# Properties

 Label 3072.1.i.f Level $3072$ Weight $1$ Character orbit 3072.i Analytic conductor $1.533$ Analytic rank $0$ Dimension $4$ Projective image $D_{2}$ CM/RM discs -8, -24, 12 Inner twists $16$

# 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: $$4$$ 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: no (minimal twist has level 384) Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{-2}, \sqrt{3})$$ Artin image $OD_{16}:C_2$ Artin field Galois closure of $$\mathbb{Q}[x]/(x^{16} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{8}^{3} q^{3} -\zeta_{8}^{2} q^{9} +O(q^{10})$$ $$q -\zeta_{8}^{3} q^{3} -\zeta_{8}^{2} q^{9} -2 \zeta_{8} q^{11} -\zeta_{8}^{2} q^{25} -\zeta_{8} q^{27} -2 q^{33} + q^{49} -2 \zeta_{8} q^{59} + 2 \zeta_{8}^{2} q^{73} -\zeta_{8} q^{75} - q^{81} -2 \zeta_{8}^{3} q^{83} + 2 q^{97} + 2 \zeta_{8}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 8q^{33} + 4q^{49} - 4q^{81} + 8q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$1025$$ $$2047$$ $$2053$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{8}^{2}$$

## 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
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
0 −0.707107 + 0.707107i 0 0 0 0 0 1.00000i 0
257.2 0 0.707107 0.707107i 0 0 0 0 0 1.00000i 0
1793.1 0 −0.707107 0.707107i 0 0 0 0 0 1.00000i 0
1793.2 0 0.707107 + 0.707107i 0 0 0 0 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
12.b even 2 1 RM by $$\Q(\sqrt{3})$$
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
16.e even 4 2 inner
16.f odd 4 2 inner
24.f even 2 1 inner
48.i odd 4 2 inner
48.k even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3072.1.i.f 4
3.b odd 2 1 inner 3072.1.i.f 4
4.b odd 2 1 inner 3072.1.i.f 4
8.b even 2 1 inner 3072.1.i.f 4
8.d odd 2 1 CM 3072.1.i.f 4
12.b even 2 1 RM 3072.1.i.f 4
16.e even 4 2 inner 3072.1.i.f 4
16.f odd 4 2 inner 3072.1.i.f 4
24.f even 2 1 inner 3072.1.i.f 4
24.h odd 2 1 CM 3072.1.i.f 4
32.g even 8 1 384.1.h.a 1
32.g even 8 1 384.1.h.b yes 1
32.g even 8 2 768.1.e.c 2
32.h odd 8 1 384.1.h.a 1
32.h odd 8 1 384.1.h.b yes 1
32.h odd 8 2 768.1.e.c 2
48.i odd 4 2 inner 3072.1.i.f 4
48.k even 4 2 inner 3072.1.i.f 4
96.o even 8 1 384.1.h.a 1
96.o even 8 1 384.1.h.b yes 1
96.o even 8 2 768.1.e.c 2
96.p odd 8 1 384.1.h.a 1
96.p odd 8 1 384.1.h.b yes 1
96.p odd 8 2 768.1.e.c 2

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

## Hecke kernels

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

 $$T_{5}$$ $$T_{11}^{4} + 16$$ $$T_{19}$$

## Hecke characteristic polynomials

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