Newspace parameters
Level: | \( N \) | \(=\) | \( 585 = 3^{2} \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 585.bu (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.67124851824\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 8.0.191102976.5 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - 6x^{6} + 6x^{4} + 36x^{2} + 36 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 195) |
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 \( x^{8} - 6x^{6} + 6x^{4} + 36x^{2} + 36 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{6} - 9\nu^{4} + 48\nu^{2} - 18 ) / 90 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{7} - 9\nu^{5} + 48\nu^{3} - 108\nu ) / 90 \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{6} + 9\nu^{4} - 18\nu^{2} - 12 ) / 30 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{7} + 9\nu^{5} - 18\nu^{3} - 12\nu ) / 30 \) |
\(\beta_{6}\) | \(=\) | \( ( -4\nu^{6} + 21\nu^{4} - 12\nu^{2} - 108 ) / 90 \) |
\(\beta_{7}\) | \(=\) | \( ( -4\nu^{7} + 21\nu^{5} - 12\nu^{3} - 108\nu ) / 90 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{4} + 3\beta_{2} + 1 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{5} + 3\beta_{3} + 4\beta_1 \) |
\(\nu^{4}\) | \(=\) | \( -6\beta_{6} + 12\beta_{4} + 12\beta_{2} \) |
\(\nu^{5}\) | \(=\) | \( -6\beta_{7} + 12\beta_{5} + 12\beta_{3} + 12\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( -54\beta_{6} + 60\beta_{4} + 54\beta_{2} - 30 \) |
\(\nu^{7}\) | \(=\) | \( -54\beta_{7} + 60\beta_{5} + 54\beta_{3} + 24\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).
\(n\) | \(326\) | \(352\) | \(496\) |
\(\chi(n)\) | \(1\) | \(1\) | \(\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
316.1 |
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−1.88389 | − | 1.08766i | 0 | 1.36603 | + | 2.36603i | − | 1.00000i | 0 | −0.383889 | + | 0.221638i | − | 1.59245i | 0 | −1.08766 | + | 1.88389i | ||||||||||||||||||||||||||||||||
316.2 | −0.975173 | − | 0.563016i | 0 | −0.366025 | − | 0.633975i | 1.00000i | 0 | 0.524827 | − | 0.303009i | 3.07638i | 0 | 0.563016 | − | 0.975173i | |||||||||||||||||||||||||||||||||||
316.3 | 0.975173 | + | 0.563016i | 0 | −0.366025 | − | 0.633975i | 1.00000i | 0 | 2.47517 | − | 1.42904i | − | 3.07638i | 0 | −0.563016 | + | 0.975173i | ||||||||||||||||||||||||||||||||||
316.4 | 1.88389 | + | 1.08766i | 0 | 1.36603 | + | 2.36603i | − | 1.00000i | 0 | 3.38389 | − | 1.95369i | 1.59245i | 0 | 1.08766 | − | 1.88389i | ||||||||||||||||||||||||||||||||||
361.1 | −1.88389 | + | 1.08766i | 0 | 1.36603 | − | 2.36603i | 1.00000i | 0 | −0.383889 | − | 0.221638i | 1.59245i | 0 | −1.08766 | − | 1.88389i | |||||||||||||||||||||||||||||||||||
361.2 | −0.975173 | + | 0.563016i | 0 | −0.366025 | + | 0.633975i | − | 1.00000i | 0 | 0.524827 | + | 0.303009i | − | 3.07638i | 0 | 0.563016 | + | 0.975173i | |||||||||||||||||||||||||||||||||
361.3 | 0.975173 | − | 0.563016i | 0 | −0.366025 | + | 0.633975i | − | 1.00000i | 0 | 2.47517 | + | 1.42904i | 3.07638i | 0 | −0.563016 | − | 0.975173i | ||||||||||||||||||||||||||||||||||
361.4 | 1.88389 | − | 1.08766i | 0 | 1.36603 | − | 2.36603i | 1.00000i | 0 | 3.38389 | + | 1.95369i | − | 1.59245i | 0 | 1.08766 | + | 1.88389i | ||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.e | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 585.2.bu.d | 8 | |
3.b | odd | 2 | 1 | 195.2.bb.b | ✓ | 8 | |
13.e | even | 6 | 1 | inner | 585.2.bu.d | 8 | |
13.f | odd | 12 | 1 | 7605.2.a.ch | 4 | ||
13.f | odd | 12 | 1 | 7605.2.a.ci | 4 | ||
15.d | odd | 2 | 1 | 975.2.bc.j | 8 | ||
15.e | even | 4 | 1 | 975.2.w.h | 8 | ||
15.e | even | 4 | 1 | 975.2.w.i | 8 | ||
39.h | odd | 6 | 1 | 195.2.bb.b | ✓ | 8 | |
39.k | even | 12 | 1 | 2535.2.a.bj | 4 | ||
39.k | even | 12 | 1 | 2535.2.a.bk | 4 | ||
195.y | odd | 6 | 1 | 975.2.bc.j | 8 | ||
195.bf | even | 12 | 1 | 975.2.w.h | 8 | ||
195.bf | even | 12 | 1 | 975.2.w.i | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
195.2.bb.b | ✓ | 8 | 3.b | odd | 2 | 1 | |
195.2.bb.b | ✓ | 8 | 39.h | odd | 6 | 1 | |
585.2.bu.d | 8 | 1.a | even | 1 | 1 | trivial | |
585.2.bu.d | 8 | 13.e | even | 6 | 1 | inner | |
975.2.w.h | 8 | 15.e | even | 4 | 1 | ||
975.2.w.h | 8 | 195.bf | even | 12 | 1 | ||
975.2.w.i | 8 | 15.e | even | 4 | 1 | ||
975.2.w.i | 8 | 195.bf | even | 12 | 1 | ||
975.2.bc.j | 8 | 15.d | odd | 2 | 1 | ||
975.2.bc.j | 8 | 195.y | odd | 6 | 1 | ||
2535.2.a.bj | 4 | 39.k | even | 12 | 1 | ||
2535.2.a.bk | 4 | 39.k | even | 12 | 1 | ||
7605.2.a.ch | 4 | 13.f | odd | 12 | 1 | ||
7605.2.a.ci | 4 | 13.f | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 6T_{2}^{6} + 30T_{2}^{4} - 36T_{2}^{2} + 36 \)
acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - 6 T^{6} + 30 T^{4} - 36 T^{2} + \cdots + 36 \)
$3$
\( T^{8} \)
$5$
\( (T^{2} + 1)^{4} \)
$7$
\( T^{8} - 12 T^{7} + 60 T^{6} - 144 T^{5} + \cdots + 9 \)
$11$
\( T^{8} - 12 T^{6} + 120 T^{4} + \cdots + 576 \)
$13$
\( T^{8} + 8 T^{7} + 16 T^{6} + \cdots + 28561 \)
$17$
\( T^{8} + 18 T^{6} + 270 T^{4} + \cdots + 2916 \)
$19$
\( T^{8} + 12 T^{7} - 576 T^{5} + \cdots + 197136 \)
$23$
\( T^{8} + 60 T^{6} + 288 T^{5} + \cdots + 5184 \)
$29$
\( T^{8} + 12 T^{7} + 114 T^{6} + \cdots + 54756 \)
$31$
\( T^{8} + 132 T^{6} + 4422 T^{4} + \cdots + 1089 \)
$37$
\( T^{8} - 96 T^{6} + 8688 T^{4} + \cdots + 278784 \)
$41$
\( T^{8} + 36 T^{7} + 570 T^{6} + \cdots + 49284 \)
$43$
\( T^{8} - 16 T^{7} + 220 T^{6} + \cdots + 2319529 \)
$47$
\( T^{8} + 180 T^{6} + 9408 T^{4} + \cdots + 427716 \)
$53$
\( (T^{4} - 144 T^{2} + 3456)^{2} \)
$59$
\( T^{8} + 36 T^{7} + 522 T^{6} + \cdots + 4435236 \)
$61$
\( (T^{4} - 16 T^{3} + 195 T^{2} - 976 T + 3721)^{2} \)
$67$
\( T^{8} + 48 T^{7} + 1020 T^{6} + \cdots + 962361 \)
$71$
\( T^{8} + 36 T^{7} + 462 T^{6} + \cdots + 1313316 \)
$73$
\( T^{8} + 264 T^{6} + 20250 T^{4} + \cdots + 1390041 \)
$79$
\( (T^{4} + 8 T^{3} - 102 T^{2} - 904 T - 1703)^{2} \)
$83$
\( (T^{4} + 132 T^{2} + 4056)^{2} \)
$89$
\( T^{8} + 36 T^{7} + 462 T^{6} + \cdots + 4356 \)
$97$
\( T^{8} - 180 T^{6} + \cdots + 26101881 \)
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