# Properties

 Label 3072.2.a.k Level $3072$ Weight $2$ Character orbit 3072.a Self dual yes Analytic conductor $24.530$ Analytic rank $1$ Dimension $4$ 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: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{5})$$ Defining polynomial: $$x^{4} - 6 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 1536) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta_{2} q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + \beta_{2} q^{5} + ( -\beta_{1} - \beta_{2} ) q^{7} + q^{9} + ( -2 + \beta_{3} ) q^{11} + \beta_{1} q^{13} -\beta_{2} q^{15} + ( -2 - \beta_{3} ) q^{17} + ( 2 - \beta_{3} ) q^{19} + ( \beta_{1} + \beta_{2} ) q^{21} + 4 \beta_{1} q^{23} + 5 q^{25} - q^{27} + ( -2 \beta_{1} + \beta_{2} ) q^{29} + ( -5 \beta_{1} - \beta_{2} ) q^{31} + ( 2 - \beta_{3} ) q^{33} + ( -10 - \beta_{3} ) q^{35} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{37} -\beta_{1} q^{39} + ( 2 + \beta_{3} ) q^{41} + ( -6 - \beta_{3} ) q^{43} + \beta_{2} q^{45} + ( 5 + 2 \beta_{3} ) q^{49} + ( 2 + \beta_{3} ) q^{51} + ( 6 \beta_{1} - \beta_{2} ) q^{53} + ( 10 \beta_{1} - 2 \beta_{2} ) q^{55} + ( -2 + \beta_{3} ) q^{57} + 2 \beta_{3} q^{59} + ( -3 \beta_{1} - 2 \beta_{2} ) q^{61} + ( -\beta_{1} - \beta_{2} ) q^{63} + \beta_{3} q^{65} -12 q^{67} -4 \beta_{1} q^{69} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -6 - 2 \beta_{3} ) q^{73} -5 q^{75} -8 \beta_{1} q^{77} + ( -3 \beta_{1} + \beta_{2} ) q^{79} + q^{81} + ( 2 - \beta_{3} ) q^{83} + ( -10 \beta_{1} - 2 \beta_{2} ) q^{85} + ( 2 \beta_{1} - \beta_{2} ) q^{87} -10 q^{89} + ( -2 - \beta_{3} ) q^{91} + ( 5 \beta_{1} + \beta_{2} ) q^{93} + ( -10 \beta_{1} + 2 \beta_{2} ) q^{95} + ( -4 + 2 \beta_{3} ) q^{97} + ( -2 + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} + 4q^{9} + O(q^{10})$$ $$4q - 4q^{3} + 4q^{9} - 8q^{11} - 8q^{17} + 8q^{19} + 20q^{25} - 4q^{27} + 8q^{33} - 40q^{35} + 8q^{41} - 24q^{43} + 20q^{49} + 8q^{51} - 8q^{57} - 48q^{67} - 24q^{73} - 20q^{75} + 4q^{81} + 8q^{83} - 40q^{89} - 8q^{91} - 16q^{97} - 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 6 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 4 \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 8 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2} + 4 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.874032 −2.28825 2.28825 0.874032
0 −1.00000 0 −3.16228 0 1.74806 0 1.00000 0
1.2 0 −1.00000 0 −3.16228 0 4.57649 0 1.00000 0
1.3 0 −1.00000 0 3.16228 0 −4.57649 0 1.00000 0
1.4 0 −1.00000 0 3.16228 0 −1.74806 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.k 4
3.b odd 2 1 9216.2.a.bj 4
4.b odd 2 1 3072.2.a.q 4
8.b even 2 1 3072.2.a.q 4
8.d odd 2 1 inner 3072.2.a.k 4
12.b even 2 1 9216.2.a.bd 4
16.e even 4 2 3072.2.d.g 8
16.f odd 4 2 3072.2.d.g 8
24.f even 2 1 9216.2.a.bj 4
24.h odd 2 1 9216.2.a.bd 4
32.g even 8 2 1536.2.j.g 8
32.g even 8 2 1536.2.j.h yes 8
32.h odd 8 2 1536.2.j.g 8
32.h odd 8 2 1536.2.j.h yes 8
96.o even 8 2 4608.2.k.bf 8
96.o even 8 2 4608.2.k.bg 8
96.p odd 8 2 4608.2.k.bf 8
96.p odd 8 2 4608.2.k.bg 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.g 8 32.g even 8 2
1536.2.j.g 8 32.h odd 8 2
1536.2.j.h yes 8 32.g even 8 2
1536.2.j.h yes 8 32.h odd 8 2
3072.2.a.k 4 1.a even 1 1 trivial
3072.2.a.k 4 8.d odd 2 1 inner
3072.2.a.q 4 4.b odd 2 1
3072.2.a.q 4 8.b even 2 1
3072.2.d.g 8 16.e even 4 2
3072.2.d.g 8 16.f odd 4 2
4608.2.k.bf 8 96.o even 8 2
4608.2.k.bf 8 96.p odd 8 2
4608.2.k.bg 8 96.o even 8 2
4608.2.k.bg 8 96.p odd 8 2
9216.2.a.bd 4 12.b even 2 1
9216.2.a.bd 4 24.h odd 2 1
9216.2.a.bj 4 3.b odd 2 1
9216.2.a.bj 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3072))$$:

 $$T_{5}^{2} - 10$$ $$T_{7}^{4} - 24 T_{7}^{2} + 64$$ $$T_{11}^{2} + 4 T_{11} - 16$$ $$T_{19}^{2} - 4 T_{19} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$( -10 + T^{2} )^{2}$$
$7$ $$64 - 24 T^{2} + T^{4}$$
$11$ $$( -16 + 4 T + T^{2} )^{2}$$
$13$ $$( -2 + T^{2} )^{2}$$
$17$ $$( -16 + 4 T + T^{2} )^{2}$$
$19$ $$( -16 - 4 T + T^{2} )^{2}$$
$23$ $$( -32 + T^{2} )^{2}$$
$29$ $$4 - 36 T^{2} + T^{4}$$
$31$ $$1600 - 120 T^{2} + T^{4}$$
$37$ $$484 - 116 T^{2} + T^{4}$$
$41$ $$( -16 - 4 T + T^{2} )^{2}$$
$43$ $$( 16 + 12 T + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$3844 - 164 T^{2} + T^{4}$$
$59$ $$( -80 + T^{2} )^{2}$$
$61$ $$484 - 116 T^{2} + T^{4}$$
$67$ $$( 12 + T )^{4}$$
$71$ $$1024 - 96 T^{2} + T^{4}$$
$73$ $$( -44 + 12 T + T^{2} )^{2}$$
$79$ $$64 - 56 T^{2} + T^{4}$$
$83$ $$( -16 - 4 T + T^{2} )^{2}$$
$89$ $$( 10 + T )^{4}$$
$97$ $$( -64 + 8 T + T^{2} )^{2}$$