# Properties

 Label 3072.1.e.b Level $3072$ Weight $1$ Character orbit 3072.e Self dual yes Analytic conductor $1.533$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$1.53312771881$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 768) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.18432.2 Artin image: $D_8$ Artin field: Galois closure of 8.0.57982058496.8

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} -\beta q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} -\beta q^{7} + q^{9} + \beta q^{13} -\beta q^{21} + q^{25} + q^{27} + \beta q^{31} -\beta q^{37} + \beta q^{39} + q^{49} + \beta q^{61} -\beta q^{63} -2 q^{67} + q^{75} + \beta q^{79} + q^{81} -2 q^{91} + \beta q^{93} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 2q^{9} + 2q^{25} + 2q^{27} + 2q^{49} - 4q^{67} + 2q^{75} + 2q^{81} - 4q^{91} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1025.1
 1.41421 −1.41421
0 1.00000 0 0 0 −1.41421 0 1.00000 0
1025.2 0 1.00000 0 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
8.d odd 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3072.1.e.b 2
3.b odd 2 1 CM 3072.1.e.b 2
4.b odd 2 1 3072.1.e.a 2
8.b even 2 1 3072.1.e.a 2
8.d odd 2 1 inner 3072.1.e.b 2
12.b even 2 1 3072.1.e.a 2
16.e even 4 2 3072.1.h.a 4
16.f odd 4 2 3072.1.h.a 4
24.f even 2 1 inner 3072.1.e.b 2
24.h odd 2 1 3072.1.e.a 2
32.g even 8 2 768.1.i.a 4
32.g even 8 2 768.1.i.b yes 4
32.h odd 8 2 768.1.i.a 4
32.h odd 8 2 768.1.i.b yes 4
48.i odd 4 2 3072.1.h.a 4
48.k even 4 2 3072.1.h.a 4
96.o even 8 2 768.1.i.a 4
96.o even 8 2 768.1.i.b yes 4
96.p odd 8 2 768.1.i.a 4
96.p odd 8 2 768.1.i.b yes 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{67} + 2$$ acting on $$S_{1}^{\mathrm{new}}(3072, [\chi])$$.

## Hecke characteristic polynomials

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