# Properties

 Label 720.2.o.b Level $720$ Weight $2$ Character orbit 720.o Analytic conductor $5.749$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.74922894553$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{8}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{24} - 2 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{5} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{7} +O(q^{10})$$ $$q + ( -\zeta_{24} - 2 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{5} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{7} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{11} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{13} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{17} + ( 4 - 8 \zeta_{24}^{4} ) q^{19} -6 \zeta_{24}^{6} q^{23} + ( 1 - 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{25} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{29} + ( 2 - 4 \zeta_{24}^{4} ) q^{31} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 6 \zeta_{24}^{6} - 6 \zeta_{24}^{7} ) q^{35} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{37} + ( -5 \zeta_{24} + 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} ) q^{41} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{43} + 11 q^{49} + ( -2 - 3 \zeta_{24} - 3 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 3 \zeta_{24}^{5} ) q^{55} + ( -5 \zeta_{24} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 10 \zeta_{24}^{7} ) q^{59} -2 q^{61} + ( -3 \zeta_{24} + 4 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{65} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{71} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{73} + ( -12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{77} + ( 6 - 12 \zeta_{24}^{4} ) q^{79} + 12 \zeta_{24}^{6} q^{83} + ( -12 + 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{85} + ( 5 \zeta_{24} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{89} + ( -6 + 12 \zeta_{24}^{4} ) q^{91} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 12 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{95} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 8q^{25} + 88q^{49} - 16q^{61} - 96q^{85} + O(q^{100})$$

## Character values

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

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
719.1
 0.965926 − 0.258819i −0.965926 − 0.258819i 0.965926 + 0.258819i −0.965926 + 0.258819i −0.258819 + 0.965926i 0.258819 + 0.965926i −0.258819 − 0.965926i 0.258819 − 0.965926i
0 0 0 −1.73205 1.41421i 0 −4.24264 0 0 0
719.2 0 0 0 −1.73205 1.41421i 0 4.24264 0 0 0
719.3 0 0 0 −1.73205 + 1.41421i 0 −4.24264 0 0 0
719.4 0 0 0 −1.73205 + 1.41421i 0 4.24264 0 0 0
719.5 0 0 0 1.73205 1.41421i 0 −4.24264 0 0 0
719.6 0 0 0 1.73205 1.41421i 0 4.24264 0 0 0
719.7 0 0 0 1.73205 + 1.41421i 0 −4.24264 0 0 0
719.8 0 0 0 1.73205 + 1.41421i 0 4.24264 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 719.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 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 720.2.o.b 8
3.b odd 2 1 inner 720.2.o.b 8
4.b odd 2 1 inner 720.2.o.b 8
5.b even 2 1 inner 720.2.o.b 8
5.c odd 4 1 3600.2.h.f 4
5.c odd 4 1 3600.2.h.g 4
8.b even 2 1 2880.2.o.d 8
8.d odd 2 1 2880.2.o.d 8
12.b even 2 1 inner 720.2.o.b 8
15.d odd 2 1 inner 720.2.o.b 8
15.e even 4 1 3600.2.h.f 4
15.e even 4 1 3600.2.h.g 4
20.d odd 2 1 inner 720.2.o.b 8
20.e even 4 1 3600.2.h.f 4
20.e even 4 1 3600.2.h.g 4
24.f even 2 1 2880.2.o.d 8
24.h odd 2 1 2880.2.o.d 8
40.e odd 2 1 2880.2.o.d 8
40.f even 2 1 2880.2.o.d 8
60.h even 2 1 inner 720.2.o.b 8
60.l odd 4 1 3600.2.h.f 4
60.l odd 4 1 3600.2.h.g 4
120.i odd 2 1 2880.2.o.d 8
120.m even 2 1 2880.2.o.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.o.b 8 1.a even 1 1 trivial
720.2.o.b 8 3.b odd 2 1 inner
720.2.o.b 8 4.b odd 2 1 inner
720.2.o.b 8 5.b even 2 1 inner
720.2.o.b 8 12.b even 2 1 inner
720.2.o.b 8 15.d odd 2 1 inner
720.2.o.b 8 20.d odd 2 1 inner
720.2.o.b 8 60.h even 2 1 inner
2880.2.o.d 8 8.b even 2 1
2880.2.o.d 8 8.d odd 2 1
2880.2.o.d 8 24.f even 2 1
2880.2.o.d 8 24.h odd 2 1
2880.2.o.d 8 40.e odd 2 1
2880.2.o.d 8 40.f even 2 1
2880.2.o.d 8 120.i odd 2 1
2880.2.o.d 8 120.m even 2 1
3600.2.h.f 4 5.c odd 4 1
3600.2.h.f 4 15.e even 4 1
3600.2.h.f 4 20.e even 4 1
3600.2.h.f 4 60.l odd 4 1
3600.2.h.g 4 5.c odd 4 1
3600.2.h.g 4 15.e even 4 1
3600.2.h.g 4 20.e even 4 1
3600.2.h.g 4 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 18$$ acting on $$S_{2}^{\mathrm{new}}(720, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 25 - 2 T^{2} + T^{4} )^{2}$$
$7$ $$( -18 + T^{2} )^{4}$$
$11$ $$( -6 + T^{2} )^{4}$$
$13$ $$( 6 + T^{2} )^{4}$$
$17$ $$( -48 + T^{2} )^{4}$$
$19$ $$( 48 + T^{2} )^{4}$$
$23$ $$( 36 + T^{2} )^{4}$$
$29$ $$( 8 + T^{2} )^{4}$$
$31$ $$( 12 + T^{2} )^{4}$$
$37$ $$( 6 + T^{2} )^{4}$$
$41$ $$( 50 + T^{2} )^{4}$$
$43$ $$( -72 + T^{2} )^{4}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$( -150 + T^{2} )^{4}$$
$61$ $$( 2 + T )^{8}$$
$67$ $$T^{8}$$
$71$ $$( -96 + T^{2} )^{4}$$
$73$ $$( 24 + T^{2} )^{4}$$
$79$ $$( 108 + T^{2} )^{4}$$
$83$ $$( 144 + T^{2} )^{4}$$
$89$ $$( 50 + T^{2} )^{4}$$
$97$ $$( 216 + T^{2} )^{4}$$