# Properties

 Label 720.2.q.c 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 90) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( -4 + 4 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( -4 + 4 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( 3 - 3 \zeta_{6} ) q^{11} + 4 \zeta_{6} q^{13} + ( 1 - 2 \zeta_{6} ) q^{15} + 3 q^{17} -5 q^{19} + ( -8 + 4 \zeta_{6} ) q^{21} + 6 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -6 + 6 \zeta_{6} ) q^{29} + 2 \zeta_{6} q^{31} + ( 6 - 3 \zeta_{6} ) q^{33} + 4 q^{35} -4 q^{37} + ( -4 + 8 \zeta_{6} ) q^{39} + 3 \zeta_{6} q^{41} + ( 11 - 11 \zeta_{6} ) q^{43} + ( 3 - 3 \zeta_{6} ) q^{45} -9 \zeta_{6} q^{49} + ( 3 + 3 \zeta_{6} ) q^{51} + 6 q^{53} -3 q^{55} + ( -5 - 5 \zeta_{6} ) q^{57} -3 \zeta_{6} q^{59} + ( 10 - 10 \zeta_{6} ) q^{61} -12 q^{63} + ( 4 - 4 \zeta_{6} ) q^{65} + 5 \zeta_{6} q^{67} + ( -6 + 12 \zeta_{6} ) q^{69} -6 q^{71} -7 q^{73} + ( -2 + \zeta_{6} ) q^{75} + 12 \zeta_{6} q^{77} + ( 14 - 14 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( 12 - 12 \zeta_{6} ) q^{83} -3 \zeta_{6} q^{85} + ( -12 + 6 \zeta_{6} ) q^{87} + 6 q^{89} -16 q^{91} + ( -2 + 4 \zeta_{6} ) q^{93} + 5 \zeta_{6} q^{95} + ( -11 + 11 \zeta_{6} ) q^{97} + 9 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} - q^{5} - 4q^{7} + 3q^{9} + O(q^{10})$$ $$2q + 3q^{3} - q^{5} - 4q^{7} + 3q^{9} + 3q^{11} + 4q^{13} + 6q^{17} - 10q^{19} - 12q^{21} + 6q^{23} - q^{25} - 6q^{29} + 2q^{31} + 9q^{33} + 8q^{35} - 8q^{37} + 3q^{41} + 11q^{43} + 3q^{45} - 9q^{49} + 9q^{51} + 12q^{53} - 6q^{55} - 15q^{57} - 3q^{59} + 10q^{61} - 24q^{63} + 4q^{65} + 5q^{67} - 12q^{71} - 14q^{73} - 3q^{75} + 12q^{77} + 14q^{79} - 9q^{81} + 12q^{83} - 3q^{85} - 18q^{87} + 12q^{89} - 32q^{91} + 5q^{95} - 11q^{97} + 18q^{99} + O(q^{100})$$

## 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 −2.00000 + 3.46410i 0 1.50000 + 2.59808i 0
481.1 0 1.50000 0.866025i 0 −0.500000 + 0.866025i 0 −2.00000 3.46410i 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.c 2
3.b odd 2 1 2160.2.q.d 2
4.b odd 2 1 90.2.e.b 2
9.c even 3 1 inner 720.2.q.c 2
9.c even 3 1 6480.2.a.z 1
9.d odd 6 1 2160.2.q.d 2
9.d odd 6 1 6480.2.a.l 1
12.b even 2 1 270.2.e.a 2
20.d odd 2 1 450.2.e.d 2
20.e even 4 2 450.2.j.a 4
36.f odd 6 1 90.2.e.b 2
36.f odd 6 1 810.2.a.a 1
36.h even 6 1 270.2.e.a 2
36.h even 6 1 810.2.a.e 1
60.h even 2 1 1350.2.e.g 2
60.l odd 4 2 1350.2.j.c 4
180.n even 6 1 1350.2.e.g 2
180.n even 6 1 4050.2.a.q 1
180.p odd 6 1 450.2.e.d 2
180.p odd 6 1 4050.2.a.bi 1
180.v odd 12 2 1350.2.j.c 4
180.v odd 12 2 4050.2.c.d 2
180.x even 12 2 450.2.j.a 4
180.x even 12 2 4050.2.c.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.b 2 4.b odd 2 1
90.2.e.b 2 36.f odd 6 1
270.2.e.a 2 12.b even 2 1
270.2.e.a 2 36.h even 6 1
450.2.e.d 2 20.d odd 2 1
450.2.e.d 2 180.p odd 6 1
450.2.j.a 4 20.e even 4 2
450.2.j.a 4 180.x even 12 2
720.2.q.c 2 1.a even 1 1 trivial
720.2.q.c 2 9.c even 3 1 inner
810.2.a.a 1 36.f odd 6 1
810.2.a.e 1 36.h even 6 1
1350.2.e.g 2 60.h even 2 1
1350.2.e.g 2 180.n even 6 1
1350.2.j.c 4 60.l odd 4 2
1350.2.j.c 4 180.v odd 12 2
2160.2.q.d 2 3.b odd 2 1
2160.2.q.d 2 9.d odd 6 1
4050.2.a.q 1 180.n even 6 1
4050.2.a.bi 1 180.p odd 6 1
4050.2.c.d 2 180.v odd 12 2
4050.2.c.p 2 180.x even 12 2
6480.2.a.l 1 9.d odd 6 1
6480.2.a.z 1 9.c even 3 1

## Hecke kernels

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

 $$T_{7}^{2} + 4 T_{7} + 16$$ $$T_{11}^{2} - 3 T_{11} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 - 3 T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$16 + 4 T + T^{2}$$
$11$ $$9 - 3 T + T^{2}$$
$13$ $$16 - 4 T + T^{2}$$
$17$ $$( -3 + T )^{2}$$
$19$ $$( 5 + T )^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$4 - 2 T + T^{2}$$
$37$ $$( 4 + T )^{2}$$
$41$ $$9 - 3 T + T^{2}$$
$43$ $$121 - 11 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$9 + 3 T + T^{2}$$
$61$ $$100 - 10 T + T^{2}$$
$67$ $$25 - 5 T + T^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$( 7 + T )^{2}$$
$79$ $$196 - 14 T + T^{2}$$
$83$ $$144 - 12 T + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$121 + 11 T + T^{2}$$