# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.359326809096$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{4}$$ Projective field Galois closure of 4.0.13500.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{8} q^{5} +O(q^{10})$$ $$q -\zeta_{8} q^{5} + ( 1 - \zeta_{8}^{2} ) q^{13} -2 \zeta_{8}^{3} q^{17} + \zeta_{8}^{2} q^{25} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{29} + ( -1 - \zeta_{8}^{2} ) q^{37} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{41} + \zeta_{8}^{2} q^{49} + 2 \zeta_{8} q^{53} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{65} + ( -1 + \zeta_{8}^{2} ) q^{73} -2 q^{85} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{89} + ( -1 - \zeta_{8}^{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 4q^{13} - 4q^{37} - 4q^{73} - 8q^{85} - 4q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{8}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
0 0 0 −0.707107 + 0.707107i 0 0 0 0 0
143.2 0 0 0 0.707107 0.707107i 0 0 0 0 0
287.1 0 0 0 −0.707107 0.707107i 0 0 0 0 0
287.2 0 0 0 0.707107 + 0.707107i 0 0 0 0 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
5.c odd 4 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
60.l odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.1.bk.a 4
3.b odd 2 1 inner 720.1.bk.a 4
4.b odd 2 1 CM 720.1.bk.a 4
5.b even 2 1 3600.1.bk.a 4
5.c odd 4 1 inner 720.1.bk.a 4
5.c odd 4 1 3600.1.bk.a 4
8.b even 2 1 2880.1.bk.a 4
8.d odd 2 1 2880.1.bk.a 4
12.b even 2 1 inner 720.1.bk.a 4
15.d odd 2 1 3600.1.bk.a 4
15.e even 4 1 inner 720.1.bk.a 4
15.e even 4 1 3600.1.bk.a 4
20.d odd 2 1 3600.1.bk.a 4
20.e even 4 1 inner 720.1.bk.a 4
20.e even 4 1 3600.1.bk.a 4
24.f even 2 1 2880.1.bk.a 4
24.h odd 2 1 2880.1.bk.a 4
40.i odd 4 1 2880.1.bk.a 4
40.k even 4 1 2880.1.bk.a 4
60.h even 2 1 3600.1.bk.a 4
60.l odd 4 1 inner 720.1.bk.a 4
60.l odd 4 1 3600.1.bk.a 4
120.q odd 4 1 2880.1.bk.a 4
120.w even 4 1 2880.1.bk.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.1.bk.a 4 1.a even 1 1 trivial
720.1.bk.a 4 3.b odd 2 1 inner
720.1.bk.a 4 4.b odd 2 1 CM
720.1.bk.a 4 5.c odd 4 1 inner
720.1.bk.a 4 12.b even 2 1 inner
720.1.bk.a 4 15.e even 4 1 inner
720.1.bk.a 4 20.e even 4 1 inner
720.1.bk.a 4 60.l odd 4 1 inner
2880.1.bk.a 4 8.b even 2 1
2880.1.bk.a 4 8.d odd 2 1
2880.1.bk.a 4 24.f even 2 1
2880.1.bk.a 4 24.h odd 2 1
2880.1.bk.a 4 40.i odd 4 1
2880.1.bk.a 4 40.k even 4 1
2880.1.bk.a 4 120.q odd 4 1
2880.1.bk.a 4 120.w even 4 1
3600.1.bk.a 4 5.b even 2 1
3600.1.bk.a 4 5.c odd 4 1
3600.1.bk.a 4 15.d odd 2 1
3600.1.bk.a 4 15.e even 4 1
3600.1.bk.a 4 20.d odd 2 1
3600.1.bk.a 4 20.e even 4 1
3600.1.bk.a 4 60.h even 2 1
3600.1.bk.a 4 60.l odd 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$1 + T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$( 2 - 2 T + T^{2} )^{2}$$
$17$ $$16 + T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( -2 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 2 + 2 T + T^{2} )^{2}$$
$41$ $$( 2 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$16 + T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( 2 + 2 T + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$( -2 + T^{2} )^{2}$$
$97$ $$( 2 + 2 T + T^{2} )^{2}$$