Newspace parameters
Level: | \( N \) | \(=\) | \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 504.cj (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.02446026187\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 8.0.12960000.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 3^{2} \) |
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 \(1,\beta_1,\ldots,\beta_{7}\) 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 \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( ( -\nu^{6} + 8\nu^{4} - 16\nu^{2} + 19 ) / 8 \) |
\(\beta_{2}\) | \(=\) | \( ( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 1 ) / 8 \) |
\(\beta_{3}\) | \(=\) | \( ( -\nu^{7} + 4\nu^{5} - 12\nu^{3} + 11\nu ) / 4 \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{7} + 2\nu^{5} - 6\nu^{3} - 3\nu ) / 2 \) |
\(\beta_{5}\) | \(=\) | \( ( -5\nu^{7} + 16\nu^{5} - 40\nu^{3} + 23\nu ) / 8 \) |
\(\beta_{6}\) | \(=\) | \( ( -7\nu^{6} + 16\nu^{4} - 40\nu^{2} - 3 ) / 8 \) |
\(\beta_{7}\) | \(=\) | \( -\nu^{7} + 3\nu^{5} - 8\nu^{3} + 2\nu \) |
\(\nu\) | \(=\) | \( ( -2\beta_{7} + 2\beta_{5} + \beta_{4} + \beta_{3} ) / 3 \) |
\(\nu^{2}\) | \(=\) | \( ( 2\beta_{6} - 5\beta_{2} + \beta _1 - 1 ) / 3 \) |
\(\nu^{3}\) | \(=\) | \( ( -\beta_{7} + 4\beta_{5} - \beta_{4} - 4\beta_{3} ) / 3 \) |
\(\nu^{4}\) | \(=\) | \( \beta_{6} - 3\beta_{2} + 2\beta _1 - 4 \) |
\(\nu^{5}\) | \(=\) | \( ( 11\beta_{7} - 2\beta_{5} - 13\beta_{4} - 13\beta_{3} ) / 3 \) |
\(\nu^{6}\) | \(=\) | \( ( -8\beta_{6} + 8\beta_{2} + 8\beta _1 - 23 ) / 3 \) |
\(\nu^{7}\) | \(=\) | \( ( 34\beta_{7} - 34\beta_{5} - 29\beta_{4} - 5\beta_{3} ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).
\(n\) | \(73\) | \(127\) | \(253\) | \(281\) |
\(\chi(n)\) | \(\beta_{2}\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 |
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−1.40126 | + | 0.190983i | 0 | 1.92705 | − | 0.535233i | −1.93649 | − | 1.11803i | 0 | −0.500000 | + | 2.59808i | −2.59808 | + | 1.11803i | 0 | 2.92705 | + | 1.19682i | ||||||||||||||||||||||||||||||
37.2 | −0.535233 | − | 1.30902i | 0 | −1.42705 | + | 1.40126i | −1.93649 | − | 1.11803i | 0 | −0.500000 | + | 2.59808i | 2.59808 | + | 1.11803i | 0 | −0.427051 | + | 3.13331i | |||||||||||||||||||||||||||||||
37.3 | 0.535233 | + | 1.30902i | 0 | −1.42705 | + | 1.40126i | 1.93649 | + | 1.11803i | 0 | −0.500000 | + | 2.59808i | −2.59808 | − | 1.11803i | 0 | −0.427051 | + | 3.13331i | |||||||||||||||||||||||||||||||
37.4 | 1.40126 | − | 0.190983i | 0 | 1.92705 | − | 0.535233i | 1.93649 | + | 1.11803i | 0 | −0.500000 | + | 2.59808i | 2.59808 | − | 1.11803i | 0 | 2.92705 | + | 1.19682i | |||||||||||||||||||||||||||||||
109.1 | −1.40126 | − | 0.190983i | 0 | 1.92705 | + | 0.535233i | −1.93649 | + | 1.11803i | 0 | −0.500000 | − | 2.59808i | −2.59808 | − | 1.11803i | 0 | 2.92705 | − | 1.19682i | |||||||||||||||||||||||||||||||
109.2 | −0.535233 | + | 1.30902i | 0 | −1.42705 | − | 1.40126i | −1.93649 | + | 1.11803i | 0 | −0.500000 | − | 2.59808i | 2.59808 | − | 1.11803i | 0 | −0.427051 | − | 3.13331i | |||||||||||||||||||||||||||||||
109.3 | 0.535233 | − | 1.30902i | 0 | −1.42705 | − | 1.40126i | 1.93649 | − | 1.11803i | 0 | −0.500000 | − | 2.59808i | −2.59808 | + | 1.11803i | 0 | −0.427051 | − | 3.13331i | |||||||||||||||||||||||||||||||
109.4 | 1.40126 | + | 0.190983i | 0 | 1.92705 | + | 0.535233i | 1.93649 | − | 1.11803i | 0 | −0.500000 | − | 2.59808i | 2.59808 | + | 1.11803i | 0 | 2.92705 | − | 1.19682i | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
8.b | even | 2 | 1 | inner |
21.h | odd | 6 | 1 | inner |
24.h | odd | 2 | 1 | inner |
56.p | even | 6 | 1 | inner |
168.s | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 504.2.cj.a | ✓ | 8 |
3.b | odd | 2 | 1 | inner | 504.2.cj.a | ✓ | 8 |
4.b | odd | 2 | 1 | 2016.2.cr.a | 8 | ||
7.c | even | 3 | 1 | inner | 504.2.cj.a | ✓ | 8 |
8.b | even | 2 | 1 | inner | 504.2.cj.a | ✓ | 8 |
8.d | odd | 2 | 1 | 2016.2.cr.a | 8 | ||
12.b | even | 2 | 1 | 2016.2.cr.a | 8 | ||
21.h | odd | 6 | 1 | inner | 504.2.cj.a | ✓ | 8 |
24.f | even | 2 | 1 | 2016.2.cr.a | 8 | ||
24.h | odd | 2 | 1 | inner | 504.2.cj.a | ✓ | 8 |
28.g | odd | 6 | 1 | 2016.2.cr.a | 8 | ||
56.k | odd | 6 | 1 | 2016.2.cr.a | 8 | ||
56.p | even | 6 | 1 | inner | 504.2.cj.a | ✓ | 8 |
84.n | even | 6 | 1 | 2016.2.cr.a | 8 | ||
168.s | odd | 6 | 1 | inner | 504.2.cj.a | ✓ | 8 |
168.v | even | 6 | 1 | 2016.2.cr.a | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
504.2.cj.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
504.2.cj.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
504.2.cj.a | ✓ | 8 | 7.c | even | 3 | 1 | inner |
504.2.cj.a | ✓ | 8 | 8.b | even | 2 | 1 | inner |
504.2.cj.a | ✓ | 8 | 21.h | odd | 6 | 1 | inner |
504.2.cj.a | ✓ | 8 | 24.h | odd | 2 | 1 | inner |
504.2.cj.a | ✓ | 8 | 56.p | even | 6 | 1 | inner |
504.2.cj.a | ✓ | 8 | 168.s | odd | 6 | 1 | inner |
2016.2.cr.a | 8 | 4.b | odd | 2 | 1 | ||
2016.2.cr.a | 8 | 8.d | odd | 2 | 1 | ||
2016.2.cr.a | 8 | 12.b | even | 2 | 1 | ||
2016.2.cr.a | 8 | 24.f | even | 2 | 1 | ||
2016.2.cr.a | 8 | 28.g | odd | 6 | 1 | ||
2016.2.cr.a | 8 | 56.k | odd | 6 | 1 | ||
2016.2.cr.a | 8 | 84.n | even | 6 | 1 | ||
2016.2.cr.a | 8 | 168.v | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 5T_{5}^{2} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - T^{6} - 3 T^{4} - 4 T^{2} + \cdots + 16 \)
$3$
\( T^{8} \)
$5$
\( (T^{4} - 5 T^{2} + 25)^{2} \)
$7$
\( (T^{2} + T + 7)^{4} \)
$11$
\( (T^{4} - 5 T^{2} + 25)^{2} \)
$13$
\( T^{8} \)
$17$
\( (T^{4} + 12 T^{2} + 144)^{2} \)
$19$
\( (T^{4} - 60 T^{2} + 3600)^{2} \)
$23$
\( (T^{4} + 48 T^{2} + 2304)^{2} \)
$29$
\( (T^{2} + 5)^{4} \)
$31$
\( (T^{2} - T + 1)^{4} \)
$37$
\( (T^{4} - 60 T^{2} + 3600)^{2} \)
$41$
\( (T^{2} - 108)^{4} \)
$43$
\( T^{8} \)
$47$
\( (T^{4} + 12 T^{2} + 144)^{2} \)
$53$
\( (T^{4} - 125 T^{2} + 15625)^{2} \)
$59$
\( (T^{4} - 5 T^{2} + 25)^{2} \)
$61$
\( (T^{4} - 60 T^{2} + 3600)^{2} \)
$67$
\( (T^{4} - 60 T^{2} + 3600)^{2} \)
$71$
\( (T^{2} - 108)^{4} \)
$73$
\( (T^{2} - 10 T + 100)^{4} \)
$79$
\( (T^{2} - 13 T + 169)^{4} \)
$83$
\( (T^{2} + 125)^{4} \)
$89$
\( (T^{4} + 192 T^{2} + 36864)^{2} \)
$97$
\( (T + 1)^{8} \)
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