Properties

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

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: $$4$$ 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.

 $$f(q)$$ $$=$$ $$q + 4 \zeta_{8}^{2} q^{5} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{7} +O(q^{10})$$ $$q + 4 \zeta_{8}^{2} q^{5} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{7} + ( -10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{11} + 20 \zeta_{8}^{2} q^{13} + 10 q^{17} + ( 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{19} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{23} + 9 q^{25} -20 \zeta_{8}^{2} q^{29} + ( 32 \zeta_{8} - 32 \zeta_{8}^{3} ) q^{35} -20 \zeta_{8}^{2} q^{37} + 30 q^{41} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{43} + ( -48 \zeta_{8} - 48 \zeta_{8}^{3} ) q^{47} -79 q^{49} -60 \zeta_{8}^{2} q^{53} + ( -40 \zeta_{8} - 40 \zeta_{8}^{3} ) q^{55} + ( 30 \zeta_{8} - 30 \zeta_{8}^{3} ) q^{59} -28 \zeta_{8}^{2} q^{61} -80 q^{65} + ( 58 \zeta_{8} - 58 \zeta_{8}^{3} ) q^{67} + ( 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{71} -10 q^{73} + 160 \zeta_{8}^{2} q^{77} + ( -80 \zeta_{8} - 80 \zeta_{8}^{3} ) q^{79} + ( -18 \zeta_{8} + 18 \zeta_{8}^{3} ) q^{83} + 40 \zeta_{8}^{2} q^{85} -22 q^{89} + ( 160 \zeta_{8} - 160 \zeta_{8}^{3} ) q^{91} + ( 40 \zeta_{8} + 40 \zeta_{8}^{3} ) q^{95} + 150 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 40q^{17} + 36q^{25} + 120q^{41} - 316q^{49} - 320q^{65} - 40q^{73} - 88q^{89} + 600q^{97} + O(q^{100})$$

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
 −0.707107 + 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
0 0 0 4.00000i 0 11.3137i 0 0 0
703.2 0 0 0 4.00000i 0 11.3137i 0 0 0
703.3 0 0 0 4.00000i 0 11.3137i 0 0 0
703.4 0 0 0 4.00000i 0 11.3137i 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.g 4
3.b odd 2 1 128.3.d.c 4
4.b odd 2 1 inner 1152.3.b.g 4
8.b even 2 1 inner 1152.3.b.g 4
8.d odd 2 1 inner 1152.3.b.g 4
12.b even 2 1 128.3.d.c 4
16.e even 4 1 2304.3.g.h 2
16.e even 4 1 2304.3.g.m 2
16.f odd 4 1 2304.3.g.h 2
16.f odd 4 1 2304.3.g.m 2
24.f even 2 1 128.3.d.c 4
24.h odd 2 1 128.3.d.c 4
48.i odd 4 1 256.3.c.c 2
48.i odd 4 1 256.3.c.f 2
48.k even 4 1 256.3.c.c 2
48.k even 4 1 256.3.c.f 2

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

 $$T_{5}^{2} + 16$$ $$T_{7}^{2} + 128$$ $$T_{17} - 10$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 16 + T^{2} )^{2}$$
$7$ $$( 128 + T^{2} )^{2}$$
$11$ $$( -200 + T^{2} )^{2}$$
$13$ $$( 400 + T^{2} )^{2}$$
$17$ $$( -10 + T )^{4}$$
$19$ $$( -200 + T^{2} )^{2}$$
$23$ $$( 128 + T^{2} )^{2}$$
$29$ $$( 400 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 400 + T^{2} )^{2}$$
$41$ $$( -30 + T )^{4}$$
$43$ $$( -8 + T^{2} )^{2}$$
$47$ $$( 4608 + T^{2} )^{2}$$
$53$ $$( 3600 + T^{2} )^{2}$$
$59$ $$( -1800 + T^{2} )^{2}$$
$61$ $$( 784 + T^{2} )^{2}$$
$67$ $$( -6728 + T^{2} )^{2}$$
$71$ $$( 3200 + T^{2} )^{2}$$
$73$ $$( 10 + T )^{4}$$
$79$ $$( 12800 + T^{2} )^{2}$$
$83$ $$( -648 + T^{2} )^{2}$$
$89$ $$( 22 + T )^{4}$$
$97$ $$( -150 + T )^{4}$$