# Properties

 Label 1152.3.h.b Level $1152$ Weight $3$ Character orbit 1152.h Analytic conductor $31.390$ Analytic rank $0$ Dimension $4$ CM discriminant -4 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} +O(q^{10})$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} -10 \zeta_{8}^{2} q^{13} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{17} -23 q^{25} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{29} -24 \zeta_{8}^{2} q^{37} + ( -31 \zeta_{8} - 31 \zeta_{8}^{3} ) q^{41} -49 q^{49} + ( 73 \zeta_{8} - 73 \zeta_{8}^{3} ) q^{53} -120 \zeta_{8}^{2} q^{61} + ( -10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{65} -96 q^{73} -14 \zeta_{8}^{2} q^{85} + ( -41 \zeta_{8} - 41 \zeta_{8}^{3} ) q^{89} -144 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 92q^{25} - 196q^{49} - 384q^{73} - 576q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i
0 0 0 −1.41421 0 0 0 0 0
449.2 0 0 0 −1.41421 0 0 0 0 0
449.3 0 0 0 1.41421 0 0 0 0 0
449.4 0 0 0 1.41421 0 0 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 CM by $$\Q(\sqrt{-1})$$
3.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.b 4
3.b odd 2 1 inner 1152.3.h.b 4
4.b odd 2 1 CM 1152.3.h.b 4
8.b even 2 1 inner 1152.3.h.b 4
8.d odd 2 1 inner 1152.3.h.b 4
12.b even 2 1 inner 1152.3.h.b 4
16.e even 4 1 2304.3.e.b 2
16.e even 4 1 2304.3.e.d 2
16.f odd 4 1 2304.3.e.b 2
16.f odd 4 1 2304.3.e.d 2
24.f even 2 1 inner 1152.3.h.b 4
24.h odd 2 1 inner 1152.3.h.b 4
48.i odd 4 1 2304.3.e.b 2
48.i odd 4 1 2304.3.e.d 2
48.k even 4 1 2304.3.e.b 2
48.k even 4 1 2304.3.e.d 2

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

 $$T_{5}^{2} - 2$$ $$T_{7}$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -2 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$( 100 + T^{2} )^{2}$$
$17$ $$( 98 + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( -2 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 576 + T^{2} )^{2}$$
$41$ $$( 1922 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$( -10658 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( 14400 + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( 96 + T )^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$( 3362 + T^{2} )^{2}$$
$97$ $$( 144 + T )^{4}$$