minimal twist has level 216) |
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 in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 2x^{4} - 32x^{2} + 256 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{6} + 14\nu^{4} - 14\nu^{2} - 16 ) / 80 \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{7} + 6\nu^{5} - 26\nu^{3} + 136\nu ) / 80 \) |
\(\beta_{3}\) | \(=\) | \( ( -3\nu^{7} - 2\nu^{5} + 42\nu^{3} + 208\nu ) / 160 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{7} + 6\nu^{5} + 18\nu^{3} - 16\nu ) / 32 \) |
\(\beta_{5}\) | \(=\) | \( ( \nu^{7} - 2\nu^{5} + 2\nu^{3} + 32\nu ) / 32 \) |
\(\beta_{6}\) | \(=\) | \( ( \nu^{6} - 2\nu^{4} + 18\nu^{2} - 32 ) / 16 \) |
\(\beta_{7}\) | \(=\) | \( ( -\nu^{6} + 2\nu^{4} + 14\nu^{2} + 16 ) / 16 \) |
\(\nu\) | \(=\) | \( ( \beta_{5} + \beta_{3} + \beta_{2} ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{7} + \beta_{6} + 1 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{4} + 3\beta_{3} - 2\beta_{2} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{7} + 5\beta_1 \) |
\(\nu^{5}\) | \(=\) | \( ( -5\beta_{5} + 6\beta_{4} - 3\beta_{3} + 7\beta_{2} ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( -7\beta_{7} + 7\beta_{6} + 10\beta _1 + 23 \) |
\(\nu^{7}\) | \(=\) | \( 19\beta_{5} + 5\beta_{4} - 14\beta_{3} + \beta_{2} \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).
\(n\) | \(325\) | \(353\) | \(703\) |
\(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
593.1 |
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0 | 0 | 0 | −5.39773 | 0 | −11.5678 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
593.2 | 0 | 0 | 0 | −5.39773 | 0 | −11.5678 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
593.3 | 0 | 0 | 0 | −2.62001 | 0 | −0.432236 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
593.4 | 0 | 0 | 0 | −2.62001 | 0 | −0.432236 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
593.5 | 0 | 0 | 0 | 2.62001 | 0 | −0.432236 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
593.6 | 0 | 0 | 0 | 2.62001 | 0 | −0.432236 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
593.7 | 0 | 0 | 0 | 5.39773 | 0 | −11.5678 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
593.8 | 0 | 0 | 0 | 5.39773 | 0 | −11.5678 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
24.h | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 864.3.h.e | 8 | |
3.b | odd | 2 | 1 | inner | 864.3.h.e | 8 | |
4.b | odd | 2 | 1 | 216.3.h.f | ✓ | 8 | |
8.b | even | 2 | 1 | inner | 864.3.h.e | 8 | |
8.d | odd | 2 | 1 | 216.3.h.f | ✓ | 8 | |
12.b | even | 2 | 1 | 216.3.h.f | ✓ | 8 | |
24.f | even | 2 | 1 | 216.3.h.f | ✓ | 8 | |
24.h | odd | 2 | 1 | inner | 864.3.h.e | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
216.3.h.f | ✓ | 8 | 4.b | odd | 2 | 1 | |
216.3.h.f | ✓ | 8 | 8.d | odd | 2 | 1 | |
216.3.h.f | ✓ | 8 | 12.b | even | 2 | 1 | |
216.3.h.f | ✓ | 8 | 24.f | even | 2 | 1 | |
864.3.h.e | 8 | 1.a | even | 1 | 1 | trivial | |
864.3.h.e | 8 | 3.b | odd | 2 | 1 | inner | |
864.3.h.e | 8 | 8.b | even | 2 | 1 | inner | |
864.3.h.e | 8 | 24.h | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 36T_{5}^{2} + 200 \)
acting on \(S_{3}^{\mathrm{new}}(864, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} \)
$5$
\( (T^{4} - 36 T^{2} + 200)^{2} \)
$7$
\( (T^{2} + 12 T + 5)^{4} \)
$11$
\( (T^{4} - 28 T^{2} + 72)^{2} \)
$13$
\( (T^{4} + 94 T^{2} + 225)^{2} \)
$17$
\( (T^{4} + 764 T^{2} + 145800)^{2} \)
$19$
\( (T^{4} + 586 T^{2} + 2025)^{2} \)
$23$
\( (T^{4} + 932 T^{2} + 8712)^{2} \)
$29$
\( (T^{4} - 3616 T^{2} + 2880000)^{2} \)
$31$
\( (T^{2} - 20 T - 1016)^{4} \)
$37$
\( (T^{4} + 3430 T^{2} + 1750329)^{2} \)
$41$
\( (T^{4} + 7952 T^{2} + 15235200)^{2} \)
$43$
\( (T^{4} + 7768 T^{2} + 5363856)^{2} \)
$47$
\( (T^{4} + 7812 T^{2} + 5604552)^{2} \)
$53$
\( (T^{4} - 5488 T^{2} + 6480000)^{2} \)
$59$
\( (T^{4} - 11196 T^{2} + 31331528)^{2} \)
$61$
\( (T^{4} + 846 T^{2} + 18225)^{2} \)
$67$
\( (T^{4} + 11106 T^{2} + 11390625)^{2} \)
$71$
\( (T^{4} + 16272 T^{2} + 58320000)^{2} \)
$73$
\( (T^{2} - 62 T + 465)^{4} \)
$79$
\( (T^{2} + 48 T - 199)^{4} \)
$83$
\( (T^{4} - 7200 T^{2} + 8000000)^{2} \)
$89$
\( (T^{4} + 19292 T^{2} + 32805000)^{2} \)
$97$
\( (T^{2} + 18 T - 5995)^{4} \)
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