# Properties

 Label 720.2.q.a Level $720$ Weight $2$ Character orbit 720.q Analytic conductor $5.749$ Analytic rank $1$ 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: $$1$$ 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 180) 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -2 + \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} + 4 \zeta_{6} q^{13} + ( 1 + \zeta_{6} ) q^{15} -6 q^{17} -2 q^{19} + ( 1 - 2 \zeta_{6} ) q^{21} -3 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -3 + 3 \zeta_{6} ) q^{29} -10 \zeta_{6} q^{31} + q^{35} -10 q^{37} + ( -4 - 4 \zeta_{6} ) q^{39} -9 \zeta_{6} q^{41} + ( -4 + 4 \zeta_{6} ) q^{43} -3 q^{45} + ( 9 - 9 \zeta_{6} ) q^{47} + 6 \zeta_{6} q^{49} + ( 12 - 6 \zeta_{6} ) q^{51} -6 q^{53} + ( 4 - 2 \zeta_{6} ) q^{57} -6 \zeta_{6} q^{59} + ( 1 - \zeta_{6} ) q^{61} + 3 \zeta_{6} q^{63} + ( 4 - 4 \zeta_{6} ) q^{65} + 11 \zeta_{6} q^{67} + ( 3 + 3 \zeta_{6} ) q^{69} -12 q^{71} -4 q^{73} + ( 1 - 2 \zeta_{6} ) q^{75} + ( -10 + 10 \zeta_{6} ) q^{79} -9 \zeta_{6} q^{81} + ( -9 + 9 \zeta_{6} ) q^{83} + 6 \zeta_{6} q^{85} + ( 3 - 6 \zeta_{6} ) q^{87} + 9 q^{89} -4 q^{91} + ( 10 + 10 \zeta_{6} ) q^{93} + 2 \zeta_{6} q^{95} + ( 10 - 10 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{3} - q^{5} - q^{7} + 3q^{9} + O(q^{10})$$ $$2q - 3q^{3} - q^{5} - q^{7} + 3q^{9} + 4q^{13} + 3q^{15} - 12q^{17} - 4q^{19} - 3q^{23} - q^{25} - 3q^{29} - 10q^{31} + 2q^{35} - 20q^{37} - 12q^{39} - 9q^{41} - 4q^{43} - 6q^{45} + 9q^{47} + 6q^{49} + 18q^{51} - 12q^{53} + 6q^{57} - 6q^{59} + q^{61} + 3q^{63} + 4q^{65} + 11q^{67} + 9q^{69} - 24q^{71} - 8q^{73} - 10q^{79} - 9q^{81} - 9q^{83} + 6q^{85} + 18q^{89} - 8q^{91} + 30q^{93} + 2q^{95} + 10q^{97} + O(q^{100})$$

## Character values

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

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

## 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 −0.500000 + 0.866025i 0 1.50000 2.59808i 0
481.1 0 −1.50000 0.866025i 0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 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.a 2
3.b odd 2 1 2160.2.q.e 2
4.b odd 2 1 180.2.i.a 2
9.c even 3 1 inner 720.2.q.a 2
9.c even 3 1 6480.2.a.t 1
9.d odd 6 1 2160.2.q.e 2
9.d odd 6 1 6480.2.a.h 1
12.b even 2 1 540.2.i.a 2
20.d odd 2 1 900.2.i.a 2
20.e even 4 2 900.2.s.a 4
36.f odd 6 1 180.2.i.a 2
36.f odd 6 1 1620.2.a.e 1
36.h even 6 1 540.2.i.a 2
36.h even 6 1 1620.2.a.b 1
60.h even 2 1 2700.2.i.a 2
60.l odd 4 2 2700.2.s.a 4
180.n even 6 1 2700.2.i.a 2
180.n even 6 1 8100.2.a.h 1
180.p odd 6 1 900.2.i.a 2
180.p odd 6 1 8100.2.a.i 1
180.v odd 12 2 2700.2.s.a 4
180.v odd 12 2 8100.2.d.f 2
180.x even 12 2 900.2.s.a 4
180.x even 12 2 8100.2.d.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.i.a 2 4.b odd 2 1
180.2.i.a 2 36.f odd 6 1
540.2.i.a 2 12.b even 2 1
540.2.i.a 2 36.h even 6 1
720.2.q.a 2 1.a even 1 1 trivial
720.2.q.a 2 9.c even 3 1 inner
900.2.i.a 2 20.d odd 2 1
900.2.i.a 2 180.p odd 6 1
900.2.s.a 4 20.e even 4 2
900.2.s.a 4 180.x even 12 2
1620.2.a.b 1 36.h even 6 1
1620.2.a.e 1 36.f odd 6 1
2160.2.q.e 2 3.b odd 2 1
2160.2.q.e 2 9.d odd 6 1
2700.2.i.a 2 60.h even 2 1
2700.2.i.a 2 180.n even 6 1
2700.2.s.a 4 60.l odd 4 2
2700.2.s.a 4 180.v odd 12 2
6480.2.a.h 1 9.d odd 6 1
6480.2.a.t 1 9.c even 3 1
8100.2.a.h 1 180.n even 6 1
8100.2.a.i 1 180.p odd 6 1
8100.2.d.e 2 180.x even 12 2
8100.2.d.f 2 180.v odd 12 2

## Hecke kernels

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

 $$T_{7}^{2} + T_{7} + 1$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 + 3 T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$1 + T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$16 - 4 T + T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$9 + 3 T + T^{2}$$
$29$ $$9 + 3 T + T^{2}$$
$31$ $$100 + 10 T + T^{2}$$
$37$ $$( 10 + T )^{2}$$
$41$ $$81 + 9 T + T^{2}$$
$43$ $$16 + 4 T + T^{2}$$
$47$ $$81 - 9 T + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$36 + 6 T + T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$121 - 11 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$( 4 + T )^{2}$$
$79$ $$100 + 10 T + T^{2}$$
$83$ $$81 + 9 T + T^{2}$$
$89$ $$( -9 + T )^{2}$$
$97$ $$100 - 10 T + T^{2}$$