Properties

Label 540.1.f.a
Level $540$
Weight $1$
Character orbit 540.f
Analytic conductor $0.269$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -15
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 540.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.269495106822\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.1166400.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{3} q^{5} -\zeta_{12}^{3} q^{8} +O(q^{10})\) \( q -\zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{3} q^{5} -\zeta_{12}^{3} q^{8} -\zeta_{12}^{4} q^{10} + \zeta_{12}^{4} q^{16} + \zeta_{12}^{3} q^{17} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{19} + \zeta_{12}^{5} q^{20} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{23} - q^{25} + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{31} -\zeta_{12}^{5} q^{32} -\zeta_{12}^{4} q^{34} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{38} + q^{40} + ( -1 - \zeta_{12}^{2} ) q^{46} - q^{49} + \zeta_{12} q^{50} -\zeta_{12}^{3} q^{53} + q^{61} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{62} - q^{64} + \zeta_{12}^{5} q^{68} + ( -1 + \zeta_{12}^{4} ) q^{76} + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{79} -\zeta_{12} q^{80} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{83} - q^{85} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{92} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{95} + \zeta_{12} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + O(q^{10}) \) \( 4 q + 2 q^{4} + 2 q^{10} - 2 q^{16} - 4 q^{25} + 2 q^{34} + 4 q^{40} - 6 q^{46} - 4 q^{49} + 4 q^{61} - 4 q^{64} - 6 q^{76} - 4 q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/540\mathbb{Z}\right)^\times\).

\(n\) \(217\) \(271\) \(461\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 0 1.00000i 0 0.500000 0.866025i
379.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.00000i 0 0 1.00000i 0 0.500000 + 0.866025i
379.3 0.866025 0.500000i 0 0.500000 0.866025i 1.00000i 0 0 1.00000i 0 0.500000 + 0.866025i
379.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.00000i 0 0 1.00000i 0 0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 540.1.f.a 4
3.b odd 2 1 inner 540.1.f.a 4
4.b odd 2 1 inner 540.1.f.a 4
5.b even 2 1 inner 540.1.f.a 4
5.c odd 4 1 2700.1.c.a 2
5.c odd 4 1 2700.1.c.b 2
9.c even 3 1 1620.1.p.a 4
9.c even 3 1 1620.1.p.d 4
9.d odd 6 1 1620.1.p.a 4
9.d odd 6 1 1620.1.p.d 4
12.b even 2 1 inner 540.1.f.a 4
15.d odd 2 1 CM 540.1.f.a 4
15.e even 4 1 2700.1.c.a 2
15.e even 4 1 2700.1.c.b 2
20.d odd 2 1 inner 540.1.f.a 4
20.e even 4 1 2700.1.c.a 2
20.e even 4 1 2700.1.c.b 2
36.f odd 6 1 1620.1.p.a 4
36.f odd 6 1 1620.1.p.d 4
36.h even 6 1 1620.1.p.a 4
36.h even 6 1 1620.1.p.d 4
45.h odd 6 1 1620.1.p.a 4
45.h odd 6 1 1620.1.p.d 4
45.j even 6 1 1620.1.p.a 4
45.j even 6 1 1620.1.p.d 4
60.h even 2 1 inner 540.1.f.a 4
60.l odd 4 1 2700.1.c.a 2
60.l odd 4 1 2700.1.c.b 2
180.n even 6 1 1620.1.p.a 4
180.n even 6 1 1620.1.p.d 4
180.p odd 6 1 1620.1.p.a 4
180.p odd 6 1 1620.1.p.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.1.f.a 4 1.a even 1 1 trivial
540.1.f.a 4 3.b odd 2 1 inner
540.1.f.a 4 4.b odd 2 1 inner
540.1.f.a 4 5.b even 2 1 inner
540.1.f.a 4 12.b even 2 1 inner
540.1.f.a 4 15.d odd 2 1 CM
540.1.f.a 4 20.d odd 2 1 inner
540.1.f.a 4 60.h even 2 1 inner
1620.1.p.a 4 9.c even 3 1
1620.1.p.a 4 9.d odd 6 1
1620.1.p.a 4 36.f odd 6 1
1620.1.p.a 4 36.h even 6 1
1620.1.p.a 4 45.h odd 6 1
1620.1.p.a 4 45.j even 6 1
1620.1.p.a 4 180.n even 6 1
1620.1.p.a 4 180.p odd 6 1
1620.1.p.d 4 9.c even 3 1
1620.1.p.d 4 9.d odd 6 1
1620.1.p.d 4 36.f odd 6 1
1620.1.p.d 4 36.h even 6 1
1620.1.p.d 4 45.h odd 6 1
1620.1.p.d 4 45.j even 6 1
1620.1.p.d 4 180.n even 6 1
1620.1.p.d 4 180.p odd 6 1
2700.1.c.a 2 5.c odd 4 1
2700.1.c.a 2 15.e even 4 1
2700.1.c.a 2 20.e even 4 1
2700.1.c.a 2 60.l odd 4 1
2700.1.c.b 2 5.c odd 4 1
2700.1.c.b 2 15.e even 4 1
2700.1.c.b 2 20.e even 4 1
2700.1.c.b 2 60.l odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(540, [\chi])\).

Hecke characteristic polynomials

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