# Properties

 Label 1152.3.h.e Level $1152$ Weight $3$ Character orbit 1152.h 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.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$31.3897264543$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.40960000.1 Defining polynomial: $$x^{8} + 7 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{18}$$ 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.

 $$f(q)$$ $$=$$ $$q + 3 \beta_{3} q^{5} -\beta_{6} q^{7} +O(q^{10})$$ $$q + 3 \beta_{3} q^{5} -\beta_{6} q^{7} + \beta_{5} q^{11} + 5 \beta_{1} q^{13} + 17 \beta_{2} q^{17} -\beta_{7} q^{19} + \beta_{4} q^{23} -7 q^{25} -11 \beta_{3} q^{29} + \beta_{6} q^{31} -3 \beta_{5} q^{35} + 32 \beta_{1} q^{37} + 9 \beta_{2} q^{41} + 2 \beta_{7} q^{43} -\beta_{4} q^{47} + 111 q^{49} -13 \beta_{3} q^{53} + 6 \beta_{6} q^{55} + 30 \beta_{2} q^{65} + 3 \beta_{7} q^{67} -7 \beta_{4} q^{71} + 96 q^{73} -160 \beta_{3} q^{77} -5 \beta_{6} q^{79} + 7 \beta_{5} q^{83} + 51 \beta_{1} q^{85} + 39 \beta_{2} q^{89} -5 \beta_{7} q^{91} + 6 \beta_{4} q^{95} + 64 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 56q^{25} + 888q^{49} + 768q^{73} + 512q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 7 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{6} + 16 \nu^{2}$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{7} + \nu^{5} + 13 \nu^{3} + 5 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{7} + \nu^{5} - 13 \nu^{3} + 5 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$-8 \nu^{6} - 48 \nu^{2}$$ $$\beta_{5}$$ $$=$$ $$($$$$16 \nu^{4} + 56$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$($$$$-16 \nu^{7} + 4 \nu^{5} - 116 \nu^{3} + 44 \nu$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$($$$$32 \nu^{7} + 8 \nu^{5} + 232 \nu^{3} + 88 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + 2 \beta_{6} - 8 \beta_{3} - 8 \beta_{2}$$$$)/32$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} + 12 \beta_{1}$$$$)/16$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} - 2 \beta_{6} + 16 \beta_{3} - 16 \beta_{2}$$$$)/16$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{5} - 56$$$$)/16$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{7} - 10 \beta_{6} + 88 \beta_{3} + 88 \beta_{2}$$$$)/32$$ $$\nu^{6}$$ $$=$$ $$($$$$-\beta_{4} - 9 \beta_{1}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-13 \beta_{7} + 26 \beta_{6} - 232 \beta_{3} + 232 \beta_{2}$$$$)/32$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 1.14412 − 1.14412i 1.14412 + 1.14412i −0.437016 + 0.437016i −0.437016 − 0.437016i 0.437016 − 0.437016i 0.437016 + 0.437016i −1.14412 + 1.14412i −1.14412 − 1.14412i
0 0 0 −4.24264 0 −12.6491 0 0 0
449.2 0 0 0 −4.24264 0 −12.6491 0 0 0
449.3 0 0 0 −4.24264 0 12.6491 0 0 0
449.4 0 0 0 −4.24264 0 12.6491 0 0 0
449.5 0 0 0 4.24264 0 −12.6491 0 0 0
449.6 0 0 0 4.24264 0 −12.6491 0 0 0
449.7 0 0 0 4.24264 0 12.6491 0 0 0
449.8 0 0 0 4.24264 0 12.6491 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.8 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.h.e 8
3.b odd 2 1 inner 1152.3.h.e 8
4.b odd 2 1 inner 1152.3.h.e 8
8.b even 2 1 inner 1152.3.h.e 8
8.d odd 2 1 inner 1152.3.h.e 8
12.b even 2 1 inner 1152.3.h.e 8
16.e even 4 1 2304.3.e.f 4
16.e even 4 1 2304.3.e.k 4
16.f odd 4 1 2304.3.e.f 4
16.f odd 4 1 2304.3.e.k 4
24.f even 2 1 inner 1152.3.h.e 8
24.h odd 2 1 inner 1152.3.h.e 8
48.i odd 4 1 2304.3.e.f 4
48.i odd 4 1 2304.3.e.k 4
48.k even 4 1 2304.3.e.f 4
48.k even 4 1 2304.3.e.k 4

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

 $$T_{5}^{2} - 18$$ $$T_{7}^{2} - 160$$ $$T_{11}^{2} - 320$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( -18 + T^{2} )^{4}$$
$7$ $$( -160 + T^{2} )^{4}$$
$11$ $$( -320 + T^{2} )^{4}$$
$13$ $$( 100 + T^{2} )^{4}$$
$17$ $$( 578 + T^{2} )^{4}$$
$19$ $$( 640 + T^{2} )^{4}$$
$23$ $$( 320 + T^{2} )^{4}$$
$29$ $$( -242 + T^{2} )^{4}$$
$31$ $$( -160 + T^{2} )^{4}$$
$37$ $$( 4096 + T^{2} )^{4}$$
$41$ $$( 162 + T^{2} )^{4}$$
$43$ $$( 2560 + T^{2} )^{4}$$
$47$ $$( 320 + T^{2} )^{4}$$
$53$ $$( -338 + T^{2} )^{4}$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$( 5760 + T^{2} )^{4}$$
$71$ $$( 15680 + T^{2} )^{4}$$
$73$ $$( -96 + T )^{8}$$
$79$ $$( -4000 + T^{2} )^{4}$$
$83$ $$( -15680 + T^{2} )^{4}$$
$89$ $$( 3042 + T^{2} )^{4}$$
$97$ $$( -64 + T )^{8}$$