# 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 $8$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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