# Properties

 Label 288.2.r.a Level $288$ Weight $2$ Character orbit 288.r Analytic conductor $2.300$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 288.r (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.29969157821$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{5} + 4 \zeta_{12}^{2} q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{5} + 4 \zeta_{12}^{2} q^{7} + ( -3 + 3 \zeta_{12}^{2} ) q^{9} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{11} -2 \zeta_{12} q^{13} + ( 2 - 4 \zeta_{12}^{2} ) q^{15} + 5 q^{17} -\zeta_{12}^{3} q^{19} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{21} + ( 2 - 2 \zeta_{12}^{2} ) q^{23} -\zeta_{12}^{2} q^{25} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( -4 + 4 \zeta_{12}^{2} ) q^{31} + ( -3 - 3 \zeta_{12}^{2} ) q^{33} + 8 \zeta_{12}^{3} q^{35} + 2 \zeta_{12}^{3} q^{37} + ( -2 + 4 \zeta_{12}^{2} ) q^{39} + ( 5 - 5 \zeta_{12}^{2} ) q^{41} + ( -11 \zeta_{12} + 11 \zeta_{12}^{3} ) q^{43} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{45} -6 \zeta_{12}^{2} q^{47} + ( -9 + 9 \zeta_{12}^{2} ) q^{49} + ( -5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{51} + 6 q^{55} + ( -2 + \zeta_{12}^{2} ) q^{57} -\zeta_{12} q^{59} + ( -12 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{61} -12 q^{63} -4 \zeta_{12}^{2} q^{65} + 3 \zeta_{12} q^{67} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{69} + 6 q^{71} + 9 q^{73} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{75} + 12 \zeta_{12} q^{77} -14 \zeta_{12}^{2} q^{79} -9 \zeta_{12}^{2} q^{81} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{83} + 10 \zeta_{12} q^{85} -14 q^{89} -8 \zeta_{12}^{3} q^{91} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{93} + ( 2 - 2 \zeta_{12}^{2} ) q^{95} -\zeta_{12}^{2} q^{97} + 9 \zeta_{12}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{7} - 6q^{9} + O(q^{10})$$ $$4q + 8q^{7} - 6q^{9} + 20q^{17} + 4q^{23} - 2q^{25} - 8q^{31} - 18q^{33} + 10q^{41} - 12q^{47} - 18q^{49} + 24q^{55} - 6q^{57} - 48q^{63} - 8q^{65} + 24q^{71} + 36q^{73} - 28q^{79} - 18q^{81} - 56q^{89} + 4q^{95} - 2q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-1 + \zeta_{12}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 −0.866025 1.50000i 0 1.73205 + 1.00000i 0 2.00000 + 3.46410i 0 −1.50000 + 2.59808i 0
49.2 0 0.866025 + 1.50000i 0 −1.73205 1.00000i 0 2.00000 + 3.46410i 0 −1.50000 + 2.59808i 0
241.1 0 −0.866025 + 1.50000i 0 1.73205 1.00000i 0 2.00000 3.46410i 0 −1.50000 2.59808i 0
241.2 0 0.866025 1.50000i 0 −1.73205 + 1.00000i 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
8.b even 2 1 inner
9.c even 3 1 inner
72.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.r.a 4
3.b odd 2 1 864.2.r.a 4
4.b odd 2 1 72.2.n.a 4
8.b even 2 1 inner 288.2.r.a 4
8.d odd 2 1 72.2.n.a 4
9.c even 3 1 inner 288.2.r.a 4
9.c even 3 1 2592.2.d.b 2
9.d odd 6 1 864.2.r.a 4
9.d odd 6 1 2592.2.d.a 2
12.b even 2 1 216.2.n.a 4
24.f even 2 1 216.2.n.a 4
24.h odd 2 1 864.2.r.a 4
36.f odd 6 1 72.2.n.a 4
36.f odd 6 1 648.2.d.d 2
36.h even 6 1 216.2.n.a 4
36.h even 6 1 648.2.d.a 2
72.j odd 6 1 864.2.r.a 4
72.j odd 6 1 2592.2.d.a 2
72.l even 6 1 216.2.n.a 4
72.l even 6 1 648.2.d.a 2
72.n even 6 1 inner 288.2.r.a 4
72.n even 6 1 2592.2.d.b 2
72.p odd 6 1 72.2.n.a 4
72.p odd 6 1 648.2.d.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.n.a 4 4.b odd 2 1
72.2.n.a 4 8.d odd 2 1
72.2.n.a 4 36.f odd 6 1
72.2.n.a 4 72.p odd 6 1
216.2.n.a 4 12.b even 2 1
216.2.n.a 4 24.f even 2 1
216.2.n.a 4 36.h even 6 1
216.2.n.a 4 72.l even 6 1
288.2.r.a 4 1.a even 1 1 trivial
288.2.r.a 4 8.b even 2 1 inner
288.2.r.a 4 9.c even 3 1 inner
288.2.r.a 4 72.n even 6 1 inner
648.2.d.a 2 36.h even 6 1
648.2.d.a 2 72.l even 6 1
648.2.d.d 2 36.f odd 6 1
648.2.d.d 2 72.p odd 6 1
864.2.r.a 4 3.b odd 2 1
864.2.r.a 4 9.d odd 6 1
864.2.r.a 4 24.h odd 2 1
864.2.r.a 4 72.j odd 6 1
2592.2.d.a 2 9.d odd 6 1
2592.2.d.a 2 72.j odd 6 1
2592.2.d.b 2 9.c even 3 1
2592.2.d.b 2 72.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 4 T_{5}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(288, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + 3 T^{2} + T^{4}$$
$5$ $$16 - 4 T^{2} + T^{4}$$
$7$ $$( 16 - 4 T + T^{2} )^{2}$$
$11$ $$81 - 9 T^{2} + T^{4}$$
$13$ $$16 - 4 T^{2} + T^{4}$$
$17$ $$( -5 + T )^{4}$$
$19$ $$( 1 + T^{2} )^{2}$$
$23$ $$( 4 - 2 T + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 16 + 4 T + T^{2} )^{2}$$
$37$ $$( 4 + T^{2} )^{2}$$
$41$ $$( 25 - 5 T + T^{2} )^{2}$$
$43$ $$14641 - 121 T^{2} + T^{4}$$
$47$ $$( 36 + 6 T + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$1 - T^{2} + T^{4}$$
$61$ $$20736 - 144 T^{2} + T^{4}$$
$67$ $$81 - 9 T^{2} + T^{4}$$
$71$ $$( -6 + T )^{4}$$
$73$ $$( -9 + T )^{4}$$
$79$ $$( 196 + 14 T + T^{2} )^{2}$$
$83$ $$256 - 16 T^{2} + T^{4}$$
$89$ $$( 14 + T )^{4}$$
$97$ $$( 1 + T + T^{2} )^{2}$$