# Properties

 Label 1152.3.b.h Level $1152$ Weight $3$ Character orbit 1152.b Analytic conductor $31.390$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

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## 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: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 384) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 \zeta_{12}^{3} q^{5} + ( -4 + 8 \zeta_{12}^{2} ) q^{7} +O(q^{10})$$ $$q + 4 \zeta_{12}^{3} q^{5} + ( -4 + 8 \zeta_{12}^{2} ) q^{7} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{11} + 18 q^{17} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{19} + ( 24 - 48 \zeta_{12}^{2} ) q^{23} + 9 q^{25} + 4 \zeta_{12}^{3} q^{29} + ( -28 + 56 \zeta_{12}^{2} ) q^{31} + ( -32 \zeta_{12} + 16 \zeta_{12}^{3} ) q^{35} + 72 \zeta_{12}^{3} q^{37} -18 q^{41} + ( -72 \zeta_{12} + 36 \zeta_{12}^{3} ) q^{43} + ( 24 - 48 \zeta_{12}^{2} ) q^{47} + q^{49} + 44 \zeta_{12}^{3} q^{53} + ( -16 + 32 \zeta_{12}^{2} ) q^{55} + ( 72 \zeta_{12} - 36 \zeta_{12}^{3} ) q^{59} + 72 \zeta_{12}^{3} q^{61} + ( 24 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{67} + ( -24 + 48 \zeta_{12}^{2} ) q^{71} -82 q^{73} + 48 \zeta_{12}^{3} q^{77} + ( 36 - 72 \zeta_{12}^{2} ) q^{79} + ( -152 \zeta_{12} + 76 \zeta_{12}^{3} ) q^{83} + 72 \zeta_{12}^{3} q^{85} -126 q^{89} + ( -48 + 96 \zeta_{12}^{2} ) q^{95} + 110 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 72q^{17} + 36q^{25} - 72q^{41} + 4q^{49} - 328q^{73} - 504q^{89} + 440q^{97} + O(q^{100})$$

## 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
 0.866025 − 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
0 0 0 4.00000i 0 6.92820i 0 0 0
703.2 0 0 0 4.00000i 0 6.92820i 0 0 0
703.3 0 0 0 4.00000i 0 6.92820i 0 0 0
703.4 0 0 0 4.00000i 0 6.92820i 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 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.h 4
3.b odd 2 1 384.3.b.a 4
4.b odd 2 1 inner 1152.3.b.h 4
8.b even 2 1 inner 1152.3.b.h 4
8.d odd 2 1 inner 1152.3.b.h 4
12.b even 2 1 384.3.b.a 4
16.e even 4 1 2304.3.g.g 2
16.e even 4 1 2304.3.g.n 2
16.f odd 4 1 2304.3.g.g 2
16.f odd 4 1 2304.3.g.n 2
24.f even 2 1 384.3.b.a 4
24.h odd 2 1 384.3.b.a 4
48.i odd 4 1 768.3.g.a 2
48.i odd 4 1 768.3.g.b 2
48.k even 4 1 768.3.g.a 2
48.k even 4 1 768.3.g.b 2

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

## Hecke kernels

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

 $$T_{5}^{2} + 16$$ $$T_{7}^{2} + 48$$ $$T_{17} - 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 16 + T^{2} )^{2}$$
$7$ $$( 48 + T^{2} )^{2}$$
$11$ $$( -48 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( -18 + T )^{4}$$
$19$ $$( -432 + T^{2} )^{2}$$
$23$ $$( 1728 + T^{2} )^{2}$$
$29$ $$( 16 + T^{2} )^{2}$$
$31$ $$( 2352 + T^{2} )^{2}$$
$37$ $$( 5184 + T^{2} )^{2}$$
$41$ $$( 18 + T )^{4}$$
$43$ $$( -3888 + T^{2} )^{2}$$
$47$ $$( 1728 + T^{2} )^{2}$$
$53$ $$( 1936 + T^{2} )^{2}$$
$59$ $$( -3888 + T^{2} )^{2}$$
$61$ $$( 5184 + T^{2} )^{2}$$
$67$ $$( -432 + T^{2} )^{2}$$
$71$ $$( 1728 + T^{2} )^{2}$$
$73$ $$( 82 + T )^{4}$$
$79$ $$( 3888 + T^{2} )^{2}$$
$83$ $$( -17328 + T^{2} )^{2}$$
$89$ $$( 126 + T )^{4}$$
$97$ $$( -110 + T )^{4}$$
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