# Properties

 Label 720.1.r.a Level 720 Weight 1 Character orbit 720.r Analytic conductor 0.359 Analytic rank 0 Dimension 4 Projective image $$D_{4}$$ CM discriminant -15 Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 720.r (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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{4}$$ Projective field Galois closure of 4.0.92160.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{5} -\zeta_{8}^{3} q^{8} +O(q^{10})$$ $$q -\zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} -\zeta_{8}^{3} q^{5} -\zeta_{8}^{3} q^{8} - q^{10} - q^{16} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{17} + ( 1 + \zeta_{8}^{2} ) q^{19} + \zeta_{8} q^{20} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{23} -\zeta_{8}^{2} q^{25} -2 \zeta_{8}^{2} q^{31} + \zeta_{8} q^{32} + ( -1 + \zeta_{8}^{2} ) q^{34} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{38} -\zeta_{8}^{2} q^{40} + ( 1 - \zeta_{8}^{2} ) q^{46} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{47} + q^{49} + \zeta_{8}^{3} q^{50} + ( -1 + \zeta_{8}^{2} ) q^{61} + 2 \zeta_{8}^{3} q^{62} -\zeta_{8}^{2} q^{64} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{68} + ( -1 + \zeta_{8}^{2} ) q^{76} + \zeta_{8}^{3} q^{80} + 2 \zeta_{8}^{3} q^{83} + ( -1 - \zeta_{8}^{2} ) q^{85} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{92} + ( 1 + \zeta_{8}^{2} ) q^{94} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{95} -\zeta_{8} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 4q^{10} - 4q^{16} + 4q^{19} - 4q^{34} + 4q^{46} + 4q^{49} - 4q^{61} - 4q^{76} - 4q^{85} + 4q^{94} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
−0.707107 + 0.707107i 0 1.00000i 0.707107 + 0.707107i 0 0 0.707107 + 0.707107i 0 −1.00000
19.2 0.707107 0.707107i 0 1.00000i −0.707107 0.707107i 0 0 −0.707107 0.707107i 0 −1.00000
379.1 −0.707107 0.707107i 0 1.00000i 0.707107 0.707107i 0 0 0.707107 0.707107i 0 −1.00000
379.2 0.707107 + 0.707107i 0 1.00000i −0.707107 + 0.707107i 0 0 −0.707107 + 0.707107i 0 −1.00000
 $$n$$: 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
5.b even 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner
80.k odd 4 1 inner
240.t even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.1.r.a 4
3.b odd 2 1 inner 720.1.r.a 4
4.b odd 2 1 2880.1.r.a 4
5.b even 2 1 inner 720.1.r.a 4
5.c odd 4 2 3600.1.bo.a 4
12.b even 2 1 2880.1.r.a 4
15.d odd 2 1 CM 720.1.r.a 4
15.e even 4 2 3600.1.bo.a 4
16.e even 4 1 2880.1.r.a 4
16.f odd 4 1 inner 720.1.r.a 4
20.d odd 2 1 2880.1.r.a 4
48.i odd 4 1 2880.1.r.a 4
48.k even 4 1 inner 720.1.r.a 4
60.h even 2 1 2880.1.r.a 4
80.j even 4 1 3600.1.bo.a 4
80.k odd 4 1 inner 720.1.r.a 4
80.q even 4 1 2880.1.r.a 4
80.s even 4 1 3600.1.bo.a 4
240.t even 4 1 inner 720.1.r.a 4
240.z odd 4 1 3600.1.bo.a 4
240.bd odd 4 1 3600.1.bo.a 4
240.bm odd 4 1 2880.1.r.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.1.r.a 4 1.a even 1 1 trivial
720.1.r.a 4 3.b odd 2 1 inner
720.1.r.a 4 5.b even 2 1 inner
720.1.r.a 4 15.d odd 2 1 CM
720.1.r.a 4 16.f odd 4 1 inner
720.1.r.a 4 48.k even 4 1 inner
720.1.r.a 4 80.k odd 4 1 inner
720.1.r.a 4 240.t even 4 1 inner
2880.1.r.a 4 4.b odd 2 1
2880.1.r.a 4 12.b even 2 1
2880.1.r.a 4 16.e even 4 1
2880.1.r.a 4 20.d odd 2 1
2880.1.r.a 4 48.i odd 4 1
2880.1.r.a 4 60.h even 2 1
2880.1.r.a 4 80.q even 4 1
2880.1.r.a 4 240.bm odd 4 1
3600.1.bo.a 4 5.c odd 4 2
3600.1.bo.a 4 15.e even 4 2
3600.1.bo.a 4 80.j even 4 1
3600.1.bo.a 4 80.s even 4 1
3600.1.bo.a 4 240.z odd 4 1
3600.1.bo.a 4 240.bd odd 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

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