# Properties

 Label 3072.2.d.a Level $3072$ Weight $2$ Character orbit 3072.d Analytic conductor $24.530$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{8}^{2} q^{3} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{7} - q^{9} +O(q^{10})$$ $$q + \zeta_{8}^{2} q^{3} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{7} - q^{9} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{13} -6 q^{17} -6 \zeta_{8}^{2} q^{19} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{21} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{23} + 5 q^{25} -\zeta_{8}^{2} q^{27} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{29} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{31} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{37} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{39} + 6 q^{41} + 6 \zeta_{8}^{2} q^{43} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{47} -5 q^{49} -6 \zeta_{8}^{2} q^{51} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{53} + 6 q^{57} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{61} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{63} + 4 \zeta_{8}^{2} q^{67} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{69} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{71} + 5 \zeta_{8}^{2} q^{75} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{79} + q^{81} + 12 \zeta_{8}^{2} q^{83} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{87} -6 q^{89} + 2 \zeta_{8}^{2} q^{91} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{93} -12 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} - 24q^{17} + 20q^{25} + 24q^{41} - 20q^{49} + 24q^{57} + 4q^{81} - 24q^{89} - 48q^{97} + O(q^{100})$$

## 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
 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
0 1.00000i 0 0 0 −1.41421 0 −1.00000 0
1537.2 0 1.00000i 0 0 0 1.41421 0 −1.00000 0
1537.3 0 1.00000i 0 0 0 −1.41421 0 −1.00000 0
1537.4 0 1.00000i 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
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.a 4
4.b odd 2 1 inner 3072.2.d.a 4
8.b even 2 1 inner 3072.2.d.a 4
8.d odd 2 1 inner 3072.2.d.a 4
16.e even 4 1 3072.2.a.a 2
16.e even 4 1 3072.2.a.g 2
16.f odd 4 1 3072.2.a.a 2
16.f odd 4 1 3072.2.a.g 2
32.g even 8 2 768.2.j.b 4
32.g even 8 2 768.2.j.c yes 4
32.h odd 8 2 768.2.j.b 4
32.h odd 8 2 768.2.j.c yes 4
48.i odd 4 1 9216.2.a.n 2
48.i odd 4 1 9216.2.a.o 2
48.k even 4 1 9216.2.a.n 2
48.k even 4 1 9216.2.a.o 2
96.o even 8 2 2304.2.k.b 4
96.o even 8 2 2304.2.k.c 4
96.p odd 8 2 2304.2.k.b 4
96.p odd 8 2 2304.2.k.c 4

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

 $$T_{5}$$ $$T_{7}^{2} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( -2 + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$( 2 + T^{2} )^{2}$$
$17$ $$( 6 + T )^{4}$$
$19$ $$( 36 + T^{2} )^{2}$$
$23$ $$( -72 + T^{2} )^{2}$$
$29$ $$( 72 + T^{2} )^{2}$$
$31$ $$( -2 + T^{2} )^{2}$$
$37$ $$( 50 + T^{2} )^{2}$$
$41$ $$( -6 + T )^{4}$$
$43$ $$( 36 + T^{2} )^{2}$$
$47$ $$( -72 + T^{2} )^{2}$$
$53$ $$( 72 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( 50 + T^{2} )^{2}$$
$67$ $$( 16 + T^{2} )^{2}$$
$71$ $$( -72 + T^{2} )^{2}$$
$73$ $$T^{4}$$
$79$ $$( -2 + T^{2} )^{2}$$
$83$ $$( 144 + T^{2} )^{2}$$
$89$ $$( 6 + T )^{4}$$
$97$ $$( 12 + T )^{4}$$