# Properties

 Label 3072.1.i.d Level $3072$ Weight $1$ Character orbit 3072.i Analytic conductor $1.533$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3072 = 2^{10} \cdot 3$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3072.i (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.53312771881$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1536) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.9216.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{9} +O(q^{10})$$ $$q + q^{3} + q^{9} + ( 1 - i ) q^{11} -2 i q^{17} + ( -1 + i ) q^{19} + i q^{25} + q^{27} + ( 1 - i ) q^{33} + ( -1 - i ) q^{43} + q^{49} -2 i q^{51} + ( -1 + i ) q^{57} + ( -1 + i ) q^{59} + ( -1 + i ) q^{67} + i q^{75} + q^{81} + ( 1 + i ) q^{83} + 2 q^{89} + ( 1 - i ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 2q^{9} + 2q^{11} - 2q^{19} + 2q^{27} + 2q^{33} - 2q^{43} + 2q^{49} - 2q^{57} - 2q^{59} - 2q^{67} + 2q^{81} + 2q^{83} + 4q^{89} + 2q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 − 1.00000i 1.00000i
0 1.00000 0 0 0 0 0 1.00000 0
1793.1 0 1.00000 0 0 0 0 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
48.i odd 4 1 inner
48.k even 4 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3072, [\chi])$$:

 $$T_{5}$$ $$T_{11}^{2} - 2 T_{11} + 2$$ $$T_{19}^{2} + 2 T_{19} + 2$$

## Hecke characteristic polynomials

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