Properties

Label 2160.1.bu.a
Level $2160$
Weight $1$
Character orbit 2160.bu
Analytic conductor $1.078$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -20
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2160.bu (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.07798042729\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 720)
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.10497600.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{4} q^{5} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{7} +O(q^{10})\) \( q + \zeta_{12}^{4} q^{5} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{7} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{23} -\zeta_{12}^{2} q^{25} -\zeta_{12}^{2} q^{29} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{35} -\zeta_{12}^{4} q^{41} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{47} + ( -1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{49} -\zeta_{12}^{2} q^{61} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{67} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{83} + q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} + O(q^{10}) \) \( 4q - 2q^{5} - 2q^{25} - 2q^{29} + 2q^{41} - 4q^{49} - 2q^{61} + 4q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(\zeta_{12}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 0 0 −0.500000 0.866025i 0 −0.866025 + 1.50000i 0 0 0
559.2 0 0 0 −0.500000 0.866025i 0 0.866025 1.50000i 0 0 0
1279.1 0 0 0 −0.500000 + 0.866025i 0 −0.866025 1.50000i 0 0 0
1279.2 0 0 0 −0.500000 + 0.866025i 0 0.866025 + 1.50000i 0 0 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner
45.j even 6 1 inner
180.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.1.bu.a 4
3.b odd 2 1 720.1.bu.a 4
4.b odd 2 1 inner 2160.1.bu.a 4
5.b even 2 1 inner 2160.1.bu.a 4
9.c even 3 1 inner 2160.1.bu.a 4
9.d odd 6 1 720.1.bu.a 4
12.b even 2 1 720.1.bu.a 4
15.d odd 2 1 720.1.bu.a 4
15.e even 4 1 3600.1.cc.a 2
15.e even 4 1 3600.1.cc.b 2
20.d odd 2 1 CM 2160.1.bu.a 4
24.f even 2 1 2880.1.bu.c 4
24.h odd 2 1 2880.1.bu.c 4
36.f odd 6 1 inner 2160.1.bu.a 4
36.h even 6 1 720.1.bu.a 4
45.h odd 6 1 720.1.bu.a 4
45.j even 6 1 inner 2160.1.bu.a 4
45.l even 12 1 3600.1.cc.a 2
45.l even 12 1 3600.1.cc.b 2
60.h even 2 1 720.1.bu.a 4
60.l odd 4 1 3600.1.cc.a 2
60.l odd 4 1 3600.1.cc.b 2
72.j odd 6 1 2880.1.bu.c 4
72.l even 6 1 2880.1.bu.c 4
120.i odd 2 1 2880.1.bu.c 4
120.m even 2 1 2880.1.bu.c 4
180.n even 6 1 720.1.bu.a 4
180.p odd 6 1 inner 2160.1.bu.a 4
180.v odd 12 1 3600.1.cc.a 2
180.v odd 12 1 3600.1.cc.b 2
360.bd even 6 1 2880.1.bu.c 4
360.bh odd 6 1 2880.1.bu.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.1.bu.a 4 3.b odd 2 1
720.1.bu.a 4 9.d odd 6 1
720.1.bu.a 4 12.b even 2 1
720.1.bu.a 4 15.d odd 2 1
720.1.bu.a 4 36.h even 6 1
720.1.bu.a 4 45.h odd 6 1
720.1.bu.a 4 60.h even 2 1
720.1.bu.a 4 180.n even 6 1
2160.1.bu.a 4 1.a even 1 1 trivial
2160.1.bu.a 4 4.b odd 2 1 inner
2160.1.bu.a 4 5.b even 2 1 inner
2160.1.bu.a 4 9.c even 3 1 inner
2160.1.bu.a 4 20.d odd 2 1 CM
2160.1.bu.a 4 36.f odd 6 1 inner
2160.1.bu.a 4 45.j even 6 1 inner
2160.1.bu.a 4 180.p odd 6 1 inner
2880.1.bu.c 4 24.f even 2 1
2880.1.bu.c 4 24.h odd 2 1
2880.1.bu.c 4 72.j odd 6 1
2880.1.bu.c 4 72.l even 6 1
2880.1.bu.c 4 120.i odd 2 1
2880.1.bu.c 4 120.m even 2 1
2880.1.bu.c 4 360.bd even 6 1
2880.1.bu.c 4 360.bh odd 6 1
3600.1.cc.a 2 15.e even 4 1
3600.1.cc.a 2 45.l even 12 1
3600.1.cc.a 2 60.l odd 4 1
3600.1.cc.a 2 180.v odd 12 1
3600.1.cc.b 2 15.e even 4 1
3600.1.cc.b 2 45.l even 12 1
3600.1.cc.b 2 60.l odd 4 1
3600.1.cc.b 2 180.v odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2160, [\chi])\).

Hecke characteristic polynomials

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