# Properties

 Label 3072.2.a.f Level $3072$ Weight $2$ Character orbit 3072.a Self dual yes Analytic conductor $24.530$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3072 = 2^{10} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3072.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.5300435009$$ 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) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{3} + 2 \beta q^{5} + 3 \beta q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + 2 \beta q^{5} + 3 \beta q^{7} + q^{9} -4 q^{11} + 3 \beta q^{13} + 2 \beta q^{15} + 6 q^{17} -2 q^{19} + 3 \beta q^{21} + 2 \beta q^{23} + 3 q^{25} + q^{27} -4 \beta q^{29} -3 \beta q^{31} -4 q^{33} + 12 q^{35} + 3 \beta q^{37} + 3 \beta q^{39} -10 q^{41} + 6 q^{43} + 2 \beta q^{45} -2 \beta q^{47} + 11 q^{49} + 6 q^{51} -4 \beta q^{53} -8 \beta q^{55} -2 q^{57} -3 \beta q^{61} + 3 \beta q^{63} + 12 q^{65} -4 q^{67} + 2 \beta q^{69} + 2 \beta q^{71} + 16 q^{73} + 3 q^{75} -12 \beta q^{77} -3 \beta q^{79} + q^{81} -16 q^{83} + 12 \beta q^{85} -4 \beta q^{87} + 14 q^{89} + 18 q^{91} -3 \beta q^{93} -4 \beta q^{95} -4 q^{97} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 2q^{9} - 8q^{11} + 12q^{17} - 4q^{19} + 6q^{25} + 2q^{27} - 8q^{33} + 24q^{35} - 20q^{41} + 12q^{43} + 22q^{49} + 12q^{51} - 4q^{57} + 24q^{65} - 8q^{67} + 32q^{73} + 6q^{75} + 2q^{81} - 32q^{83} + 28q^{89} + 36q^{91} - 8q^{97} - 8q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
0 1.00000 0 −2.82843 0 −4.24264 0 1.00000 0
1.2 0 1.00000 0 2.82843 0 4.24264 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.a.f 2
3.b odd 2 1 9216.2.a.q 2
4.b odd 2 1 3072.2.a.d 2
8.b even 2 1 3072.2.a.d 2
8.d odd 2 1 inner 3072.2.a.f 2
12.b even 2 1 9216.2.a.e 2
16.e even 4 2 3072.2.d.d 4
16.f odd 4 2 3072.2.d.d 4
24.f even 2 1 9216.2.a.q 2
24.h odd 2 1 9216.2.a.e 2
32.g even 8 2 768.2.j.a 4
32.g even 8 2 768.2.j.d yes 4
32.h odd 8 2 768.2.j.a 4
32.h odd 8 2 768.2.j.d yes 4
96.o even 8 2 2304.2.k.a 4
96.o even 8 2 2304.2.k.d 4
96.p odd 8 2 2304.2.k.a 4
96.p odd 8 2 2304.2.k.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
768.2.j.a 4 32.g even 8 2
768.2.j.a 4 32.h odd 8 2
768.2.j.d yes 4 32.g even 8 2
768.2.j.d yes 4 32.h odd 8 2
2304.2.k.a 4 96.o even 8 2
2304.2.k.a 4 96.p odd 8 2
2304.2.k.d 4 96.o even 8 2
2304.2.k.d 4 96.p odd 8 2
3072.2.a.d 2 4.b odd 2 1
3072.2.a.d 2 8.b even 2 1
3072.2.a.f 2 1.a even 1 1 trivial
3072.2.a.f 2 8.d odd 2 1 inner
3072.2.d.d 4 16.e even 4 2
3072.2.d.d 4 16.f odd 4 2
9216.2.a.e 2 12.b even 2 1
9216.2.a.e 2 24.h odd 2 1
9216.2.a.q 2 3.b odd 2 1
9216.2.a.q 2 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3072))$$:

 $$T_{5}^{2} - 8$$ $$T_{7}^{2} - 18$$ $$T_{11} + 4$$ $$T_{19} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-8 + T^{2}$$
$7$ $$-18 + T^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$-18 + T^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$( 2 + T )^{2}$$
$23$ $$-8 + T^{2}$$
$29$ $$-32 + T^{2}$$
$31$ $$-18 + T^{2}$$
$37$ $$-18 + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$( -6 + T )^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$-32 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$-18 + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$-8 + T^{2}$$
$73$ $$( -16 + T )^{2}$$
$79$ $$-18 + T^{2}$$
$83$ $$( 16 + T )^{2}$$
$89$ $$( -14 + T )^{2}$$
$97$ $$( 4 + T )^{2}$$