# Properties

 Label 1152.4.a.d Level $1152$ Weight $4$ Character orbit 1152.a Self dual yes Analytic conductor $67.970$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1152.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.9702003266$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 128) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 6q^{5} + 20q^{7} + O(q^{10})$$ $$q - 6q^{5} + 20q^{7} - 14q^{11} + 54q^{13} + 66q^{17} + 162q^{19} - 172q^{23} - 89q^{25} + 2q^{29} - 128q^{31} - 120q^{35} + 158q^{37} - 202q^{41} - 298q^{43} + 408q^{47} + 57q^{49} + 690q^{53} + 84q^{55} + 322q^{59} - 298q^{61} - 324q^{65} + 202q^{67} + 700q^{71} - 418q^{73} - 280q^{77} + 744q^{79} + 678q^{83} - 396q^{85} + 82q^{89} + 1080q^{91} - 972q^{95} - 1122q^{97} + O(q^{100})$$

## 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 −6.00000 0 20.0000 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 1152.4.a.d 1
3.b odd 2 1 128.4.a.d yes 1
4.b odd 2 1 1152.4.a.c 1
8.b even 2 1 1152.4.a.j 1
8.d odd 2 1 1152.4.a.i 1
12.b even 2 1 128.4.a.b yes 1
24.f even 2 1 128.4.a.c yes 1
24.h odd 2 1 128.4.a.a 1
48.i odd 4 2 256.4.b.b 2
48.k even 4 2 256.4.b.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.a.a 1 24.h odd 2 1
128.4.a.b yes 1 12.b even 2 1
128.4.a.c yes 1 24.f even 2 1
128.4.a.d yes 1 3.b odd 2 1
256.4.b.b 2 48.i odd 4 2
256.4.b.f 2 48.k even 4 2
1152.4.a.c 1 4.b odd 2 1
1152.4.a.d 1 1.a even 1 1 trivial
1152.4.a.i 1 8.d odd 2 1
1152.4.a.j 1 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1152))$$:

 $$T_{5} + 6$$ $$T_{7} - 20$$ $$T_{13} - 54$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$6 + T$$
$7$ $$-20 + T$$
$11$ $$14 + T$$
$13$ $$-54 + T$$
$17$ $$-66 + T$$
$19$ $$-162 + T$$
$23$ $$172 + T$$
$29$ $$-2 + T$$
$31$ $$128 + T$$
$37$ $$-158 + T$$
$41$ $$202 + T$$
$43$ $$298 + T$$
$47$ $$-408 + T$$
$53$ $$-690 + T$$
$59$ $$-322 + T$$
$61$ $$298 + T$$
$67$ $$-202 + T$$
$71$ $$-700 + T$$
$73$ $$418 + T$$
$79$ $$-744 + T$$
$83$ $$-678 + T$$
$89$ $$-82 + T$$
$97$ $$1122 + T$$