# Properties

 Label 1152.3.g.a Level $1152$ Weight $3$ Character orbit 1152.g Analytic conductor $31.390$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 128) 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 $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{5} + ( -4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{7} +O(q^{10})$$ $$q + ( -2 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{5} + ( -4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{7} + ( 6 \zeta_{8} + 10 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{11} + ( -6 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{13} + ( 2 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{17} + ( -2 \zeta_{8} + 18 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{19} + ( -4 \zeta_{8} + 28 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{23} + ( 11 - 16 \zeta_{8} + 16 \zeta_{8}^{3} ) q^{25} + ( -34 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{29} + ( 32 \zeta_{8} + 32 \zeta_{8}^{3} ) q^{31} + ( -8 \zeta_{8} - 24 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{35} + ( 10 + 28 \zeta_{8} - 28 \zeta_{8}^{3} ) q^{37} + ( -2 + 16 \zeta_{8} - 16 \zeta_{8}^{3} ) q^{41} + ( 34 \zeta_{8} - 2 \zeta_{8}^{2} + 34 \zeta_{8}^{3} ) q^{43} + ( 8 \zeta_{8} - 24 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{47} + ( 1 - 32 \zeta_{8} + 32 \zeta_{8}^{3} ) q^{49} + ( 22 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{53} + ( 28 \zeta_{8} + 28 \zeta_{8}^{2} + 28 \zeta_{8}^{3} ) q^{55} + ( 22 \zeta_{8} - 22 \zeta_{8}^{2} + 22 \zeta_{8}^{3} ) q^{59} + ( 74 + 28 \zeta_{8} - 28 \zeta_{8}^{3} ) q^{61} + ( -20 - 16 \zeta_{8} + 16 \zeta_{8}^{3} ) q^{65} + ( 6 \zeta_{8} - 54 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{67} + ( 20 \zeta_{8} - 12 \zeta_{8}^{2} + 20 \zeta_{8}^{3} ) q^{71} + ( 22 - 24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{73} + ( 88 + 64 \zeta_{8} - 64 \zeta_{8}^{3} ) q^{77} + ( 24 \zeta_{8} - 104 \zeta_{8}^{2} + 24 \zeta_{8}^{3} ) q^{79} + ( 74 \zeta_{8} - 10 \zeta_{8}^{2} + 74 \zeta_{8}^{3} ) q^{83} + ( -68 + 24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{85} + ( -54 - 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{89} + ( 40 \zeta_{8} + 56 \zeta_{8}^{2} + 40 \zeta_{8}^{3} ) q^{91} + ( 76 \zeta_{8} - 52 \zeta_{8}^{2} + 76 \zeta_{8}^{3} ) q^{95} + ( -82 + 40 \zeta_{8} - 40 \zeta_{8}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{5} + O(q^{10})$$ $$4q - 8q^{5} - 24q^{13} + 8q^{17} + 44q^{25} - 136q^{29} + 40q^{37} - 8q^{41} + 4q^{49} + 88q^{53} + 296q^{61} - 80q^{65} + 88q^{73} + 352q^{77} - 272q^{85} - 216q^{89} - 328q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i
0 0 0 −7.65685 0 1.65685i 0 0 0
127.2 0 0 0 −7.65685 0 1.65685i 0 0 0
127.3 0 0 0 3.65685 0 9.65685i 0 0 0
127.4 0 0 0 3.65685 0 9.65685i 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.3.g.a 4
3.b odd 2 1 128.3.c.b yes 4
4.b odd 2 1 inner 1152.3.g.a 4
8.b even 2 1 1152.3.g.b 4
8.d odd 2 1 1152.3.g.b 4
12.b even 2 1 128.3.c.b yes 4
16.e even 4 1 2304.3.b.j 4
16.e even 4 1 2304.3.b.p 4
16.f odd 4 1 2304.3.b.j 4
16.f odd 4 1 2304.3.b.p 4
24.f even 2 1 128.3.c.a 4
24.h odd 2 1 128.3.c.a 4
48.i odd 4 1 256.3.d.d 4
48.i odd 4 1 256.3.d.e 4
48.k even 4 1 256.3.d.d 4
48.k even 4 1 256.3.d.e 4

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

 $$T_{5}^{2} + 4 T_{5} - 28$$ $$T_{13}^{2} + 12 T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -28 + 4 T + T^{2} )^{2}$$
$7$ $$256 + 96 T^{2} + T^{4}$$
$11$ $$784 + 344 T^{2} + T^{4}$$
$13$ $$( 4 + 12 T + T^{2} )^{2}$$
$17$ $$( -124 - 4 T + T^{2} )^{2}$$
$19$ $$99856 + 664 T^{2} + T^{4}$$
$23$ $$565504 + 1632 T^{2} + T^{4}$$
$29$ $$( 1124 + 68 T + T^{2} )^{2}$$
$31$ $$( 2048 + T^{2} )^{2}$$
$37$ $$( -1468 - 20 T + T^{2} )^{2}$$
$41$ $$( -508 + 4 T + T^{2} )^{2}$$
$43$ $$5326864 + 4632 T^{2} + T^{4}$$
$47$ $$200704 + 1408 T^{2} + T^{4}$$
$53$ $$( 452 - 44 T + T^{2} )^{2}$$
$59$ $$234256 + 2904 T^{2} + T^{4}$$
$61$ $$( 3908 - 148 T + T^{2} )^{2}$$
$67$ $$8088336 + 5976 T^{2} + T^{4}$$
$71$ $$430336 + 1888 T^{2} + T^{4}$$
$73$ $$( -668 - 44 T + T^{2} )^{2}$$
$79$ $$93392896 + 23936 T^{2} + T^{4}$$
$83$ $$117765904 + 22104 T^{2} + T^{4}$$
$89$ $$( -284 + 108 T + T^{2} )^{2}$$
$97$ $$( 3524 + 164 T + T^{2} )^{2}$$