# Properties

 Label 720.2.q.f Level $720$ Weight $2$ Character orbit 720.q Analytic conductor $5.749$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{3} ) q^{3} + ( 1 - \beta_{2} ) q^{5} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( -3 + \beta_{1} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{3} ) q^{3} + ( 1 - \beta_{2} ) q^{5} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( -3 + \beta_{1} - \beta_{3} ) q^{9} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{11} + ( 2 - 2 \beta_{1} + 4 \beta_{3} ) q^{13} -\beta_{1} q^{15} + ( -5 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{17} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{19} + ( 6 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{21} + ( -1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{23} -\beta_{2} q^{25} + ( 3 + 2 \beta_{1} - 2 \beta_{3} ) q^{27} + ( 1 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{29} + ( -2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{31} + ( -6 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{33} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{35} -4 q^{37} + ( -6 - 6 \beta_{2} - 2 \beta_{3} ) q^{39} + ( 3 - 3 \beta_{2} ) q^{41} + ( -1 + 2 \beta_{1} - 8 \beta_{2} - \beta_{3} ) q^{43} + ( -3 + \beta_{1} + 3 \beta_{2} ) q^{45} + ( 1 - 2 \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{47} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{49} + ( 3 + 4 \beta_{1} - 6 \beta_{2} - 5 \beta_{3} ) q^{51} + ( 2 - 4 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} ) q^{53} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{55} + ( 3 - 6 \beta_{2} - \beta_{3} ) q^{57} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{59} + ( -3 + 6 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{61} + ( -3 - 5 \beta_{1} + 2 \beta_{3} ) q^{63} + ( 2 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{65} + ( -7 + 7 \beta_{2} ) q^{67} + ( -3 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{69} + 6 q^{71} + ( -7 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{73} -\beta_{3} q^{75} + ( 10 - 2 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{77} + 2 \beta_{2} q^{79} + ( 6 - 5 \beta_{1} + 5 \beta_{3} ) q^{81} + ( -1 + 2 \beta_{1} + 5 \beta_{2} - \beta_{3} ) q^{83} + ( -4 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{85} + ( -6 + \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{87} + ( 6 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{89} + ( 16 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{91} + ( -6 + 2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{93} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{95} + ( -1 + 2 \beta_{1} + 6 \beta_{2} - \beta_{3} ) q^{97} + ( 3 + 5 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} + 2q^{5} + q^{7} - 10q^{9} + O(q^{10})$$ $$4q - 2q^{3} + 2q^{5} + q^{7} - 10q^{9} - 3q^{11} + 2q^{13} - q^{15} - 18q^{17} - 2q^{19} + 16q^{21} - 3q^{23} - 2q^{25} + 16q^{27} + 3q^{29} - 2q^{31} - 15q^{33} + 2q^{35} - 16q^{37} - 34q^{39} + 6q^{41} - 17q^{43} - 5q^{45} - 9q^{47} - 3q^{49} + 9q^{51} - 6q^{55} + q^{57} + 3q^{59} - q^{61} - 19q^{63} - 2q^{65} - 14q^{67} - 15q^{69} + 24q^{71} - 22q^{73} + q^{75} + 18q^{77} + 4q^{79} + 14q^{81} + 9q^{83} - 9q^{85} - 18q^{87} + 30q^{89} + 68q^{91} - 32q^{93} - q^{95} + 11q^{97} + 24q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 3$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu + 3$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + 2 \beta_{1} + 3$$

## Character values

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

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1 + \beta_{2}$$

## 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
 −1.18614 + 1.26217i 1.68614 − 0.396143i 1.68614 + 0.396143i −1.18614 − 1.26217i
0 −0.500000 1.65831i 0 0.500000 + 0.866025i 0 −1.18614 + 2.05446i 0 −2.50000 + 1.65831i 0
241.2 0 −0.500000 + 1.65831i 0 0.500000 + 0.866025i 0 1.68614 2.92048i 0 −2.50000 1.65831i 0
481.1 0 −0.500000 1.65831i 0 0.500000 0.866025i 0 1.68614 + 2.92048i 0 −2.50000 + 1.65831i 0
481.2 0 −0.500000 + 1.65831i 0 0.500000 0.866025i 0 −1.18614 2.05446i 0 −2.50000 1.65831i 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.f 4
3.b odd 2 1 2160.2.q.f 4
4.b odd 2 1 90.2.e.c 4
9.c even 3 1 inner 720.2.q.f 4
9.c even 3 1 6480.2.a.be 2
9.d odd 6 1 2160.2.q.f 4
9.d odd 6 1 6480.2.a.bn 2
12.b even 2 1 270.2.e.c 4
20.d odd 2 1 450.2.e.j 4
20.e even 4 2 450.2.j.g 8
36.f odd 6 1 90.2.e.c 4
36.f odd 6 1 810.2.a.i 2
36.h even 6 1 270.2.e.c 4
36.h even 6 1 810.2.a.k 2
60.h even 2 1 1350.2.e.l 4
60.l odd 4 2 1350.2.j.f 8
180.n even 6 1 1350.2.e.l 4
180.n even 6 1 4050.2.a.bo 2
180.p odd 6 1 450.2.e.j 4
180.p odd 6 1 4050.2.a.bw 2
180.v odd 12 2 1350.2.j.f 8
180.v odd 12 2 4050.2.c.ba 4
180.x even 12 2 450.2.j.g 8
180.x even 12 2 4050.2.c.v 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.c 4 4.b odd 2 1
90.2.e.c 4 36.f odd 6 1
270.2.e.c 4 12.b even 2 1
270.2.e.c 4 36.h even 6 1
450.2.e.j 4 20.d odd 2 1
450.2.e.j 4 180.p odd 6 1
450.2.j.g 8 20.e even 4 2
450.2.j.g 8 180.x even 12 2
720.2.q.f 4 1.a even 1 1 trivial
720.2.q.f 4 9.c even 3 1 inner
810.2.a.i 2 36.f odd 6 1
810.2.a.k 2 36.h even 6 1
1350.2.e.l 4 60.h even 2 1
1350.2.e.l 4 180.n even 6 1
1350.2.j.f 8 60.l odd 4 2
1350.2.j.f 8 180.v odd 12 2
2160.2.q.f 4 3.b odd 2 1
2160.2.q.f 4 9.d odd 6 1
4050.2.a.bo 2 180.n even 6 1
4050.2.a.bw 2 180.p odd 6 1
4050.2.c.v 4 180.x even 12 2
4050.2.c.ba 4 180.v odd 12 2
6480.2.a.be 2 9.c even 3 1
6480.2.a.bn 2 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 $$S_{2}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{4} - T_{7}^{3} + 9 T_{7}^{2} + 8 T_{7} + 64$$ $$T_{11}^{4} + 3 T_{11}^{3} + 15 T_{11}^{2} - 18 T_{11} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T + T^{2} )^{2}$$
$5$ $$( 1 - T + T^{2} )^{2}$$
$7$ $$64 + 8 T + 9 T^{2} - T^{3} + T^{4}$$
$11$ $$36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4}$$
$13$ $$1024 + 64 T + 36 T^{2} - 2 T^{3} + T^{4}$$
$17$ $$( 12 + 9 T + T^{2} )^{2}$$
$19$ $$( -8 + T + T^{2} )^{2}$$
$23$ $$36 - 18 T + 15 T^{2} + 3 T^{3} + T^{4}$$
$29$ $$36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4}$$
$31$ $$1024 - 64 T + 36 T^{2} + 2 T^{3} + T^{4}$$
$37$ $$( 4 + T )^{4}$$
$41$ $$( 9 - 3 T + T^{2} )^{2}$$
$43$ $$4096 + 1088 T + 225 T^{2} + 17 T^{3} + T^{4}$$
$47$ $$144 + 108 T + 69 T^{2} + 9 T^{3} + T^{4}$$
$53$ $$( -132 + T^{2} )^{2}$$
$59$ $$36 + 18 T + 15 T^{2} - 3 T^{3} + T^{4}$$
$61$ $$5476 - 74 T + 75 T^{2} + T^{3} + T^{4}$$
$67$ $$( 49 + 7 T + T^{2} )^{2}$$
$71$ $$( -6 + T )^{4}$$
$73$ $$( -44 + 11 T + T^{2} )^{2}$$
$79$ $$( 4 - 2 T + T^{2} )^{2}$$
$83$ $$144 - 108 T + 69 T^{2} - 9 T^{3} + T^{4}$$
$89$ $$( -18 - 15 T + T^{2} )^{2}$$
$97$ $$484 - 242 T + 99 T^{2} - 11 T^{3} + T^{4}$$