# Properties

 Label 720.2.q.k 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ 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 basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{3} + ( 1 + \beta_{2} ) q^{5} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{7} + ( 1 + \beta_{2} - \beta_{5} ) q^{9} +O(q^{10})$$ $$q -\beta_{4} q^{3} + ( 1 + \beta_{2} ) q^{5} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{7} + ( 1 + \beta_{2} - \beta_{5} ) q^{9} + ( -1 + 2 \beta_{1} - 2 \beta_{4} - \beta_{5} ) q^{11} + ( -2 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{5} ) q^{13} -\beta_{1} q^{15} + ( 1 + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{17} + ( -1 + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{19} + ( -5 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{21} + ( -1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{23} + \beta_{2} q^{25} + ( -3 - \beta_{1} - 5 \beta_{2} + \beta_{3} - \beta_{4} ) q^{27} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{29} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{31} + ( -1 - 5 \beta_{2} - \beta_{3} - 2 \beta_{5} ) q^{33} + ( 1 - \beta_{1} + \beta_{4} - \beta_{5} ) q^{35} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{37} + ( 5 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{39} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{41} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{43} + ( \beta_{2} + \beta_{3} ) q^{45} + ( 1 - 3 \beta_{1} - 5 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{47} + ( -6 - 2 \beta_{1} - 7 \beta_{2} + \beta_{3} + \beta_{5} ) q^{49} + ( -4 + \beta_{2} - \beta_{3} + \beta_{5} ) q^{51} + ( -2 - 2 \beta_{1} + 2 \beta_{4} - 2 \beta_{5} ) q^{53} + ( -1 - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{55} + ( -4 + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{57} + ( 2 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{59} + ( 1 - 4 \beta_{1} + 7 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{61} + ( 1 - 8 \beta_{2} + 3 \beta_{4} - \beta_{5} ) q^{63} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{5} ) q^{65} + ( 3 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{67} + ( 1 - 4 \beta_{2} + \beta_{3} + 2 \beta_{5} ) q^{69} + ( -7 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{71} + 8 q^{73} + ( -\beta_{1} + \beta_{4} ) q^{75} + ( -2 + 6 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{77} -2 \beta_{2} q^{79} + ( 3 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{81} + ( -\beta_{1} + 7 \beta_{2} - \beta_{3} - \beta_{5} ) q^{83} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} ) q^{85} + ( -10 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{87} -3 q^{89} + ( -3 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 10 \beta_{4} + 4 \beta_{5} ) q^{91} + ( 5 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{93} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} ) q^{95} + ( 2 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{97} + ( -8 + 6 \beta_{1} - 7 \beta_{2} + \beta_{3} - 6 \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + q^{3} + 3q^{5} + 3q^{7} + 5q^{9} + O(q^{10})$$ $$6q + q^{3} + 3q^{5} + 3q^{7} + 5q^{9} - 6q^{13} - q^{15} - 12q^{19} - 20q^{21} - 3q^{23} - 3q^{25} - 2q^{27} + 3q^{29} + 6q^{31} + 12q^{33} + 6q^{35} + 24q^{37} + 20q^{39} - 3q^{41} + 6q^{43} - 2q^{45} + 15q^{47} - 18q^{49} - 30q^{51} - 12q^{53} - 32q^{57} + 6q^{59} - 21q^{61} + 29q^{63} + 6q^{65} + 9q^{67} + 15q^{69} - 48q^{71} + 48q^{73} - 2q^{75} - 6q^{77} + 6q^{79} + 29q^{81} - 21q^{83} - 42q^{87} - 18q^{89} + 16q^{93} - 6q^{95} - 18q^{97} - 12q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} - 5 \nu^{4} - \nu^{3} - 9 \nu^{2} + 6 \nu + 45$$$$)/27$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{5} + \nu^{4} + 2 \nu^{3} - 12 \nu - 36$$$$)/27$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} - \nu^{4} + \nu^{3} + 6 \nu^{2} + 3 \nu - 9$$$$)/9$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} - \nu^{4} - 2 \nu^{3} + 3 \nu^{2} - 6 \nu - 9$$$$)/9$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{5} + 7 \nu^{4} - 13 \nu^{3} - 9 \nu^{2} - 3 \nu - 90$$$$)/27$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_{1} - 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{5} + 2 \beta_{4} + \beta_{3} - 4 \beta_{2} - 4 \beta_{1} + 1$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} + 10 \beta_{2} + \beta_{1} + 2$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} - \beta_{2} - 10 \beta_{1} + 16$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$7 \beta_{5} - 5 \beta_{4} + 11 \beta_{3} + 19 \beta_{2} + 10 \beta_{1} + 38$$$$)/3$$

## 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
 −1.62241 + 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 2.05042 3.55142i 0 2.26442 + 1.96784i 0
241.2 0 0.403374 + 1.68443i 0 0.500000 + 0.866025i 0 −1.91751 + 3.32123i 0 −2.67458 + 1.35891i 0
241.3 0 1.71903 0.211943i 0 0.500000 + 0.866025i 0 1.36710 2.36788i 0 2.91016 0.728674i 0
481.1 0 −1.62241 + 0.606458i 0 0.500000 0.866025i 0 2.05042 + 3.55142i 0 2.26442 1.96784i 0
481.2 0 0.403374 1.68443i 0 0.500000 0.866025i 0 −1.91751 3.32123i 0 −2.67458 1.35891i 0
481.3 0 1.71903 + 0.211943i 0 0.500000 0.866025i 0 1.36710 + 2.36788i 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.k 6
3.b odd 2 1 2160.2.q.i 6
4.b odd 2 1 180.2.i.b 6
9.c even 3 1 inner 720.2.q.k 6
9.c even 3 1 6480.2.a.bt 3
9.d odd 6 1 2160.2.q.i 6
9.d odd 6 1 6480.2.a.bw 3
12.b even 2 1 540.2.i.b 6
20.d odd 2 1 900.2.i.c 6
20.e even 4 2 900.2.s.c 12
36.f odd 6 1 180.2.i.b 6
36.f odd 6 1 1620.2.a.i 3
36.h even 6 1 540.2.i.b 6
36.h even 6 1 1620.2.a.j 3
60.h even 2 1 2700.2.i.c 6
60.l odd 4 2 2700.2.s.c 12
180.n even 6 1 2700.2.i.c 6
180.n even 6 1 8100.2.a.u 3
180.p odd 6 1 900.2.i.c 6
180.p odd 6 1 8100.2.a.v 3
180.v odd 12 2 2700.2.s.c 12
180.v odd 12 2 8100.2.d.o 6
180.x even 12 2 900.2.s.c 12
180.x even 12 2 8100.2.d.p 6

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

## Hecke kernels

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

 $$T_{7}^{6} - 3 T_{7}^{5} + 24 T_{7}^{4} - 41 T_{7}^{3} + 354 T_{7}^{2} - 645 T_{7} + 1849$$ $$T_{11}^{6} + 24 T_{11}^{4} + 72 T_{11}^{3} + 576 T_{11}^{2} + 864 T_{11} + 1296$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$27 - 9 T - 6 T^{2} + 3 T^{3} - 2 T^{4} - T^{5} + T^{6}$$
$5$ $$( 1 - T + T^{2} )^{3}$$
$7$ $$1849 - 645 T + 354 T^{2} - 41 T^{3} + 24 T^{4} - 3 T^{5} + T^{6}$$
$11$ $$1296 + 864 T + 576 T^{2} + 72 T^{3} + 24 T^{4} + T^{6}$$
$13$ $$5776 + 912 T + 600 T^{2} + 80 T^{3} + 48 T^{4} + 6 T^{5} + T^{6}$$
$17$ $$( 36 - 24 T + T^{3} )^{2}$$
$19$ $$( -4 - 12 T + 6 T^{2} + T^{3} )^{2}$$
$23$ $$81 - 135 T + 198 T^{2} - 63 T^{3} + 24 T^{4} + 3 T^{5} + T^{6}$$
$29$ $$77841 - 19251 T + 5598 T^{2} - 351 T^{3} + 78 T^{4} - 3 T^{5} + T^{6}$$
$31$ $$16 - 48 T + 168 T^{2} + 64 T^{3} + 48 T^{4} - 6 T^{5} + T^{6}$$
$37$ $$( 436 - 36 T - 12 T^{2} + T^{3} )^{2}$$
$41$ $$6561 - 6561 T + 6318 T^{2} - 405 T^{3} + 90 T^{4} + 3 T^{5} + T^{6}$$
$43$ $$5776 - 912 T + 600 T^{2} - 80 T^{3} + 48 T^{4} - 6 T^{5} + T^{6}$$
$47$ $$729 - 1053 T + 1116 T^{2} - 531 T^{3} + 186 T^{4} - 15 T^{5} + T^{6}$$
$53$ $$( 72 - 60 T + 6 T^{2} + T^{3} )^{2}$$
$59$ $$5184 + 4320 T + 3168 T^{2} + 504 T^{3} + 96 T^{4} - 6 T^{5} + T^{6}$$
$61$ $$167281 - 25767 T + 12558 T^{2} + 2141 T^{3} + 378 T^{4} + 21 T^{5} + T^{6}$$
$67$ $$22801 - 3171 T + 1800 T^{2} - 113 T^{3} + 102 T^{4} - 9 T^{5} + T^{6}$$
$71$ $$( -324 + 108 T + 24 T^{2} + T^{3} )^{2}$$
$73$ $$( -8 + T )^{6}$$
$79$ $$( 4 - 2 T + T^{2} )^{3}$$
$83$ $$59049 + 31347 T + 11538 T^{2} + 2223 T^{3} + 312 T^{4} + 21 T^{5} + T^{6}$$
$89$ $$( 3 + T )^{6}$$
$97$ $$179776 - 15264 T + 8928 T^{2} + 1496 T^{3} + 288 T^{4} + 18 T^{5} + T^{6}$$