# Properties

 Label 1152.3.g.d Level $1152$ Weight $3$ Character orbit 1152.g Analytic conductor $31.390$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1152.g (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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{5} -\beta_{5} q^{7} +O(q^{10})$$ $$q -\beta_{3} q^{5} -\beta_{5} q^{7} + \beta_{1} q^{11} + ( -6 + \beta_{7} ) q^{13} + ( -\beta_{3} - \beta_{4} ) q^{17} + ( -\beta_{2} - 2 \beta_{5} ) q^{19} + ( -2 \beta_{1} - \beta_{6} ) q^{23} + ( 11 - 2 \beta_{7} ) q^{25} + ( -\beta_{3} + 2 \beta_{4} ) q^{29} + ( -2 \beta_{2} - \beta_{5} ) q^{31} + ( -3 \beta_{1} - 2 \beta_{6} ) q^{35} + ( -2 - \beta_{7} ) q^{37} + ( 5 \beta_{3} - 3 \beta_{4} ) q^{41} + ( 3 \beta_{2} - 2 \beta_{5} ) q^{43} + ( 6 \beta_{1} - \beta_{6} ) q^{47} + ( -11 + 2 \beta_{7} ) q^{49} + ( -\beta_{3} + 6 \beta_{4} ) q^{53} + ( 6 \beta_{2} + 4 \beta_{5} ) q^{55} + ( 2 \beta_{1} - 2 \beta_{6} ) q^{59} + ( 14 - \beta_{7} ) q^{61} + ( 19 \beta_{3} - 5 \beta_{4} ) q^{65} + ( -6 \beta_{2} + 12 \beta_{5} ) q^{67} -4 \beta_{6} q^{71} + ( -38 + 6 \beta_{7} ) q^{73} + ( -10 \beta_{3} + 6 \beta_{4} ) q^{77} + ( -8 \beta_{2} + 3 \beta_{5} ) q^{79} + ( -\beta_{1} - 4 \beta_{6} ) q^{83} + 40 q^{85} + ( -2 \beta_{3} - 10 \beta_{4} ) q^{89} + ( 13 \beta_{2} + 10 \beta_{5} ) q^{91} + ( -10 \beta_{1} - 5 \beta_{6} ) q^{95} + ( 14 - 4 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 48q^{13} + 88q^{25} - 16q^{37} - 88q^{49} + 112q^{61} - 304q^{73} + 320q^{85} + 112q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$8 \nu^{4} + 4$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{6} - 10 \nu^{2}$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - 6 \nu^{6} + 4 \nu^{5} - 7 \nu^{3} + 18 \nu^{2} - 4 \nu$$$$)/12$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 2 \nu^{6} - 4 \nu^{5} + 7 \nu^{3} + 6 \nu^{2} + 4 \nu$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{7} - 2 \nu^{6} + 4 \nu^{5} + 13 \nu^{3} - 10 \nu^{2} + 44 \nu$$$$)/12$$ $$\beta_{6}$$ $$=$$ $$($$$$2 \nu^{7} - 8 \nu^{5} - 14 \nu^{3} + 8 \nu$$$$)/3$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{7} + 4 \nu^{5} - 13 \nu^{3} + 44 \nu$$$$)/6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$4 \beta_{7} + \beta_{6} + 8 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2}$$$$)/64$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{4} + 3 \beta_{3} - 3 \beta_{2}$$$$)/16$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{7} - 5 \beta_{6} + 8 \beta_{5} + 10 \beta_{4} - 10 \beta_{3} - 2 \beta_{2}$$$$)/64$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{1} - 4$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$4 \beta_{7} - 11 \beta_{6} + 8 \beta_{5} - 22 \beta_{4} + 22 \beta_{3} - 2 \beta_{2}$$$$)/64$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{4} - 15 \beta_{3} - 9 \beta_{2}$$$$)/16$$ $$\nu^{7}$$ $$=$$ $$($$$$-28 \beta_{7} + 13 \beta_{6} + 56 \beta_{5} - 26 \beta_{4} + 26 \beta_{3} - 14 \beta_{2}$$$$)/64$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 −1.28897 + 0.581861i −1.28897 − 0.581861i 1.28897 + 0.581861i 1.28897 − 0.581861i 0.581861 + 1.28897i 0.581861 − 1.28897i −0.581861 + 1.28897i −0.581861 − 1.28897i
0 0 0 −8.11993 0 9.48331i 0 0 0
127.2 0 0 0 −8.11993 0 9.48331i 0 0 0
127.3 0 0 0 −2.46308 0 5.48331i 0 0 0
127.4 0 0 0 −2.46308 0 5.48331i 0 0 0
127.5 0 0 0 2.46308 0 5.48331i 0 0 0
127.6 0 0 0 2.46308 0 5.48331i 0 0 0
127.7 0 0 0 8.11993 0 9.48331i 0 0 0
127.8 0 0 0 8.11993 0 9.48331i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 127.8 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
12.b 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.g.d 8
3.b odd 2 1 inner 1152.3.g.d 8
4.b odd 2 1 inner 1152.3.g.d 8
8.b even 2 1 1152.3.g.e yes 8
8.d odd 2 1 1152.3.g.e yes 8
12.b even 2 1 inner 1152.3.g.d 8
16.e even 4 1 2304.3.b.r 8
16.e even 4 1 2304.3.b.s 8
16.f odd 4 1 2304.3.b.r 8
16.f odd 4 1 2304.3.b.s 8
24.f even 2 1 1152.3.g.e yes 8
24.h odd 2 1 1152.3.g.e yes 8
48.i odd 4 1 2304.3.b.r 8
48.i odd 4 1 2304.3.b.s 8
48.k even 4 1 2304.3.b.r 8
48.k even 4 1 2304.3.b.s 8

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

 $$T_{5}^{4} - 72 T_{5}^{2} + 400$$ $$T_{13}^{2} + 12 T_{13} - 188$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$( 400 - 72 T^{2} + T^{4} )^{2}$$
$7$ $$( 2704 + 120 T^{2} + T^{4} )^{2}$$
$11$ $$( 112 + T^{2} )^{4}$$
$13$ $$( -188 + 12 T + T^{2} )^{4}$$
$17$ $$( 6400 - 288 T^{2} + T^{4} )^{2}$$
$19$ $$( 6400 + 736 T^{2} + T^{4} )^{2}$$
$23$ $$( 4096 + 1920 T^{2} + T^{4} )^{2}$$
$29$ $$( 132496 - 840 T^{2} + T^{4} )^{2}$$
$31$ $$( 71824 + 760 T^{2} + T^{4} )^{2}$$
$37$ $$( -220 + 4 T + T^{2} )^{4}$$
$41$ $$( 2119936 - 3360 T^{2} + T^{4} )^{2}$$
$43$ $$( 30976 + 1248 T^{2} + T^{4} )^{2}$$
$47$ $$( 12390400 + 9088 T^{2} + T^{4} )^{2}$$
$53$ $$( 4787344 - 7176 T^{2} + T^{4} )^{2}$$
$59$ $$( 2560000 + 4992 T^{2} + T^{4} )^{2}$$
$61$ $$( -28 - 28 T + T^{2} )^{4}$$
$67$ $$( 56070144 + 17280 T^{2} + T^{4} )^{2}$$
$71$ $$( 8192 + T^{2} )^{4}$$
$73$ $$( -6620 + 76 T + T^{2} )^{4}$$
$79$ $$( 8179600 + 7736 T^{2} + T^{4} )^{2}$$
$83$ $$( 65286400 + 16608 T^{2} + T^{4} )^{2}$$
$89$ $$( 5017600 - 20608 T^{2} + T^{4} )^{2}$$
$97$ $$( -3388 - 28 T + T^{2} )^{4}$$