# Properties

 Label 1152.3.b.f Level $1152$ Weight $3$ Character orbit 1152.b Analytic conductor $31.390$ Analytic rank $0$ Dimension $4$ CM discriminant -24 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$31.3897264543$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{5} + \beta_{2} q^{7} +O(q^{10})$$ $$q + \beta_{1} q^{5} + \beta_{2} q^{7} + \beta_{3} q^{11} + 21 q^{25} + 25 \beta_{1} q^{29} -5 \beta_{2} q^{31} -\beta_{3} q^{35} -47 q^{49} + 47 \beta_{1} q^{53} + 4 \beta_{2} q^{55} -6 \beta_{3} q^{59} + 50 q^{73} + 96 \beta_{1} q^{77} + 15 \beta_{2} q^{79} + 5 \beta_{3} q^{83} -190 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 84q^{25} - 188q^{49} + 200q^{73} - 760q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{2}$$$$/3$$ $$\beta_{2}$$ $$=$$ $$($$$$4 \nu^{3} + 12 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-8 \nu^{3} + 24 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2}$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{1}$$$$/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{3} + 6 \beta_{2}$$$$)/16$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
703.1
 1.22474 − 1.22474i −1.22474 + 1.22474i −1.22474 − 1.22474i 1.22474 + 1.22474i
0 0 0 2.00000i 0 9.79796i 0 0 0
703.2 0 0 0 2.00000i 0 9.79796i 0 0 0
703.3 0 0 0 2.00000i 0 9.79796i 0 0 0
703.4 0 0 0 2.00000i 0 9.79796i 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 1152.3.b.f 4
3.b odd 2 1 inner 1152.3.b.f 4
4.b odd 2 1 inner 1152.3.b.f 4
8.b even 2 1 inner 1152.3.b.f 4
8.d odd 2 1 inner 1152.3.b.f 4
12.b even 2 1 inner 1152.3.b.f 4
16.e even 4 1 2304.3.g.i 2
16.e even 4 1 2304.3.g.l 2
16.f odd 4 1 2304.3.g.i 2
16.f odd 4 1 2304.3.g.l 2
24.f even 2 1 inner 1152.3.b.f 4
24.h odd 2 1 CM 1152.3.b.f 4
48.i odd 4 1 2304.3.g.i 2
48.i odd 4 1 2304.3.g.l 2
48.k even 4 1 2304.3.g.i 2
48.k even 4 1 2304.3.g.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.b.f 4 1.a even 1 1 trivial
1152.3.b.f 4 3.b odd 2 1 inner
1152.3.b.f 4 4.b odd 2 1 inner
1152.3.b.f 4 8.b even 2 1 inner
1152.3.b.f 4 8.d odd 2 1 inner
1152.3.b.f 4 12.b even 2 1 inner
1152.3.b.f 4 24.f even 2 1 inner
1152.3.b.f 4 24.h odd 2 1 CM
2304.3.g.i 2 16.e even 4 1
2304.3.g.i 2 16.f odd 4 1
2304.3.g.i 2 48.i odd 4 1
2304.3.g.i 2 48.k even 4 1
2304.3.g.l 2 16.e even 4 1
2304.3.g.l 2 16.f odd 4 1
2304.3.g.l 2 48.i odd 4 1
2304.3.g.l 2 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1152, [\chi])$$:

 $$T_{5}^{2} + 4$$ $$T_{7}^{2} + 96$$ $$T_{17}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 4 + T^{2} )^{2}$$
$7$ $$( 96 + T^{2} )^{2}$$
$11$ $$( -384 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( 2500 + T^{2} )^{2}$$
$31$ $$( 2400 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$( 8836 + T^{2} )^{2}$$
$59$ $$( -13824 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -50 + T )^{4}$$
$79$ $$( 21600 + T^{2} )^{2}$$
$83$ $$( -9600 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$( 190 + T )^{4}$$