# Properties

 Label 720.2.q.j Level $720$ Weight $2$ Character orbit 720.q Analytic conductor $5.749$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.954288.1 Defining polynomial: $$x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 6 x^{2} - 9 x + 27$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} - \beta_{5} ) q^{3} + \beta_{3} q^{5} + ( -2 - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} - \beta_{5} ) q^{3} + \beta_{3} q^{5} + ( -2 - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{7} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{9} + ( -\beta_{1} + \beta_{2} ) q^{11} + ( 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{13} -\beta_{2} q^{15} + ( 2 - \beta_{1} + 2 \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{17} + ( 2 - \beta_{1} - \beta_{4} + \beta_{5} ) q^{19} + ( 5 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{5} ) q^{21} + ( -3 \beta_{1} + \beta_{2} + 4 \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{23} + ( -1 - \beta_{3} ) q^{25} + ( 2 + \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - \beta_{4} ) q^{27} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{29} + ( \beta_{2} + 6 \beta_{3} + \beta_{4} - \beta_{5} ) q^{31} + ( -4 + \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{33} + ( 2 - \beta_{1} - \beta_{2} + \beta_{5} ) q^{35} + ( \beta_{1} + \beta_{4} - \beta_{5} ) q^{37} + ( 3 \beta_{1} - \beta_{2} - 6 \beta_{3} + 2 \beta_{5} ) q^{39} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{41} + ( 2 - 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{43} + ( 1 - \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{45} + ( -2 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{47} + ( 2 \beta_{1} + 5 \beta_{2} + 6 \beta_{3} + 5 \beta_{4} - 3 \beta_{5} ) q^{49} + ( 2 + 4 \beta_{1} + \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{5} ) q^{51} + ( -4 - 2 \beta_{2} + 2 \beta_{4} ) q^{53} + ( \beta_{1} + \beta_{4} - \beta_{5} ) q^{55} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{57} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} ) q^{59} + ( 3 + 3 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{61} + ( -6 + 3 \beta_{1} - 3 \beta_{5} ) q^{63} + ( -\beta_{1} + \beta_{2} - 2 \beta_{4} - 2 \beta_{5} ) q^{65} + ( 5 \beta_{1} - 2 \beta_{3} + 5 \beta_{5} ) q^{67} + ( -5 - 7 \beta_{1} - \beta_{3} - 5 \beta_{4} - 3 \beta_{5} ) q^{69} + ( 6 - 3 \beta_{1} - 2 \beta_{2} - \beta_{4} + 3 \beta_{5} ) q^{71} + ( 4 - 4 \beta_{2} + 4 \beta_{4} ) q^{73} + \beta_{5} q^{75} + ( -3 \beta_{2} - 6 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{77} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{79} + ( 4 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{81} + ( -6 - 2 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} - \beta_{4} - \beta_{5} ) q^{83} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{85} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} ) q^{87} -11 q^{89} + ( 12 - 3 \beta_{1} - 3 \beta_{4} + 3 \beta_{5} ) q^{91} + ( -2 - \beta_{1} - 5 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} ) q^{93} + ( \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{95} + ( -8 - 2 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{97} + ( 4 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + q^{3} - 3q^{5} - 5q^{7} + 5q^{9} + O(q^{10})$$ $$6q + q^{3} - 3q^{5} - 5q^{7} + 5q^{9} - 2q^{11} + q^{15} + 4q^{17} + 8q^{19} + 12q^{21} - 7q^{23} - 3q^{25} - 2q^{27} + 7q^{29} - 16q^{31} - 20q^{33} + 10q^{35} + 4q^{37} + 18q^{39} + q^{41} + 2q^{43} + 2q^{45} - 13q^{47} - 10q^{49} - 20q^{53} + 4q^{55} - 14q^{57} - 6q^{59} + 11q^{61} - 27q^{63} + q^{67} - 33q^{69} + 28q^{71} + 32q^{73} - 2q^{75} + 12q^{77} - 6q^{79} + 29q^{81} - 21q^{83} - 2q^{85} - 2q^{87} - 66q^{89} + 60q^{91} + 2q^{93} - 4q^{95} - 30q^{97} + 14q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 2 x^{4} + 3 x^{3} - 6 x^{2} - 9 x + 27$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 5 \nu^{4} + \nu^{3} + 9 \nu^{2} - 6 \nu - 45$$$$)/27$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{5} - \nu^{4} - 2 \nu^{3} + 12 \nu + 9$$$$)/27$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} + 2 \nu^{4} - 2 \nu^{3} - 6 \nu - 18$$$$)/9$$ $$\beta_{5}$$ $$=$$ $$($$$$4 \nu^{5} + 2 \nu^{4} - 5 \nu^{3} + 18 \nu^{2} - 24 \nu - 72$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2$$ $$\nu^{3}$$ $$=$$ $$-\beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 2$$ $$\nu^{4}$$ $$=$$ $$-2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 4 \beta_{2} + 5$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{5} + \beta_{4} - 10 \beta_{3} - 4 \beta_{2} + 6 \beta_{1} + 4$$

## Character values

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

 $$n$$ $$181$$ $$271$$ $$577$$ $$641$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$\beta_{3}$$

## 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
 −1.62241 + 0.606458i 0.403374 − 1.68443i 1.71903 + 0.211943i −1.62241 − 0.606458i 0.403374 + 1.68443i 1.71903 − 0.211943i
0 −1.62241 0.606458i 0 −0.500000 0.866025i 0 −2.62241 + 4.54214i 0 2.26442 + 1.96784i 0
241.2 0 0.403374 + 1.68443i 0 −0.500000 0.866025i 0 −0.596626 + 1.03339i 0 −2.67458 + 1.35891i 0
241.3 0 1.71903 0.211943i 0 −0.500000 0.866025i 0 0.719035 1.24540i 0 2.91016 0.728674i 0
481.1 0 −1.62241 + 0.606458i 0 −0.500000 + 0.866025i 0 −2.62241 4.54214i 0 2.26442 1.96784i 0
481.2 0 0.403374 1.68443i 0 −0.500000 + 0.866025i 0 −0.596626 1.03339i 0 −2.67458 1.35891i 0
481.3 0 1.71903 + 0.211943i 0 −0.500000 + 0.866025i 0 0.719035 + 1.24540i 0 2.91016 + 0.728674i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 481.3 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.j 6
3.b odd 2 1 2160.2.q.j 6
4.b odd 2 1 360.2.q.d 6
9.c even 3 1 inner 720.2.q.j 6
9.c even 3 1 6480.2.a.bx 3
9.d odd 6 1 2160.2.q.j 6
9.d odd 6 1 6480.2.a.bu 3
12.b even 2 1 1080.2.q.d 6
36.f odd 6 1 360.2.q.d 6
36.f odd 6 1 3240.2.a.r 3
36.h even 6 1 1080.2.q.d 6
36.h even 6 1 3240.2.a.q 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.q.d 6 4.b odd 2 1
360.2.q.d 6 36.f odd 6 1
720.2.q.j 6 1.a even 1 1 trivial
720.2.q.j 6 9.c even 3 1 inner
1080.2.q.d 6 12.b even 2 1
1080.2.q.d 6 36.h even 6 1
2160.2.q.j 6 3.b odd 2 1
2160.2.q.j 6 9.d odd 6 1
3240.2.a.q 3 36.h even 6 1
3240.2.a.r 3 36.f odd 6 1
6480.2.a.bu 3 9.d odd 6 1
6480.2.a.bx 3 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 $$S_{2}^{\mathrm{new}}(720, [\chi])$$:

 $$T_{7}^{6} + 5 T_{7}^{5} + 28 T_{7}^{4} + 3 T_{7}^{3} + 54 T_{7}^{2} + 27 T_{7} + 81$$ $$T_{11}^{6} + 2 T_{11}^{5} + 12 T_{11}^{4} + 8 T_{11}^{3} + 88 T_{11}^{2} + 96 T_{11} + 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$27 - 9 T - 6 T^{2} + 3 T^{3} - 2 T^{4} - T^{5} + T^{6}$$
$5$ $$( 1 + T + T^{2} )^{3}$$
$7$ $$81 + 27 T + 54 T^{2} + 3 T^{3} + 28 T^{4} + 5 T^{5} + T^{6}$$
$11$ $$144 + 96 T + 88 T^{2} + 8 T^{3} + 12 T^{4} + 2 T^{5} + T^{6}$$
$13$ $$1296 + 864 T + 576 T^{2} + 72 T^{3} + 24 T^{4} + T^{6}$$
$17$ $$( 108 - 36 T - 2 T^{2} + T^{3} )^{2}$$
$19$ $$( 4 - 4 T - 4 T^{2} + T^{3} )^{2}$$
$23$ $$91809 + 15453 T + 4722 T^{2} + 249 T^{3} + 100 T^{4} + 7 T^{5} + T^{6}$$
$29$ $$729 - 135 T + 214 T^{2} - 19 T^{3} + 54 T^{4} - 7 T^{5} + T^{6}$$
$31$ $$11664 + 8208 T + 4048 T^{2} + 1000 T^{3} + 180 T^{4} + 16 T^{5} + T^{6}$$
$37$ $$( 12 - 8 T - 2 T^{2} + T^{3} )^{2}$$
$41$ $$9 + 27 T + 78 T^{2} + 15 T^{3} + 10 T^{4} - T^{5} + T^{6}$$
$43$ $$11664 - 3888 T + 1512 T^{2} - 144 T^{3} + 40 T^{4} - 2 T^{5} + T^{6}$$
$47$ $$1261129 + 95455 T + 21824 T^{2} + 1141 T^{3} + 254 T^{4} + 13 T^{5} + T^{6}$$
$53$ $$( -24 + 12 T + 10 T^{2} + T^{3} )^{2}$$
$59$ $$5184 - 4320 T + 3168 T^{2} - 504 T^{3} + 96 T^{4} + 6 T^{5} + T^{6}$$
$61$ $$269361 - 21279 T + 7390 T^{2} - 587 T^{3} + 162 T^{4} - 11 T^{5} + T^{6}$$
$67$ $$12769 + 15029 T + 17576 T^{2} + 359 T^{3} + 134 T^{4} - T^{5} + T^{6}$$
$71$ $$( 36 - 20 T - 14 T^{2} + T^{3} )^{2}$$
$73$ $$( 384 - 16 T^{2} + T^{3} )^{2}$$
$79$ $$10816 - 8736 T + 6432 T^{2} - 712 T^{3} + 120 T^{4} + 6 T^{5} + T^{6}$$
$83$ $$59049 + 31347 T + 11538 T^{2} + 2223 T^{3} + 312 T^{4} + 21 T^{5} + T^{6}$$
$89$ $$( 11 + T )^{6}$$
$97$ $$1201216 - 170976 T + 57216 T^{2} + 6872 T^{3} + 744 T^{4} + 30 T^{5} + T^{6}$$