# Properties

 Label 3072.2.a.o Level $3072$ Weight $2$ Character orbit 3072.a Self dual yes Analytic conductor $24.530$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# 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: 4.4.4352.1 Defining polynomial: $$x^{4} - 6 x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 48) 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( -1 - \beta_{3} ) q^{5} + ( -1 - \beta_{2} ) q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( -1 - \beta_{3} ) q^{5} + ( -1 - \beta_{2} ) q^{7} + q^{9} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{11} + ( -2 + \beta_{2} + \beta_{3} ) q^{13} + ( -1 - \beta_{3} ) q^{15} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{19} + ( -1 - \beta_{2} ) q^{21} + 2 \beta_{1} q^{23} + ( 1 + 2 \beta_{1} + 2 \beta_{3} ) q^{25} + q^{27} + ( -3 + 2 \beta_{1} + \beta_{3} ) q^{29} + ( -3 + \beta_{2} ) q^{31} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{33} + ( 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{35} + ( -4 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{37} + ( -2 + \beta_{2} + \beta_{3} ) q^{39} + ( 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{41} + ( -\beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{43} + ( -1 - \beta_{3} ) q^{45} -2 \beta_{1} q^{47} + ( 1 - 4 \beta_{1} + 2 \beta_{2} ) q^{49} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{51} + ( -5 - 2 \beta_{1} - \beta_{3} ) q^{53} + ( -4 - 4 \beta_{1} ) q^{55} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{57} -4 \beta_{1} q^{59} + ( -4 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{61} + ( -1 - \beta_{2} ) q^{63} + ( -2 - 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{65} + ( -4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{67} + 2 \beta_{1} q^{69} + ( 2 + 2 \beta_{2} ) q^{71} + ( -2 + 4 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 1 + 2 \beta_{1} + 2 \beta_{3} ) q^{75} + ( -6 + 2 \beta_{1} - 2 \beta_{2} ) q^{77} + ( -3 + 4 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{79} + q^{81} + ( 5 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{83} + ( -6 + 2 \beta_{1} + 2 \beta_{2} ) q^{85} + ( -3 + 2 \beta_{1} + \beta_{3} ) q^{87} + ( -2 + 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{89} + ( -4 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{91} + ( -3 + \beta_{2} ) q^{93} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{95} + ( -6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{97} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} - 4q^{5} - 4q^{7} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{3} - 4q^{5} - 4q^{7} + 4q^{9} - 8q^{13} - 4q^{15} - 4q^{21} + 4q^{25} + 4q^{27} - 12q^{29} - 12q^{31} - 16q^{37} - 8q^{39} - 4q^{45} + 4q^{49} - 20q^{53} - 16q^{55} - 16q^{61} - 4q^{63} - 8q^{65} - 16q^{67} + 8q^{71} - 8q^{73} + 4q^{75} - 24q^{77} - 12q^{79} + 4q^{81} - 24q^{85} - 12q^{87} - 8q^{89} - 16q^{91} - 12q^{93} + 24q^{95} + O(q^{100})$$

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

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

## 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
 2.68554 −1.74912 0.334904 −1.27133
0 1.00000 0 −3.79793 0 −2.15894 0 1.00000 0
1.2 0 1.00000 0 −2.47363 0 2.55765 0 1.00000 0
1.3 0 1.00000 0 0.473626 0 −4.55765 0 1.00000 0
1.4 0 1.00000 0 1.79793 0 0.158942 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3072.2.a.o 4
3.b odd 2 1 9216.2.a.bn 4
4.b odd 2 1 3072.2.a.i 4
8.b even 2 1 3072.2.a.n 4
8.d odd 2 1 3072.2.a.t 4
12.b even 2 1 9216.2.a.bo 4
16.e even 4 2 3072.2.d.i 8
16.f odd 4 2 3072.2.d.f 8
24.f even 2 1 9216.2.a.y 4
24.h odd 2 1 9216.2.a.x 4
32.g even 8 2 192.2.j.a 8
32.g even 8 2 384.2.j.a 8
32.h odd 8 2 48.2.j.a 8
32.h odd 8 2 384.2.j.b 8
96.o even 8 2 144.2.k.b 8
96.o even 8 2 1152.2.k.c 8
96.p odd 8 2 576.2.k.b 8
96.p odd 8 2 1152.2.k.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.j.a 8 32.h odd 8 2
144.2.k.b 8 96.o even 8 2
192.2.j.a 8 32.g even 8 2
384.2.j.a 8 32.g even 8 2
384.2.j.b 8 32.h odd 8 2
576.2.k.b 8 96.p odd 8 2
1152.2.k.c 8 96.o even 8 2
1152.2.k.f 8 96.p odd 8 2
3072.2.a.i 4 4.b odd 2 1
3072.2.a.n 4 8.b even 2 1
3072.2.a.o 4 1.a even 1 1 trivial
3072.2.a.t 4 8.d odd 2 1
3072.2.d.f 8 16.f odd 4 2
3072.2.d.i 8 16.e even 4 2
9216.2.a.x 4 24.h odd 2 1
9216.2.a.y 4 24.f even 2 1
9216.2.a.bn 4 3.b odd 2 1
9216.2.a.bo 4 12.b even 2 1

## Hecke kernels

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

 $$T_{5}^{4} + 4 T_{5}^{3} - 4 T_{5}^{2} - 16 T_{5} + 8$$ $$T_{7}^{4} + 4 T_{7}^{3} - 8 T_{7}^{2} - 24 T_{7} + 4$$ $$T_{11}^{4} - 24 T_{11}^{2} + 32 T_{11} + 32$$ $$T_{19}^{4} - 32 T_{19}^{2} + 64 T_{19} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$8 - 16 T - 4 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$4 - 24 T - 8 T^{2} + 4 T^{3} + T^{4}$$
$11$ $$32 + 32 T - 24 T^{2} + T^{4}$$
$13$ $$4 - 48 T + 4 T^{2} + 8 T^{3} + T^{4}$$
$17$ $$16 - 64 T - 32 T^{2} + T^{4}$$
$19$ $$16 + 64 T - 32 T^{2} + T^{4}$$
$23$ $$( -8 + T^{2} )^{2}$$
$29$ $$-248 - 80 T + 28 T^{2} + 12 T^{3} + T^{4}$$
$31$ $$-28 + 24 T + 40 T^{2} + 12 T^{3} + T^{4}$$
$37$ $$-1052 - 224 T + 52 T^{2} + 16 T^{3} + T^{4}$$
$41$ $$-112 + 192 T - 64 T^{2} + T^{4}$$
$43$ $$-112 + 256 T - 96 T^{2} + T^{4}$$
$47$ $$( -8 + T^{2} )^{2}$$
$53$ $$136 + 272 T + 124 T^{2} + 20 T^{3} + T^{4}$$
$59$ $$( -32 + T^{2} )^{2}$$
$61$ $$-1052 - 224 T + 52 T^{2} + 16 T^{3} + T^{4}$$
$67$ $$256 - 256 T + 16 T^{3} + T^{4}$$
$71$ $$64 + 192 T - 32 T^{2} - 8 T^{3} + T^{4}$$
$73$ $$64 + 64 T - 96 T^{2} + 8 T^{3} + T^{4}$$
$79$ $$-10108 - 2888 T - 168 T^{2} + 12 T^{3} + T^{4}$$
$83$ $$32 + 160 T - 216 T^{2} + T^{4}$$
$89$ $$-1904 - 1632 T - 200 T^{2} + 8 T^{3} + T^{4}$$
$97$ $$512 + 768 T - 224 T^{2} + T^{4}$$