# Properties

 Label 1152.3.b.b Level $1152$ Weight $3$ Character orbit 1152.b Analytic conductor $31.390$ Analytic rank $0$ Dimension $2$ CM discriminant -4 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 6 i q^{5} +O(q^{10})$$ $$q + 6 i q^{5} + 24 i q^{13} -16 q^{17} -11 q^{25} -42 i q^{29} + 24 i q^{37} -80 q^{41} + 49 q^{49} + 90 i q^{53} -120 i q^{61} -144 q^{65} -110 q^{73} -96 i q^{85} -160 q^{89} + 130 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 32q^{17} - 22q^{25} - 160q^{41} + 98q^{49} - 288q^{65} - 220q^{73} - 320q^{89} + 260q^{97} + O(q^{100})$$

## 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
 − 1.00000i 1.00000i
0 0 0 6.00000i 0 0 0 0 0
703.2 0 0 0 6.00000i 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 $$\Q(\sqrt{-1})$$
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.b 2
3.b odd 2 1 1152.3.b.d yes 2
4.b odd 2 1 CM 1152.3.b.b 2
8.b even 2 1 inner 1152.3.b.b 2
8.d odd 2 1 inner 1152.3.b.b 2
12.b even 2 1 1152.3.b.d yes 2
16.e even 4 1 2304.3.g.c 1
16.e even 4 1 2304.3.g.d 1
16.f odd 4 1 2304.3.g.c 1
16.f odd 4 1 2304.3.g.d 1
24.f even 2 1 1152.3.b.d yes 2
24.h odd 2 1 1152.3.b.d yes 2
48.i odd 4 1 2304.3.g.b 1
48.i odd 4 1 2304.3.g.e 1
48.k even 4 1 2304.3.g.b 1
48.k even 4 1 2304.3.g.e 1

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

 $$T_{5}^{2} + 36$$ $$T_{7}$$ $$T_{17} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$36 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$576 + T^{2}$$
$17$ $$( 16 + T )^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$1764 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$576 + T^{2}$$
$41$ $$( 80 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$8100 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$14400 + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 110 + T )^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$( 160 + T )^{2}$$
$97$ $$( -130 + T )^{2}$$