# Properties

 Label 3072.1.p.c Level $3072$ Weight $1$ Character orbit 3072.p Analytic conductor $1.533$ Analytic rank $0$ Dimension $8$ Projective image $D_{8}$ CM discriminant -3 Inner twists $8$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$3072 = 2^{10} \cdot 3$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3072.p (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.53312771881$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{8})$$ Coefficient field: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{8}$$ Projective field: Galois closure of 8.0.57982058496.8

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{16} q^{3} + ( -\zeta_{16} + \zeta_{16}^{3} ) q^{7} + \zeta_{16}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{16} q^{3} + ( -\zeta_{16} + \zeta_{16}^{3} ) q^{7} + \zeta_{16}^{2} q^{9} + ( \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{13} + ( -\zeta_{16}^{3} - \zeta_{16}^{7} ) q^{19} + ( -\zeta_{16}^{2} + \zeta_{16}^{4} ) q^{21} + \zeta_{16}^{6} q^{25} + \zeta_{16}^{3} q^{27} + ( \zeta_{16} - \zeta_{16}^{7} ) q^{31} + ( 1 + \zeta_{16}^{6} ) q^{37} + ( \zeta_{16}^{5} + \zeta_{16}^{7} ) q^{39} + ( -\zeta_{16} + \zeta_{16}^{5} ) q^{43} + ( \zeta_{16}^{2} - \zeta_{16}^{4} + \zeta_{16}^{6} ) q^{49} + ( 1 - \zeta_{16}^{4} ) q^{57} + ( -1 - \zeta_{16}^{2} ) q^{61} + ( -\zeta_{16}^{3} + \zeta_{16}^{5} ) q^{63} + ( 1 + \zeta_{16}^{4} ) q^{73} + \zeta_{16}^{7} q^{75} + ( -\zeta_{16}^{3} - \zeta_{16}^{5} ) q^{79} + \zeta_{16}^{4} q^{81} + ( -\zeta_{16} - \zeta_{16}^{5} ) q^{91} + ( 1 + \zeta_{16}^{2} ) q^{93} + ( \zeta_{16}^{2} - \zeta_{16}^{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 8q^{37} + 8q^{57} - 8q^{61} + 8q^{73} + 8q^{93} + O(q^{100})$$

## Character values

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

 $$n$$ $$1025$$ $$2047$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$1$$ $$\zeta_{16}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
641.1
 −0.923880 − 0.382683i 0.923880 + 0.382683i −0.923880 + 0.382683i 0.923880 − 0.382683i −0.382683 + 0.923880i 0.382683 − 0.923880i −0.382683 − 0.923880i 0.382683 + 0.923880i
0 −0.923880 0.382683i 0 0 0 0.541196 0.541196i 0 0.707107 + 0.707107i 0
641.2 0 0.923880 + 0.382683i 0 0 0 −0.541196 + 0.541196i 0 0.707107 + 0.707107i 0
1409.1 0 −0.923880 + 0.382683i 0 0 0 0.541196 + 0.541196i 0 0.707107 0.707107i 0
1409.2 0 0.923880 0.382683i 0 0 0 −0.541196 0.541196i 0 0.707107 0.707107i 0
2177.1 0 −0.382683 + 0.923880i 0 0 0 1.30656 1.30656i 0 −0.707107 0.707107i 0
2177.2 0 0.382683 0.923880i 0 0 0 −1.30656 + 1.30656i 0 −0.707107 0.707107i 0
2945.1 0 −0.382683 0.923880i 0 0 0 1.30656 + 1.30656i 0 −0.707107 + 0.707107i 0
2945.2 0 0.382683 + 0.923880i 0 0 0 −1.30656 1.30656i 0 −0.707107 + 0.707107i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2945.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
12.b even 2 1 inner
32.g even 8 1 inner
32.h odd 8 1 inner
96.o even 8 1 inner
96.p odd 8 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{4} + 2 T_{13}^{2} - 4 T_{13} + 2$$ acting on $$S_{1}^{\mathrm{new}}(3072, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$1 + T^{8}$$
$5$ $$T^{8}$$
$7$ $$4 + 12 T^{4} + T^{8}$$
$11$ $$T^{8}$$
$13$ $$( 2 - 4 T + 2 T^{2} + T^{4} )^{2}$$
$17$ $$T^{8}$$
$19$ $$16 + T^{8}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$( 2 - 4 T^{2} + T^{4} )^{2}$$
$37$ $$( 2 - 4 T + 6 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$41$ $$T^{8}$$
$43$ $$16 + T^{8}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$( 2 + 4 T + 6 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$67$ $$T^{8}$$
$71$ $$T^{8}$$
$73$ $$( 2 - 2 T + T^{2} )^{4}$$
$79$ $$( 2 + 4 T^{2} + T^{4} )^{2}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$( -2 + T^{2} )^{4}$$
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