Properties

Label 1584.1.bf.b
Level $1584$
Weight $1$
Character orbit 1584.bf
Analytic conductor $0.791$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -11
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1584.bf (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 99)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.891.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} + \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{6}^{2} q^{3} - \zeta_{6}^{2} q^{5} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6}^{2} q^{3} - \zeta_{6}^{2} q^{5} - \zeta_{6} q^{9} + \zeta_{6} q^{11} - \zeta_{6} q^{15} - \zeta_{6}^{2} q^{23} - q^{27} + \zeta_{6}^{2} q^{31} + q^{33} - q^{37} - q^{45} - \zeta_{6} q^{47} + \zeta_{6}^{2} q^{49} - q^{53} + q^{55} + \zeta_{6}^{2} q^{59} + \zeta_{6}^{2} q^{67} - 2 \zeta_{6} q^{69} + q^{71} + \zeta_{6}^{2} q^{81} + q^{89} + \zeta_{6} q^{93} + \zeta_{6} q^{97} - \zeta_{6}^{2} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + q^{5} - q^{9} + q^{11} - q^{15} + 2 q^{23} - 2 q^{27} - q^{31} + 2 q^{33} - 2 q^{37} - 2 q^{45} - q^{47} - q^{49} - 2 q^{53} + 2 q^{55} - q^{59} - q^{67} - 2 q^{69} + 2 q^{71} - q^{81} + 4 q^{89} + q^{93} + q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(353\) \(991\) \(1189\)
\(\chi(n)\) \(-1\) \(\zeta_{6}^{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
241.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 0 0 −0.500000 + 0.866025i 0
769.1 0 0.500000 0.866025i 0 0.500000 0.866025i 0 0 0 −0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
9.c even 3 1 inner
99.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1584.1.bf.b 2
4.b odd 2 1 99.1.h.a 2
9.c even 3 1 inner 1584.1.bf.b 2
11.b odd 2 1 CM 1584.1.bf.b 2
12.b even 2 1 297.1.h.a 2
20.d odd 2 1 2475.1.y.a 2
20.e even 4 2 2475.1.t.a 4
36.f odd 6 1 99.1.h.a 2
36.f odd 6 1 891.1.c.a 1
36.h even 6 1 297.1.h.a 2
36.h even 6 1 891.1.c.b 1
44.c even 2 1 99.1.h.a 2
44.g even 10 4 1089.1.s.a 8
44.h odd 10 4 1089.1.s.a 8
99.h odd 6 1 inner 1584.1.bf.b 2
132.d odd 2 1 297.1.h.a 2
132.n odd 10 4 3267.1.w.a 8
132.o even 10 4 3267.1.w.a 8
180.p odd 6 1 2475.1.y.a 2
180.x even 12 2 2475.1.t.a 4
220.g even 2 1 2475.1.y.a 2
220.i odd 4 2 2475.1.t.a 4
396.k even 6 1 99.1.h.a 2
396.k even 6 1 891.1.c.a 1
396.o odd 6 1 297.1.h.a 2
396.o odd 6 1 891.1.c.b 1
396.ba even 30 4 3267.1.w.a 8
396.bb odd 30 4 3267.1.w.a 8
396.be odd 30 4 1089.1.s.a 8
396.bf even 30 4 1089.1.s.a 8
1980.bk even 6 1 2475.1.y.a 2
1980.cf odd 12 2 2475.1.t.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.1.h.a 2 4.b odd 2 1
99.1.h.a 2 36.f odd 6 1
99.1.h.a 2 44.c even 2 1
99.1.h.a 2 396.k even 6 1
297.1.h.a 2 12.b even 2 1
297.1.h.a 2 36.h even 6 1
297.1.h.a 2 132.d odd 2 1
297.1.h.a 2 396.o odd 6 1
891.1.c.a 1 36.f odd 6 1
891.1.c.a 1 396.k even 6 1
891.1.c.b 1 36.h even 6 1
891.1.c.b 1 396.o odd 6 1
1089.1.s.a 8 44.g even 10 4
1089.1.s.a 8 44.h odd 10 4
1089.1.s.a 8 396.be odd 30 4
1089.1.s.a 8 396.bf even 30 4
1584.1.bf.b 2 1.a even 1 1 trivial
1584.1.bf.b 2 9.c even 3 1 inner
1584.1.bf.b 2 11.b odd 2 1 CM
1584.1.bf.b 2 99.h odd 6 1 inner
2475.1.t.a 4 20.e even 4 2
2475.1.t.a 4 180.x even 12 2
2475.1.t.a 4 220.i odd 4 2
2475.1.t.a 4 1980.cf odd 12 2
2475.1.y.a 2 20.d odd 2 1
2475.1.y.a 2 180.p odd 6 1
2475.1.y.a 2 220.g even 2 1
2475.1.y.a 2 1980.bk even 6 1
3267.1.w.a 8 132.n odd 10 4
3267.1.w.a 8 132.o even 10 4
3267.1.w.a 8 396.ba even 30 4
3267.1.w.a 8 396.bb odd 30 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1584, [\chi])\):

\( T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display
\( T_{23}^{2} - 2T_{23} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$53$ \( (T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$71$ \( (T - 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - T + 1 \) Copy content Toggle raw display
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