# Properties

 Label 1152.2.a.b Level $1152$ Weight $2$ Character orbit 1152.a Self dual yes Analytic conductor $9.199$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 384) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 4q^{5} + 2q^{7} + O(q^{10})$$ $$q - 4q^{5} + 2q^{7} + 4q^{11} - 2q^{13} + 2q^{17} - 8q^{19} - 4q^{23} + 11q^{25} - 6q^{31} - 8q^{35} + 2q^{37} - 6q^{41} - 4q^{47} - 3q^{49} - 16q^{55} - 4q^{59} - 14q^{61} + 8q^{65} - 4q^{67} - 12q^{71} - 10q^{73} + 8q^{77} + 10q^{79} - 12q^{83} - 8q^{85} + 14q^{89} - 4q^{91} + 32q^{95} + 10q^{97} + O(q^{100})$$

## 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 −4.00000 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.b 1
3.b odd 2 1 384.2.a.h yes 1
4.b odd 2 1 1152.2.a.a 1
8.b even 2 1 1152.2.a.t 1
8.d odd 2 1 1152.2.a.s 1
12.b even 2 1 384.2.a.d yes 1
15.d odd 2 1 9600.2.a.e 1
16.e even 4 2 2304.2.d.f 2
16.f odd 4 2 2304.2.d.o 2
24.f even 2 1 384.2.a.e yes 1
24.h odd 2 1 384.2.a.a 1
48.i odd 4 2 768.2.d.c 2
48.k even 4 2 768.2.d.f 2
60.h even 2 1 9600.2.a.bz 1
120.i odd 2 1 9600.2.a.bk 1
120.m even 2 1 9600.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.2.a.a 1 24.h odd 2 1
384.2.a.d yes 1 12.b even 2 1
384.2.a.e yes 1 24.f even 2 1
384.2.a.h yes 1 3.b odd 2 1
768.2.d.c 2 48.i odd 4 2
768.2.d.f 2 48.k even 4 2
1152.2.a.a 1 4.b odd 2 1
1152.2.a.b 1 1.a even 1 1 trivial
1152.2.a.s 1 8.d odd 2 1
1152.2.a.t 1 8.b even 2 1
2304.2.d.f 2 16.e even 4 2
2304.2.d.o 2 16.f odd 4 2
9600.2.a.e 1 15.d odd 2 1
9600.2.a.t 1 120.m even 2 1
9600.2.a.bk 1 120.i odd 2 1
9600.2.a.bz 1 60.h even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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