# Properties

 Label 1152.2.d.g Level $1152$ Weight $2$ Character orbit 1152.d Analytic conductor $9.199$ Analytic rank $0$ Dimension $4$ CM discriminant -24 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1152.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.19876631285$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + \beta_{1} q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + \beta_{1} q^{7} -\beta_{3} q^{11} -7 q^{25} + 3 \beta_{2} q^{29} -\beta_{1} q^{31} -3 \beta_{3} q^{35} + 17 q^{49} + \beta_{2} q^{53} -4 \beta_{1} q^{55} -2 \beta_{3} q^{59} -14 q^{73} -8 \beta_{2} q^{77} + 3 \beta_{1} q^{79} -\beta_{3} q^{83} + 2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 28q^{25} + 68q^{49} - 56q^{73} + 8q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{3} + 4 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} - 2$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{3}$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3}$$$$/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$641$$ $$901$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
577.1
 −1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
0 0 0 3.46410i 0 −4.89898 0 0 0
577.2 0 0 0 3.46410i 0 4.89898 0 0 0
577.3 0 0 0 3.46410i 0 −4.89898 0 0 0
577.4 0 0 0 3.46410i 0 4.89898 0 0 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{5}^{2} + 12$$ $$T_{7}^{2} - 24$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 12 + T^{2} )^{2}$$
$7$ $$( -24 + T^{2} )^{2}$$
$11$ $$( 32 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( 108 + T^{2} )^{2}$$
$31$ $$( -24 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$( 12 + T^{2} )^{2}$$
$59$ $$( 128 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( 14 + T )^{4}$$
$79$ $$( -216 + T^{2} )^{2}$$
$83$ $$( 32 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$( -2 + T )^{4}$$