# Properties

 Label 1536.1.e.a Level $1536$ Weight $1$ Character orbit 1536.e Analytic conductor $0.767$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -24 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1536 = 2^{9} \cdot 3$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1536.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.766563859404$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.4608.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{8}^{2} q^{3} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{7} - q^{9} +O(q^{10})$$ $$q -\zeta_{8}^{2} q^{3} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{7} - q^{9} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{15} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{21} - q^{25} + \zeta_{8}^{2} q^{27} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{29} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{31} + 2 \zeta_{8}^{2} q^{35} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{45} + q^{49} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{53} -2 \zeta_{8}^{2} q^{59} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{63} + \zeta_{8}^{2} q^{75} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{79} + q^{81} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{87} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{93} -2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} - 4q^{25} + 4q^{49} + 4q^{81} - 8q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times$$.

 $$n$$ $$511$$ $$517$$ $$1025$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1025.1
 0.707107 + 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 1.00000i 0 1.41421i 0 −1.41421 0 −1.00000 0
1025.2 0 1.00000i 0 1.41421i 0 1.41421 0 −1.00000 0
1025.3 0 1.00000i 0 1.41421i 0 1.41421 0 −1.00000 0
1025.4 0 1.00000i 0 1.41421i 0 −1.41421 0 −1.00000 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
4.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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1536.1.e.a 4
3.b odd 2 1 inner 1536.1.e.a 4
4.b odd 2 1 inner 1536.1.e.a 4
8.b even 2 1 inner 1536.1.e.a 4
8.d odd 2 1 inner 1536.1.e.a 4
12.b even 2 1 inner 1536.1.e.a 4
16.e even 4 1 1536.1.h.a 2
16.e even 4 1 1536.1.h.b 2
16.f odd 4 1 1536.1.h.a 2
16.f odd 4 1 1536.1.h.b 2
24.f even 2 1 inner 1536.1.e.a 4
24.h odd 2 1 CM 1536.1.e.a 4
32.g even 8 2 3072.1.i.e 4
32.g even 8 2 3072.1.i.h 4
32.h odd 8 2 3072.1.i.e 4
32.h odd 8 2 3072.1.i.h 4
48.i odd 4 1 1536.1.h.a 2
48.i odd 4 1 1536.1.h.b 2
48.k even 4 1 1536.1.h.a 2
48.k even 4 1 1536.1.h.b 2
96.o even 8 2 3072.1.i.e 4
96.o even 8 2 3072.1.i.h 4
96.p odd 8 2 3072.1.i.e 4
96.p odd 8 2 3072.1.i.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.1.e.a 4 1.a even 1 1 trivial
1536.1.e.a 4 3.b odd 2 1 inner
1536.1.e.a 4 4.b odd 2 1 inner
1536.1.e.a 4 8.b even 2 1 inner
1536.1.e.a 4 8.d odd 2 1 inner
1536.1.e.a 4 12.b even 2 1 inner
1536.1.e.a 4 24.f even 2 1 inner
1536.1.e.a 4 24.h odd 2 1 CM
1536.1.h.a 2 16.e even 4 1
1536.1.h.a 2 16.f odd 4 1
1536.1.h.a 2 48.i odd 4 1
1536.1.h.a 2 48.k even 4 1
1536.1.h.b 2 16.e even 4 1
1536.1.h.b 2 16.f odd 4 1
1536.1.h.b 2 48.i odd 4 1
1536.1.h.b 2 48.k even 4 1
3072.1.i.e 4 32.g even 8 2
3072.1.i.e 4 32.h odd 8 2
3072.1.i.e 4 96.o even 8 2
3072.1.i.e 4 96.p odd 8 2
3072.1.i.h 4 32.g even 8 2
3072.1.i.h 4 32.h odd 8 2
3072.1.i.h 4 96.o even 8 2
3072.1.i.h 4 96.p odd 8 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 2$$ acting on $$S_{1}^{\mathrm{new}}(1536, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$( 2 + T^{2} )^{2}$$
$7$ $$( -2 + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( 2 + T^{2} )^{2}$$
$31$ $$( -2 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$( 2 + T^{2} )^{2}$$
$59$ $$( 4 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$( -2 + T^{2} )^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$( 2 + T )^{4}$$