# Properties

 Label 1536.2.d.d Level $1536$ Weight $2$ Character orbit 1536.d Analytic conductor $12.265$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1536 = 2^{9} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1536.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.2650217505$$ 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 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{8}^{2} q^{3} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{5} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{7} - q^{9} +O(q^{10})$$ $$q -\zeta_{8}^{2} q^{3} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{5} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{7} - q^{9} + 2 \zeta_{8}^{2} q^{11} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{15} + 2 q^{17} -4 \zeta_{8}^{2} q^{19} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{21} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{23} + 3 q^{25} + \zeta_{8}^{2} q^{27} + ( 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{29} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{31} + 2 q^{33} -2 \zeta_{8}^{2} q^{35} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{37} -6 q^{41} + 8 \zeta_{8}^{2} q^{43} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{45} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{47} -5 q^{49} -2 \zeta_{8}^{2} q^{51} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{53} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{55} -4 q^{57} + 12 \zeta_{8}^{2} q^{59} + ( 10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{61} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{63} -8 \zeta_{8}^{2} q^{67} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{69} + ( 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{71} + 8 q^{73} -3 \zeta_{8}^{2} q^{75} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{77} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{79} + q^{81} -6 \zeta_{8}^{2} q^{83} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{85} + ( 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{87} -2 q^{89} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{93} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{95} -14 q^{97} -2 \zeta_{8}^{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} + 8q^{17} + 12q^{25} + 8q^{33} - 24q^{41} - 20q^{49} - 16q^{57} + 32q^{73} + 4q^{81} - 8q^{89} - 56q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
769.1
 −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
769.2 0 1.00000i 0 1.41421i 0 −1.41421 0 −1.00000 0
769.3 0 1.00000i 0 1.41421i 0 −1.41421 0 −1.00000 0
769.4 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
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 1536.2.d.d 4
3.b odd 2 1 4608.2.d.g 4
4.b odd 2 1 inner 1536.2.d.d 4
8.b even 2 1 inner 1536.2.d.d 4
8.d odd 2 1 inner 1536.2.d.d 4
12.b even 2 1 4608.2.d.g 4
16.e even 4 1 1536.2.a.c 2
16.e even 4 1 1536.2.a.j yes 2
16.f odd 4 1 1536.2.a.c 2
16.f odd 4 1 1536.2.a.j yes 2
24.f even 2 1 4608.2.d.g 4
24.h odd 2 1 4608.2.d.g 4
48.i odd 4 1 4608.2.a.h 2
48.i odd 4 1 4608.2.a.j 2
48.k even 4 1 4608.2.a.h 2
48.k even 4 1 4608.2.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.c 2 16.e even 4 1
1536.2.a.c 2 16.f odd 4 1
1536.2.a.j yes 2 16.e even 4 1
1536.2.a.j yes 2 16.f odd 4 1
1536.2.d.d 4 1.a even 1 1 trivial
1536.2.d.d 4 4.b odd 2 1 inner
1536.2.d.d 4 8.b even 2 1 inner
1536.2.d.d 4 8.d odd 2 1 inner
4608.2.a.h 2 48.i odd 4 1
4608.2.a.h 2 48.k even 4 1
4608.2.a.j 2 48.i odd 4 1
4608.2.a.j 2 48.k even 4 1
4608.2.d.g 4 3.b odd 2 1
4608.2.d.g 4 12.b even 2 1
4608.2.d.g 4 24.f even 2 1
4608.2.d.g 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1536, [\chi])$$:

 $$T_{5}^{2} + 2$$ $$T_{7}^{2} - 2$$ $$T_{23}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$( 2 + T^{2} )^{2}$$
$7$ $$( -2 + T^{2} )^{2}$$
$11$ $$( 4 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( -2 + T )^{4}$$
$19$ $$( 16 + T^{2} )^{2}$$
$23$ $$( -8 + T^{2} )^{2}$$
$29$ $$( 98 + T^{2} )^{2}$$
$31$ $$( -50 + T^{2} )^{2}$$
$37$ $$( 72 + T^{2} )^{2}$$
$41$ $$( 6 + T )^{4}$$
$43$ $$( 64 + T^{2} )^{2}$$
$47$ $$( -8 + T^{2} )^{2}$$
$53$ $$( 2 + T^{2} )^{2}$$
$59$ $$( 144 + T^{2} )^{2}$$
$61$ $$( 200 + T^{2} )^{2}$$
$67$ $$( 64 + T^{2} )^{2}$$
$71$ $$( -200 + T^{2} )^{2}$$
$73$ $$( -8 + T )^{4}$$
$79$ $$( -18 + T^{2} )^{2}$$
$83$ $$( 36 + T^{2} )^{2}$$
$89$ $$( 2 + T )^{4}$$
$97$ $$( 14 + T )^{4}$$