# Properties

 Label 1536.2.j.c Level $1536$ Weight $2$ Character orbit 1536.j 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.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.2650217505$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{8} q^{3} + ( 1 - \zeta_{8}^{2} ) q^{5} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{7} + \zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{8} q^{3} + ( 1 - \zeta_{8}^{2} ) q^{5} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{7} + \zeta_{8}^{2} q^{9} + ( -3 - 3 \zeta_{8}^{2} ) q^{13} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{15} -4 q^{17} -8 \zeta_{8} q^{19} + ( 2 - 2 \zeta_{8}^{2} ) q^{21} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{23} + 3 \zeta_{8}^{2} q^{25} + \zeta_{8}^{3} q^{27} + ( 1 + \zeta_{8}^{2} ) q^{29} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{31} -4 \zeta_{8} q^{35} + ( -3 + 3 \zeta_{8}^{2} ) q^{37} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{39} -4 \zeta_{8}^{2} q^{41} + ( 1 + \zeta_{8}^{2} ) q^{45} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{47} - q^{49} -4 \zeta_{8} q^{51} + ( -5 + 5 \zeta_{8}^{2} ) q^{53} -8 \zeta_{8}^{2} q^{57} + 12 \zeta_{8}^{3} q^{59} + ( 1 + \zeta_{8}^{2} ) q^{61} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{63} -6 q^{65} -4 \zeta_{8} q^{67} + ( 4 - 4 \zeta_{8}^{2} ) q^{69} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{71} + 14 \zeta_{8}^{2} q^{73} + 3 \zeta_{8}^{3} q^{75} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{79} - q^{81} -16 \zeta_{8} q^{83} + ( -4 + 4 \zeta_{8}^{2} ) q^{85} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{87} -6 \zeta_{8}^{2} q^{89} + 12 \zeta_{8}^{3} q^{91} + ( -2 - 2 \zeta_{8}^{2} ) q^{93} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{95} + 16 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{5} + O(q^{10})$$ $$4q + 4q^{5} - 12q^{13} - 16q^{17} + 8q^{21} + 4q^{29} - 12q^{37} + 4q^{45} - 4q^{49} - 20q^{53} + 4q^{61} - 24q^{65} + 16q^{69} - 4q^{81} - 16q^{85} - 8q^{93} + 64q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$511$$ $$517$$ $$1025$$ $$\chi(n)$$ $$1$$ $$-\zeta_{8}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
385.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
0 −0.707107 + 0.707107i 0 1.00000 + 1.00000i 0 2.82843i 0 1.00000i 0
385.2 0 0.707107 0.707107i 0 1.00000 + 1.00000i 0 2.82843i 0 1.00000i 0
1153.1 0 −0.707107 0.707107i 0 1.00000 1.00000i 0 2.82843i 0 1.00000i 0
1153.2 0 0.707107 + 0.707107i 0 1.00000 1.00000i 0 2.82843i 0 1.00000i 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
16.e even 4 1 inner
16.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1536.2.j.c yes 4
3.b odd 2 1 4608.2.k.y 4
4.b odd 2 1 inner 1536.2.j.c yes 4
8.b even 2 1 1536.2.j.b 4
8.d odd 2 1 1536.2.j.b 4
12.b even 2 1 4608.2.k.y 4
16.e even 4 1 1536.2.j.b 4
16.e even 4 1 inner 1536.2.j.c yes 4
16.f odd 4 1 1536.2.j.b 4
16.f odd 4 1 inner 1536.2.j.c yes 4
24.f even 2 1 4608.2.k.bb 4
24.h odd 2 1 4608.2.k.bb 4
32.g even 8 1 3072.2.a.b 2
32.g even 8 1 3072.2.a.h 2
32.g even 8 2 3072.2.d.c 4
32.h odd 8 1 3072.2.a.b 2
32.h odd 8 1 3072.2.a.h 2
32.h odd 8 2 3072.2.d.c 4
48.i odd 4 1 4608.2.k.y 4
48.i odd 4 1 4608.2.k.bb 4
48.k even 4 1 4608.2.k.y 4
48.k even 4 1 4608.2.k.bb 4
96.o even 8 1 9216.2.a.h 2
96.o even 8 1 9216.2.a.i 2
96.p odd 8 1 9216.2.a.h 2
96.p odd 8 1 9216.2.a.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.b 4 8.b even 2 1
1536.2.j.b 4 8.d odd 2 1
1536.2.j.b 4 16.e even 4 1
1536.2.j.b 4 16.f odd 4 1
1536.2.j.c yes 4 1.a even 1 1 trivial
1536.2.j.c yes 4 4.b odd 2 1 inner
1536.2.j.c yes 4 16.e even 4 1 inner
1536.2.j.c yes 4 16.f odd 4 1 inner
3072.2.a.b 2 32.g even 8 1
3072.2.a.b 2 32.h odd 8 1
3072.2.a.h 2 32.g even 8 1
3072.2.a.h 2 32.h odd 8 1
3072.2.d.c 4 32.g even 8 2
3072.2.d.c 4 32.h odd 8 2
4608.2.k.y 4 3.b odd 2 1
4608.2.k.y 4 12.b even 2 1
4608.2.k.y 4 48.i odd 4 1
4608.2.k.y 4 48.k even 4 1
4608.2.k.bb 4 24.f even 2 1
4608.2.k.bb 4 24.h odd 2 1
4608.2.k.bb 4 48.i odd 4 1
4608.2.k.bb 4 48.k even 4 1
9216.2.a.h 2 96.o even 8 1
9216.2.a.h 2 96.p odd 8 1
9216.2.a.i 2 96.o even 8 1
9216.2.a.i 2 96.p odd 8 1

## Hecke kernels

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

 $$T_{5}^{2} - 2 T_{5} + 2$$ $$T_{7}^{2} + 8$$ $$T_{13}^{2} + 6 T_{13} + 18$$ $$T_{19}^{4} + 4096$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 + T^{4}$$
$5$ $$( 2 - 2 T + T^{2} )^{2}$$
$7$ $$( 8 + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$( 18 + 6 T + T^{2} )^{2}$$
$17$ $$( 4 + T )^{4}$$
$19$ $$4096 + T^{4}$$
$23$ $$( 32 + T^{2} )^{2}$$
$29$ $$( 2 - 2 T + T^{2} )^{2}$$
$31$ $$( -8 + T^{2} )^{2}$$
$37$ $$( 18 + 6 T + T^{2} )^{2}$$
$41$ $$( 16 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$( -128 + T^{2} )^{2}$$
$53$ $$( 50 + 10 T + T^{2} )^{2}$$
$59$ $$20736 + T^{4}$$
$61$ $$( 2 - 2 T + T^{2} )^{2}$$
$67$ $$256 + T^{4}$$
$71$ $$( 128 + T^{2} )^{2}$$
$73$ $$( 196 + T^{2} )^{2}$$
$79$ $$( -72 + T^{2} )^{2}$$
$83$ $$65536 + T^{4}$$
$89$ $$( 36 + T^{2} )^{2}$$
$97$ $$( -16 + T )^{4}$$