# Properties

 Label 3072.2.d.h Level $3072$ Weight $2$ Character orbit 3072.d Analytic conductor $24.530$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3072 = 2^{10} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3072.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5300435009$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 768) 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{24}^{6} q^{3} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{5} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} - q^{9} +O(q^{10})$$ $$q + \zeta_{24}^{6} q^{3} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{5} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{7} - q^{9} + ( 2 - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{11} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{13} + ( 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{15} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{17} + ( 2 - 4 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{19} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{21} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{23} + ( -3 + 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{25} -\zeta_{24}^{6} q^{27} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{29} + ( -3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{31} + ( 2 + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{33} + ( -2 + 4 \zeta_{24}^{4} - 6 \zeta_{24}^{6} ) q^{35} + ( \zeta_{24} - \zeta_{24}^{3} - 5 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{37} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{39} + ( -8 - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{41} + ( 2 - 4 \zeta_{24}^{4} ) q^{43} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{45} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{47} - q^{49} + ( -2 + 4 \zeta_{24}^{4} ) q^{51} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{53} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{55} + ( -4 + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{57} + ( -8 + 16 \zeta_{24}^{4} ) q^{59} + ( \zeta_{24} - \zeta_{24}^{3} - 5 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{61} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{63} + ( 6 - 12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{65} + ( 4 - 8 \zeta_{24}^{4} + 8 \zeta_{24}^{6} ) q^{67} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{69} + ( -10 \zeta_{24} - 10 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{71} -4 q^{73} + ( -4 + 8 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{75} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 8 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{77} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{79} + q^{81} + ( 2 - 4 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{83} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{85} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{87} + ( -2 - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{89} + ( 6 - 12 \zeta_{24}^{4} ) q^{91} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{93} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 16 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{95} + ( 8 + 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{97} + ( -2 + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{9} + O(q^{10})$$ $$8q - 8q^{9} - 24q^{25} + 16q^{33} - 64q^{41} - 8q^{49} - 32q^{57} + 48q^{65} - 32q^{73} + 8q^{81} - 16q^{89} + 64q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times$$.

 $$n$$ $$1025$$ $$2047$$ $$2053$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1537.1
 −0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 − 0.258819i 0.258819 − 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i −0.965926 − 0.258819i −0.258819 − 0.965926i
0 1.00000i 0 3.86370i 0 −2.44949 0 −1.00000 0
1537.2 0 1.00000i 0 1.03528i 0 −2.44949 0 −1.00000 0
1537.3 0 1.00000i 0 1.03528i 0 2.44949 0 −1.00000 0
1537.4 0 1.00000i 0 3.86370i 0 2.44949 0 −1.00000 0
1537.5 0 1.00000i 0 3.86370i 0 2.44949 0 −1.00000 0
1537.6 0 1.00000i 0 1.03528i 0 2.44949 0 −1.00000 0
1537.7 0 1.00000i 0 1.03528i 0 −2.44949 0 −1.00000 0
1537.8 0 1.00000i 0 3.86370i 0 −2.44949 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1537.8 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 3072.2.d.h 8
4.b odd 2 1 inner 3072.2.d.h 8
8.b even 2 1 inner 3072.2.d.h 8
8.d odd 2 1 inner 3072.2.d.h 8
16.e even 4 1 3072.2.a.l 4
16.e even 4 1 3072.2.a.r 4
16.f odd 4 1 3072.2.a.l 4
16.f odd 4 1 3072.2.a.r 4
32.g even 8 2 768.2.j.e 8
32.g even 8 2 768.2.j.f yes 8
32.h odd 8 2 768.2.j.e 8
32.h odd 8 2 768.2.j.f yes 8
48.i odd 4 1 9216.2.a.bc 4
48.i odd 4 1 9216.2.a.bi 4
48.k even 4 1 9216.2.a.bc 4
48.k even 4 1 9216.2.a.bi 4
96.o even 8 2 2304.2.k.e 8
96.o even 8 2 2304.2.k.l 8
96.p odd 8 2 2304.2.k.e 8
96.p odd 8 2 2304.2.k.l 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.e 8 32.g even 8 2
768.2.j.e 8 32.h odd 8 2
768.2.j.f yes 8 32.g even 8 2
768.2.j.f yes 8 32.h odd 8 2
2304.2.k.e 8 96.o even 8 2
2304.2.k.e 8 96.p odd 8 2
2304.2.k.l 8 96.o even 8 2
2304.2.k.l 8 96.p odd 8 2
3072.2.a.l 4 16.e even 4 1
3072.2.a.l 4 16.f odd 4 1
3072.2.a.r 4 16.e even 4 1
3072.2.a.r 4 16.f odd 4 1
3072.2.d.h 8 1.a even 1 1 trivial
3072.2.d.h 8 4.b odd 2 1 inner
3072.2.d.h 8 8.b even 2 1 inner
3072.2.d.h 8 8.d odd 2 1 inner
9216.2.a.bc 4 48.i odd 4 1
9216.2.a.bc 4 48.k even 4 1
9216.2.a.bi 4 48.i odd 4 1
9216.2.a.bi 4 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 $$S_{2}^{\mathrm{new}}(3072, [\chi])$$:

 $$T_{5}^{4} + 16 T_{5}^{2} + 16$$ $$T_{7}^{2} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 1 + T^{2} )^{4}$$
$5$ $$( 16 + 16 T^{2} + T^{4} )^{2}$$
$7$ $$( -6 + T^{2} )^{4}$$
$11$ $$( 64 + 32 T^{2} + T^{4} )^{2}$$
$13$ $$( 18 + T^{2} )^{4}$$
$17$ $$( -12 + T^{2} )^{4}$$
$19$ $$( 16 + 56 T^{2} + T^{4} )^{2}$$
$23$ $$( -8 + T^{2} )^{4}$$
$29$ $$( 2704 + 112 T^{2} + T^{4} )^{2}$$
$31$ $$( -54 + T^{2} )^{4}$$
$37$ $$( 36 + 84 T^{2} + T^{4} )^{2}$$
$41$ $$( 52 + 16 T + T^{2} )^{4}$$
$43$ $$( 12 + T^{2} )^{4}$$
$47$ $$( -8 + T^{2} )^{4}$$
$53$ $$( 1936 + 112 T^{2} + T^{4} )^{2}$$
$59$ $$( 192 + T^{2} )^{4}$$
$61$ $$( 36 + 84 T^{2} + T^{4} )^{2}$$
$67$ $$( 256 + 224 T^{2} + T^{4} )^{2}$$
$71$ $$( 10816 - 304 T^{2} + T^{4} )^{2}$$
$73$ $$( 4 + T )^{8}$$
$79$ $$( -6 + T^{2} )^{4}$$
$83$ $$( 64 + 32 T^{2} + T^{4} )^{2}$$
$89$ $$( -44 + 4 T + T^{2} )^{4}$$
$97$ $$( 16 - 16 T + T^{2} )^{4}$$