# Properties

 Label 720.1.bu.a Level $720$ Weight $1$ Character orbit 720.bu Analytic conductor $0.359$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -20 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 720.bu (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.359326809096$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{6}$$ Projective field Galois closure of 6.0.10497600.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{12} q^{3} -\zeta_{12}^{4} q^{5} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{12} q^{3} -\zeta_{12}^{4} q^{5} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{2} q^{9} + \zeta_{12}^{5} q^{15} + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{21} + ( -\zeta_{12}^{3} - \zeta_{12}^{5} ) q^{23} -\zeta_{12}^{2} q^{25} -\zeta_{12}^{3} q^{27} + \zeta_{12}^{2} q^{29} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{35} + \zeta_{12}^{4} q^{41} + q^{45} + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{47} + ( -1 + \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{49} -\zeta_{12}^{2} q^{61} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{63} + ( \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{67} + ( -1 + \zeta_{12}^{4} ) q^{69} + \zeta_{12}^{3} q^{75} + \zeta_{12}^{4} q^{81} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{83} -\zeta_{12}^{3} q^{87} - q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{5} + 2q^{9} + O(q^{10})$$ $$4q + 2q^{5} + 2q^{9} - 2q^{25} + 2q^{29} - 2q^{41} + 4q^{45} - 4q^{49} - 2q^{61} - 6q^{69} - 2q^{81} - 4q^{89} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$$.

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$\zeta_{12}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 −0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0.866025 1.50000i 0 0.500000 0.866025i 0
79.2 0 0.866025 0.500000i 0 0.500000 + 0.866025i 0 −0.866025 + 1.50000i 0 0.500000 0.866025i 0
319.1 0 −0.866025 0.500000i 0 0.500000 0.866025i 0 0.866025 + 1.50000i 0 0.500000 + 0.866025i 0
319.2 0 0.866025 + 0.500000i 0 0.500000 0.866025i 0 −0.866025 1.50000i 0 0.500000 + 0.866025i 0
 $$n$$: 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
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 720.1.bu.a 4
3.b odd 2 1 2160.1.bu.a 4
4.b odd 2 1 inner 720.1.bu.a 4
5.b even 2 1 inner 720.1.bu.a 4
5.c odd 4 1 3600.1.cc.a 2
5.c odd 4 1 3600.1.cc.b 2
8.b even 2 1 2880.1.bu.c 4
8.d odd 2 1 2880.1.bu.c 4
9.c even 3 1 inner 720.1.bu.a 4
9.d odd 6 1 2160.1.bu.a 4
12.b even 2 1 2160.1.bu.a 4
15.d odd 2 1 2160.1.bu.a 4
20.d odd 2 1 CM 720.1.bu.a 4
20.e even 4 1 3600.1.cc.a 2
20.e even 4 1 3600.1.cc.b 2
36.f odd 6 1 inner 720.1.bu.a 4
36.h even 6 1 2160.1.bu.a 4
40.e odd 2 1 2880.1.bu.c 4
40.f even 2 1 2880.1.bu.c 4
45.h odd 6 1 2160.1.bu.a 4
45.j even 6 1 inner 720.1.bu.a 4
45.k odd 12 1 3600.1.cc.a 2
45.k odd 12 1 3600.1.cc.b 2
60.h even 2 1 2160.1.bu.a 4
72.n even 6 1 2880.1.bu.c 4
72.p odd 6 1 2880.1.bu.c 4
180.n even 6 1 2160.1.bu.a 4
180.p odd 6 1 inner 720.1.bu.a 4
180.x even 12 1 3600.1.cc.a 2
180.x even 12 1 3600.1.cc.b 2
360.z odd 6 1 2880.1.bu.c 4
360.bk even 6 1 2880.1.bu.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.1.bu.a 4 1.a even 1 1 trivial
720.1.bu.a 4 4.b odd 2 1 inner
720.1.bu.a 4 5.b even 2 1 inner
720.1.bu.a 4 9.c even 3 1 inner
720.1.bu.a 4 20.d odd 2 1 CM
720.1.bu.a 4 36.f odd 6 1 inner
720.1.bu.a 4 45.j even 6 1 inner
720.1.bu.a 4 180.p odd 6 1 inner
2160.1.bu.a 4 3.b odd 2 1
2160.1.bu.a 4 9.d odd 6 1
2160.1.bu.a 4 12.b even 2 1
2160.1.bu.a 4 15.d odd 2 1
2160.1.bu.a 4 36.h even 6 1
2160.1.bu.a 4 45.h odd 6 1
2160.1.bu.a 4 60.h even 2 1
2160.1.bu.a 4 180.n even 6 1
2880.1.bu.c 4 8.b even 2 1
2880.1.bu.c 4 8.d odd 2 1
2880.1.bu.c 4 40.e odd 2 1
2880.1.bu.c 4 40.f even 2 1
2880.1.bu.c 4 72.n even 6 1
2880.1.bu.c 4 72.p odd 6 1
2880.1.bu.c 4 360.z odd 6 1
2880.1.bu.c 4 360.bk even 6 1
3600.1.cc.a 2 5.c odd 4 1
3600.1.cc.a 2 20.e even 4 1
3600.1.cc.a 2 45.k odd 12 1
3600.1.cc.a 2 180.x even 12 1
3600.1.cc.b 2 5.c odd 4 1
3600.1.cc.b 2 20.e even 4 1
3600.1.cc.b 2 45.k odd 12 1
3600.1.cc.b 2 180.x even 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(720, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$9 + 3 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$9 + 3 T^{2} + T^{4}$$
$29$ $$( 1 - T + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( 1 + T + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$9 + 3 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 1 + T + T^{2} )^{2}$$
$67$ $$9 + 3 T^{2} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$9 + 3 T^{2} + T^{4}$$
$89$ $$( 1 + T )^{4}$$
$97$ $$T^{4}$$