# Properties

 Label 1152.3.b.i Level $1152$ Weight $3$ Character orbit 1152.b Analytic conductor $31.390$ Analytic rank $0$ Dimension $8$ CM no 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: $$8$$ Coefficient field: 8.0.157351936.1 Defining polynomial: $$x^{8} + x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{22}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{5} -\beta_{5} q^{7} +O(q^{10})$$ $$q -\beta_{1} q^{5} -\beta_{5} q^{7} -\beta_{3} q^{11} -\beta_{2} q^{13} -\beta_{4} q^{17} -\beta_{7} q^{19} -\beta_{6} q^{23} -3 q^{25} -\beta_{1} q^{29} -3 \beta_{5} q^{31} -7 \beta_{3} q^{35} + 7 \beta_{2} q^{37} + 3 \beta_{4} q^{41} + \beta_{7} q^{43} -3 \beta_{6} q^{47} -7 q^{49} + 9 \beta_{1} q^{53} + 4 \beta_{5} q^{55} -18 \beta_{3} q^{59} -19 \beta_{2} q^{61} -\beta_{4} q^{65} + 2 \beta_{7} q^{67} -4 \beta_{6} q^{71} + 26 q^{73} + 8 \beta_{1} q^{77} + 17 \beta_{5} q^{79} -21 \beta_{3} q^{83} + 28 \beta_{2} q^{85} -2 \beta_{4} q^{89} -\beta_{7} q^{91} -7 \beta_{6} q^{95} + 18 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 24q^{25} - 56q^{49} + 208q^{73} + 144q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$4 \nu^{4} + 2$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{2}$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} - 4 \nu^{5} + 7 \nu^{3} + 4 \nu$$$$)/6$$ $$\beta_{4}$$ $$=$$ $$-2 \nu^{6} + 6 \nu^{2}$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{7} + 4 \nu^{5} + 13 \nu^{3} + 44 \nu$$$$)/12$$ $$\beta_{6}$$ $$=$$ $$($$$$2 \nu^{7} - 8 \nu^{5} - 14 \nu^{3} + 8 \nu$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{7} + 4 \nu^{5} - 13 \nu^{3} + 44 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{7} + \beta_{6} + 8 \beta_{5} + 4 \beta_{3}$$$$)/64$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} + 6 \beta_{2}$$$$)/16$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{7} - 5 \beta_{6} + 8 \beta_{5} + 20 \beta_{3}$$$$)/64$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{1} - 2$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$2 \beta_{7} - 11 \beta_{6} + 8 \beta_{5} - 44 \beta_{3}$$$$)/64$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{4} + 18 \beta_{2}$$$$)/16$$ $$\nu^{7}$$ $$=$$ $$($$$$-14 \beta_{7} + 13 \beta_{6} + 56 \beta_{5} - 52 \beta_{3}$$$$)/64$$

## 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
 1.28897 + 0.581861i −0.581861 + 1.28897i −1.28897 − 0.581861i 0.581861 − 1.28897i 0.581861 + 1.28897i −1.28897 + 0.581861i −0.581861 − 1.28897i 1.28897 − 0.581861i
0 0 0 5.29150i 0 7.48331i 0 0 0
703.2 0 0 0 5.29150i 0 7.48331i 0 0 0
703.3 0 0 0 5.29150i 0 7.48331i 0 0 0
703.4 0 0 0 5.29150i 0 7.48331i 0 0 0
703.5 0 0 0 5.29150i 0 7.48331i 0 0 0
703.6 0 0 0 5.29150i 0 7.48331i 0 0 0
703.7 0 0 0 5.29150i 0 7.48331i 0 0 0
703.8 0 0 0 5.29150i 0 7.48331i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 703.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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
24.h 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.i 8
3.b odd 2 1 inner 1152.3.b.i 8
4.b odd 2 1 inner 1152.3.b.i 8
8.b even 2 1 inner 1152.3.b.i 8
8.d odd 2 1 inner 1152.3.b.i 8
12.b even 2 1 inner 1152.3.b.i 8
16.e even 4 1 2304.3.g.p 4
16.e even 4 1 2304.3.g.w 4
16.f odd 4 1 2304.3.g.p 4
16.f odd 4 1 2304.3.g.w 4
24.f even 2 1 inner 1152.3.b.i 8
24.h odd 2 1 inner 1152.3.b.i 8
48.i odd 4 1 2304.3.g.p 4
48.i odd 4 1 2304.3.g.w 4
48.k even 4 1 2304.3.g.p 4
48.k even 4 1 2304.3.g.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.b.i 8 1.a even 1 1 trivial
1152.3.b.i 8 3.b odd 2 1 inner
1152.3.b.i 8 4.b odd 2 1 inner
1152.3.b.i 8 8.b even 2 1 inner
1152.3.b.i 8 8.d odd 2 1 inner
1152.3.b.i 8 12.b even 2 1 inner
1152.3.b.i 8 24.f even 2 1 inner
1152.3.b.i 8 24.h odd 2 1 inner
2304.3.g.p 4 16.e even 4 1
2304.3.g.p 4 16.f odd 4 1
2304.3.g.p 4 48.i odd 4 1
2304.3.g.p 4 48.k even 4 1
2304.3.g.w 4 16.e even 4 1
2304.3.g.w 4 16.f odd 4 1
2304.3.g.w 4 48.i odd 4 1
2304.3.g.w 4 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 $$S_{3}^{\mathrm{new}}(1152, [\chi])$$:

 $$T_{5}^{2} + 28$$ $$T_{7}^{2} + 56$$ $$T_{17}^{2} - 448$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 28 + T^{2} )^{4}$$
$7$ $$( 56 + T^{2} )^{4}$$
$11$ $$( -32 + T^{2} )^{4}$$
$13$ $$( 16 + T^{2} )^{4}$$
$17$ $$( -448 + T^{2} )^{4}$$
$19$ $$( -896 + T^{2} )^{4}$$
$23$ $$( 512 + T^{2} )^{4}$$
$29$ $$( 28 + T^{2} )^{4}$$
$31$ $$( 504 + T^{2} )^{4}$$
$37$ $$( 784 + T^{2} )^{4}$$
$41$ $$( -4032 + T^{2} )^{4}$$
$43$ $$( -896 + T^{2} )^{4}$$
$47$ $$( 4608 + T^{2} )^{4}$$
$53$ $$( 2268 + T^{2} )^{4}$$
$59$ $$( -10368 + T^{2} )^{4}$$
$61$ $$( 5776 + T^{2} )^{4}$$
$67$ $$( -3584 + T^{2} )^{4}$$
$71$ $$( 8192 + T^{2} )^{4}$$
$73$ $$( -26 + T )^{8}$$
$79$ $$( 16184 + T^{2} )^{4}$$
$83$ $$( -14112 + T^{2} )^{4}$$
$89$ $$( -1792 + T^{2} )^{4}$$
$97$ $$( -18 + T )^{8}$$