# Properties

 Label 1152.2.f.b Level $1152$ Weight $2$ Character orbit 1152.f Analytic conductor $9.199$ Analytic rank $0$ Dimension $4$ CM discriminant -4 Inner twists $8$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1152 = 2^{7} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1152.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.19876631285$$ 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: $$2^{2}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} +O(q^{10})$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{5} + 6 \zeta_{8}^{2} q^{13} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{17} -3 q^{25} + ( 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{29} + 12 \zeta_{8}^{2} q^{37} + ( -9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{41} + 7 q^{49} + ( 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{53} + 12 \zeta_{8}^{2} q^{61} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{65} -16 q^{73} + 6 \zeta_{8}^{2} q^{85} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{89} + 8 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 12q^{25} + 28q^{49} - 64q^{73} + 32q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
575.1
 −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
575.2 0 0 0 −1.41421 0 0 0 0 0
575.3 0 0 0 1.41421 0 0 0 0 0
575.4 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 1152.2.f.b 4
3.b odd 2 1 inner 1152.2.f.b 4
4.b odd 2 1 CM 1152.2.f.b 4
8.b even 2 1 inner 1152.2.f.b 4
8.d odd 2 1 inner 1152.2.f.b 4
12.b even 2 1 inner 1152.2.f.b 4
16.e even 4 1 2304.2.c.d 2
16.e even 4 1 2304.2.c.f 2
16.f odd 4 1 2304.2.c.d 2
16.f odd 4 1 2304.2.c.f 2
24.f even 2 1 inner 1152.2.f.b 4
24.h odd 2 1 inner 1152.2.f.b 4
48.i odd 4 1 2304.2.c.d 2
48.i odd 4 1 2304.2.c.f 2
48.k even 4 1 2304.2.c.d 2
48.k even 4 1 2304.2.c.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.f.b 4 1.a even 1 1 trivial
1152.2.f.b 4 3.b odd 2 1 inner
1152.2.f.b 4 4.b odd 2 1 CM
1152.2.f.b 4 8.b even 2 1 inner
1152.2.f.b 4 8.d odd 2 1 inner
1152.2.f.b 4 12.b even 2 1 inner
1152.2.f.b 4 24.f even 2 1 inner
1152.2.f.b 4 24.h odd 2 1 inner
2304.2.c.d 2 16.e even 4 1
2304.2.c.d 2 16.f odd 4 1
2304.2.c.d 2 48.i odd 4 1
2304.2.c.d 2 48.k even 4 1
2304.2.c.f 2 16.e even 4 1
2304.2.c.f 2 16.f odd 4 1
2304.2.c.f 2 48.i odd 4 1
2304.2.c.f 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 $$S_{2}^{\mathrm{new}}(1152, [\chi])$$:

 $$T_{5}^{2} - 2$$ $$T_{7}$$ $$T_{23}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -2 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$( 36 + T^{2} )^{2}$$
$17$ $$( 18 + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( -98 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 144 + T^{2} )^{2}$$
$41$ $$( 162 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$( -50 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( 144 + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( 16 + T )^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$( 18 + T^{2} )^{2}$$
$97$ $$( -8 + T )^{4}$$
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