# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$288 = 2^{5} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 288.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.29969157821$$ Analytic rank: $$0$$ 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: yes 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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{6} ) q^{3} -4 \zeta_{6} q^{5} + ( 2 - 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 + \zeta_{6} ) q^{3} -4 \zeta_{6} q^{5} + ( 2 - 2 \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( 5 - 5 \zeta_{6} ) q^{11} + 2 \zeta_{6} q^{13} + ( 4 - 8 \zeta_{6} ) q^{15} -3 q^{17} + q^{19} + ( 4 - 2 \zeta_{6} ) q^{21} + 6 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 2 - 2 \zeta_{6} ) q^{29} + 4 \zeta_{6} q^{31} + ( 10 - 5 \zeta_{6} ) q^{33} -8 q^{35} -8 q^{37} + ( -2 + 4 \zeta_{6} ) q^{39} -\zeta_{6} q^{41} + ( 7 - 7 \zeta_{6} ) q^{43} + ( 12 - 12 \zeta_{6} ) q^{45} + ( -2 + 2 \zeta_{6} ) q^{47} + 3 \zeta_{6} q^{49} + ( -3 - 3 \zeta_{6} ) q^{51} -4 q^{53} -20 q^{55} + ( 1 + \zeta_{6} ) q^{57} -5 \zeta_{6} q^{59} + 6 q^{63} + ( 8 - 8 \zeta_{6} ) q^{65} + 13 \zeta_{6} q^{67} + ( -6 + 12 \zeta_{6} ) q^{69} + 8 q^{71} + 3 q^{73} + ( -22 + 11 \zeta_{6} ) q^{75} -10 \zeta_{6} q^{77} + ( -8 + 8 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( -12 + 12 \zeta_{6} ) q^{83} + 12 \zeta_{6} q^{85} + ( 4 - 2 \zeta_{6} ) q^{87} -10 q^{89} + 4 q^{91} + ( -4 + 8 \zeta_{6} ) q^{93} -4 \zeta_{6} q^{95} + ( 11 - 11 \zeta_{6} ) q^{97} + 15 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} - 4q^{5} + 2q^{7} + 3q^{9} + O(q^{10})$$ $$2q + 3q^{3} - 4q^{5} + 2q^{7} + 3q^{9} + 5q^{11} + 2q^{13} - 6q^{17} + 2q^{19} + 6q^{21} + 6q^{23} - 11q^{25} + 2q^{29} + 4q^{31} + 15q^{33} - 16q^{35} - 16q^{37} - q^{41} + 7q^{43} + 12q^{45} - 2q^{47} + 3q^{49} - 9q^{51} - 8q^{53} - 40q^{55} + 3q^{57} - 5q^{59} + 12q^{63} + 8q^{65} + 13q^{67} + 16q^{71} + 6q^{73} - 33q^{75} - 10q^{77} - 8q^{79} - 9q^{81} - 12q^{83} + 12q^{85} + 6q^{87} - 20q^{89} + 8q^{91} - 4q^{95} + 11q^{97} + 30q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 + 0.866025i 0 −2.00000 3.46410i 0 1.00000 1.73205i 0 1.50000 + 2.59808i 0
193.1 0 1.50000 0.866025i 0 −2.00000 + 3.46410i 0 1.00000 + 1.73205i 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 288.2.i.b yes 2
3.b odd 2 1 864.2.i.b 2
4.b odd 2 1 288.2.i.a 2
8.b even 2 1 576.2.i.b 2
8.d odd 2 1 576.2.i.h 2
9.c even 3 1 inner 288.2.i.b yes 2
9.c even 3 1 2592.2.a.g 1
9.d odd 6 1 864.2.i.b 2
9.d odd 6 1 2592.2.a.a 1
12.b even 2 1 864.2.i.a 2
24.f even 2 1 1728.2.i.a 2
24.h odd 2 1 1728.2.i.b 2
36.f odd 6 1 288.2.i.a 2
36.f odd 6 1 2592.2.a.h 1
36.h even 6 1 864.2.i.a 2
36.h even 6 1 2592.2.a.b 1
72.j odd 6 1 1728.2.i.b 2
72.j odd 6 1 5184.2.a.be 1
72.l even 6 1 1728.2.i.a 2
72.l even 6 1 5184.2.a.bf 1
72.n even 6 1 576.2.i.b 2
72.n even 6 1 5184.2.a.a 1
72.p odd 6 1 576.2.i.h 2
72.p odd 6 1 5184.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.a 2 4.b odd 2 1
288.2.i.a 2 36.f odd 6 1
288.2.i.b yes 2 1.a even 1 1 trivial
288.2.i.b yes 2 9.c even 3 1 inner
576.2.i.b 2 8.b even 2 1
576.2.i.b 2 72.n even 6 1
576.2.i.h 2 8.d odd 2 1
576.2.i.h 2 72.p odd 6 1
864.2.i.a 2 12.b even 2 1
864.2.i.a 2 36.h even 6 1
864.2.i.b 2 3.b odd 2 1
864.2.i.b 2 9.d odd 6 1
1728.2.i.a 2 24.f even 2 1
1728.2.i.a 2 72.l even 6 1
1728.2.i.b 2 24.h odd 2 1
1728.2.i.b 2 72.j odd 6 1
2592.2.a.a 1 9.d odd 6 1
2592.2.a.b 1 36.h even 6 1
2592.2.a.g 1 9.c even 3 1
2592.2.a.h 1 36.f odd 6 1
5184.2.a.a 1 72.n even 6 1
5184.2.a.b 1 72.p odd 6 1
5184.2.a.be 1 72.j odd 6 1
5184.2.a.bf 1 72.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(288, [\chi])$$:

 $$T_{5}^{2} + 4 T_{5} + 16$$ $$T_{7}^{2} - 2 T_{7} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 - 3 T + T^{2}$$
$5$ $$16 + 4 T + T^{2}$$
$7$ $$4 - 2 T + T^{2}$$
$11$ $$25 - 5 T + T^{2}$$
$13$ $$4 - 2 T + T^{2}$$
$17$ $$( 3 + T )^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$4 - 2 T + T^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$( 8 + T )^{2}$$
$41$ $$1 + T + T^{2}$$
$43$ $$49 - 7 T + T^{2}$$
$47$ $$4 + 2 T + T^{2}$$
$53$ $$( 4 + T )^{2}$$
$59$ $$25 + 5 T + T^{2}$$
$61$ $$T^{2}$$
$67$ $$169 - 13 T + T^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$( -3 + T )^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$144 + 12 T + T^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$121 - 11 T + T^{2}$$