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: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).
\(n\) | \(326\) | \(352\) | \(496\) |
\(\chi(n)\) | \(1\) | \(1\) | \(1 - \zeta_{12}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
316.1 |
|
0.633975 | + | 0.366025i | 0 | −0.732051 | − | 1.26795i | 1.00000i | 0 | −3.86603 | + | 2.23205i | − | 2.53590i | 0 | −0.366025 | + | 0.633975i | |||||||||||||||||||||
316.2 | 2.36603 | + | 1.36603i | 0 | 2.73205 | + | 4.73205i | − | 1.00000i | 0 | −2.13397 | + | 1.23205i | 9.46410i | 0 | 1.36603 | − | 2.36603i | ||||||||||||||||||||||
361.1 | 0.633975 | − | 0.366025i | 0 | −0.732051 | + | 1.26795i | − | 1.00000i | 0 | −3.86603 | − | 2.23205i | 2.53590i | 0 | −0.366025 | − | 0.633975i | ||||||||||||||||||||||
361.2 | 2.36603 | − | 1.36603i | 0 | 2.73205 | − | 4.73205i | 1.00000i | 0 | −2.13397 | − | 1.23205i | − | 9.46410i | 0 | 1.36603 | + | 2.36603i | ||||||||||||||||||||||
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.a | 4 | |
3.b | odd | 2 | 1 | 195.2.bb.a | ✓ | 4 | |
13.e | even | 6 | 1 | inner | 585.2.bu.a | 4 | |
13.f | odd | 12 | 1 | 7605.2.a.y | 2 | ||
13.f | odd | 12 | 1 | 7605.2.a.bk | 2 | ||
15.d | odd | 2 | 1 | 975.2.bc.h | 4 | ||
15.e | even | 4 | 1 | 975.2.w.a | 4 | ||
15.e | even | 4 | 1 | 975.2.w.f | 4 | ||
39.h | odd | 6 | 1 | 195.2.bb.a | ✓ | 4 | |
39.k | even | 12 | 1 | 2535.2.a.n | 2 | ||
39.k | even | 12 | 1 | 2535.2.a.s | 2 | ||
195.y | odd | 6 | 1 | 975.2.bc.h | 4 | ||
195.bf | even | 12 | 1 | 975.2.w.a | 4 | ||
195.bf | even | 12 | 1 | 975.2.w.f | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
195.2.bb.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
195.2.bb.a | ✓ | 4 | 39.h | odd | 6 | 1 | |
585.2.bu.a | 4 | 1.a | even | 1 | 1 | trivial | |
585.2.bu.a | 4 | 13.e | even | 6 | 1 | inner | |
975.2.w.a | 4 | 15.e | even | 4 | 1 | ||
975.2.w.a | 4 | 195.bf | even | 12 | 1 | ||
975.2.w.f | 4 | 15.e | even | 4 | 1 | ||
975.2.w.f | 4 | 195.bf | even | 12 | 1 | ||
975.2.bc.h | 4 | 15.d | odd | 2 | 1 | ||
975.2.bc.h | 4 | 195.y | odd | 6 | 1 | ||
2535.2.a.n | 2 | 39.k | even | 12 | 1 | ||
2535.2.a.s | 2 | 39.k | even | 12 | 1 | ||
7605.2.a.y | 2 | 13.f | odd | 12 | 1 | ||
7605.2.a.bk | 2 | 13.f | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 6T_{2}^{3} + 14T_{2}^{2} - 12T_{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 6 T^{3} + 14 T^{2} - 12 T + 4 \)
$3$
\( T^{4} \)
$5$
\( (T^{2} + 1)^{2} \)
$7$
\( T^{4} + 12 T^{3} + 59 T^{2} + \cdots + 121 \)
$11$
\( (T^{2} - 6 T + 12)^{2} \)
$13$
\( T^{4} + 23T^{2} + 169 \)
$17$
\( T^{4} - 10 T^{3} + 78 T^{2} + \cdots + 484 \)
$19$
\( T^{4} + 12 T^{3} + 56 T^{2} + 96 T + 64 \)
$23$
\( T^{4} - 8 T^{3} + 60 T^{2} - 32 T + 16 \)
$29$
\( T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4 \)
$31$
\( T^{4} + 62T^{2} + 529 \)
$37$
\( T^{4} - 16T^{2} + 256 \)
$41$
\( T^{4} - 6 T^{3} - 34 T^{2} + \cdots + 2116 \)
$43$
\( T^{4} + 4 T^{3} + 15 T^{2} + 4 T + 1 \)
$47$
\( T^{4} + 104T^{2} + 4 \)
$53$
\( (T^{2} - 48)^{2} \)
$59$
\( T^{4} - 6 T^{3} - 66 T^{2} + \cdots + 6084 \)
$61$
\( T^{4} + 2 T^{3} + 15 T^{2} - 22 T + 121 \)
$67$
\( T^{4} + 12 T^{3} - 21 T^{2} + \cdots + 4761 \)
$71$
\( T^{4} - 6 T^{3} - 106 T^{2} + \cdots + 13924 \)
$73$
\( T^{4} + 266T^{2} + 6889 \)
$79$
\( (T^{2} + 10 T - 23)^{2} \)
$83$
\( T^{4} + 96T^{2} + 576 \)
$89$
\( T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36 \)
$97$
\( T^{4} + 12 T^{3} - 109 T^{2} + \cdots + 24649 \)
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