# Properties

 Label 1152.2.a.o Level $1152$ Weight $2$ Character orbit 1152.a Self dual yes Analytic conductor $9.199$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1152.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.19876631285$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{5} - 2q^{7} + O(q^{10})$$ $$q + 2q^{5} - 2q^{7} - 4q^{11} + 2q^{13} + 4q^{17} + 4q^{19} + 8q^{23} - q^{25} + 6q^{29} + 6q^{31} - 4q^{35} - 2q^{37} + 12q^{41} - 12q^{43} + 8q^{47} - 3q^{49} + 6q^{53} - 8q^{55} - 8q^{59} - 10q^{61} + 4q^{65} + 8q^{67} + 2q^{73} + 8q^{77} + 14q^{79} - 12q^{83} + 8q^{85} - 8q^{89} - 4q^{91} + 8q^{95} - 2q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 2.00000 0 −2.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.2.a.o yes 1
3.b odd 2 1 1152.2.a.e yes 1
4.b odd 2 1 1152.2.a.q yes 1
8.b even 2 1 1152.2.a.d 1
8.d odd 2 1 1152.2.a.f yes 1
12.b even 2 1 1152.2.a.g yes 1
16.e even 4 2 2304.2.d.p 2
16.f odd 4 2 2304.2.d.g 2
24.f even 2 1 1152.2.a.p yes 1
24.h odd 2 1 1152.2.a.n yes 1
48.i odd 4 2 2304.2.d.n 2
48.k even 4 2 2304.2.d.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.a.d 1 8.b even 2 1
1152.2.a.e yes 1 3.b odd 2 1
1152.2.a.f yes 1 8.d odd 2 1
1152.2.a.g yes 1 12.b even 2 1
1152.2.a.n yes 1 24.h odd 2 1
1152.2.a.o yes 1 1.a even 1 1 trivial
1152.2.a.p yes 1 24.f even 2 1
1152.2.a.q yes 1 4.b odd 2 1
2304.2.d.e 2 48.k even 4 2
2304.2.d.g 2 16.f odd 4 2
2304.2.d.n 2 48.i odd 4 2
2304.2.d.p 2 16.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1152))$$:

 $$T_{5} - 2$$ $$T_{7} + 2$$ $$T_{11} + 4$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-2 + T$$
$7$ $$2 + T$$
$11$ $$4 + T$$
$13$ $$-2 + T$$
$17$ $$-4 + T$$
$19$ $$-4 + T$$
$23$ $$-8 + T$$
$29$ $$-6 + T$$
$31$ $$-6 + T$$
$37$ $$2 + T$$
$41$ $$-12 + T$$
$43$ $$12 + T$$
$47$ $$-8 + T$$
$53$ $$-6 + T$$
$59$ $$8 + T$$
$61$ $$10 + T$$
$67$ $$-8 + T$$
$71$ $$T$$
$73$ $$-2 + T$$
$79$ $$-14 + T$$
$83$ $$12 + T$$
$89$ $$8 + T$$
$97$ $$2 + T$$