# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 720.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.74922894553$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 360) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 + \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} + ( -5 + 5 \zeta_{6} ) q^{11} + ( -1 + 2 \zeta_{6} ) q^{15} + 3 q^{17} -5 q^{19} + 6 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 10 - 10 \zeta_{6} ) q^{29} -2 \zeta_{6} q^{31} + ( -10 + 5 \zeta_{6} ) q^{33} + 4 q^{37} + 3 \zeta_{6} q^{41} + ( 3 - 3 \zeta_{6} ) q^{43} + ( -3 + 3 \zeta_{6} ) q^{45} + ( 4 - 4 \zeta_{6} ) q^{47} + 7 \zeta_{6} q^{49} + ( 3 + 3 \zeta_{6} ) q^{51} -6 q^{53} -5 q^{55} + ( -5 - 5 \zeta_{6} ) q^{57} -3 \zeta_{6} q^{59} + ( -2 + 2 \zeta_{6} ) q^{61} -11 \zeta_{6} q^{67} + ( -6 + 12 \zeta_{6} ) q^{69} + 14 q^{71} -15 q^{73} + ( -2 + \zeta_{6} ) q^{75} + ( 10 - 10 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( -12 + 12 \zeta_{6} ) q^{83} + 3 \zeta_{6} q^{85} + ( 20 - 10 \zeta_{6} ) q^{87} + 14 q^{89} + ( 2 - 4 \zeta_{6} ) q^{93} -5 \zeta_{6} q^{95} + ( 13 - 13 \zeta_{6} ) q^{97} -15 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} + q^{5} + 3q^{9} + O(q^{10})$$ $$2q + 3q^{3} + q^{5} + 3q^{9} - 5q^{11} + 6q^{17} - 10q^{19} + 6q^{23} - q^{25} + 10q^{29} - 2q^{31} - 15q^{33} + 8q^{37} + 3q^{41} + 3q^{43} - 3q^{45} + 4q^{47} + 7q^{49} + 9q^{51} - 12q^{53} - 10q^{55} - 15q^{57} - 3q^{59} - 2q^{61} - 11q^{67} + 28q^{71} - 30q^{73} - 3q^{75} + 10q^{79} - 9q^{81} - 12q^{83} + 3q^{85} + 30q^{87} + 28q^{89} - 5q^{95} + 13q^{97} - 30q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
241.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 + 0.866025i 0 0.500000 + 0.866025i 0 0 0 1.50000 + 2.59808i 0
481.1 0 1.50000 0.866025i 0 0.500000 0.866025i 0 0 0 1.50000 2.59808i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.q.e 2
3.b odd 2 1 2160.2.q.c 2
4.b odd 2 1 360.2.q.a 2
9.c even 3 1 inner 720.2.q.e 2
9.c even 3 1 6480.2.a.e 1
9.d odd 6 1 2160.2.q.c 2
9.d odd 6 1 6480.2.a.q 1
12.b even 2 1 1080.2.q.a 2
36.f odd 6 1 360.2.q.a 2
36.f odd 6 1 3240.2.a.b 1
36.h even 6 1 1080.2.q.a 2
36.h even 6 1 3240.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.a 2 4.b odd 2 1
360.2.q.a 2 36.f odd 6 1
720.2.q.e 2 1.a even 1 1 trivial
720.2.q.e 2 9.c even 3 1 inner
1080.2.q.a 2 12.b even 2 1
1080.2.q.a 2 36.h even 6 1
2160.2.q.c 2 3.b odd 2 1
2160.2.q.c 2 9.d odd 6 1
3240.2.a.b 1 36.f odd 6 1
3240.2.a.f 1 36.h even 6 1
6480.2.a.e 1 9.c even 3 1
6480.2.a.q 1 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}$$ $$T_{11}^{2} + 5 T_{11} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 - 3 T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$25 + 5 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$( -3 + T )^{2}$$
$19$ $$( 5 + T )^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$100 - 10 T + T^{2}$$
$31$ $$4 + 2 T + T^{2}$$
$37$ $$( -4 + T )^{2}$$
$41$ $$9 - 3 T + T^{2}$$
$43$ $$9 - 3 T + T^{2}$$
$47$ $$16 - 4 T + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$9 + 3 T + T^{2}$$
$61$ $$4 + 2 T + T^{2}$$
$67$ $$121 + 11 T + T^{2}$$
$71$ $$( -14 + T )^{2}$$
$73$ $$( 15 + T )^{2}$$
$79$ $$100 - 10 T + T^{2}$$
$83$ $$144 + 12 T + T^{2}$$
$89$ $$( -14 + T )^{2}$$
$97$ $$169 - 13 T + T^{2}$$