# Properties

 Label 3072.2.a.r Level $3072$ Weight $2$ Character orbit 3072.a Self dual yes Analytic conductor $24.530$ Analytic rank $0$ 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} -\beta_{2} q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} -\beta_{2} q^{7} + q^{9} + ( 2 - \beta_{3} ) q^{11} + 3 \beta_{1} q^{13} + ( \beta_{1} + \beta_{2} ) q^{15} -\beta_{3} q^{17} + ( 4 + \beta_{3} ) q^{19} -\beta_{2} q^{21} -2 \beta_{1} q^{23} + ( 3 + 2 \beta_{3} ) q^{25} + q^{27} + ( \beta_{1} + 3 \beta_{2} ) q^{29} -3 \beta_{2} q^{31} + ( 2 - \beta_{3} ) q^{33} + ( -6 - \beta_{3} ) q^{35} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{37} + 3 \beta_{1} q^{39} + ( 8 - \beta_{3} ) q^{41} -\beta_{3} q^{43} + ( \beta_{1} + \beta_{2} ) q^{45} + 2 \beta_{1} q^{47} - q^{49} -\beta_{3} q^{51} + ( -5 \beta_{1} + \beta_{2} ) q^{53} -4 \beta_{1} q^{55} + ( 4 + \beta_{3} ) q^{57} + 4 \beta_{3} q^{59} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{61} -\beta_{2} q^{63} + ( 6 + 3 \beta_{3} ) q^{65} + ( 8 + 2 \beta_{3} ) q^{67} -2 \beta_{1} q^{69} + ( -8 \beta_{1} + 2 \beta_{2} ) q^{71} + 4 q^{73} + ( 3 + 2 \beta_{3} ) q^{75} + ( 6 \beta_{1} - 2 \beta_{2} ) q^{77} + \beta_{2} q^{79} + q^{81} + ( 2 + \beta_{3} ) q^{83} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{85} + ( \beta_{1} + 3 \beta_{2} ) q^{87} + ( 2 - 2 \beta_{3} ) q^{89} -3 \beta_{3} q^{91} -3 \beta_{2} q^{93} + ( 10 \beta_{1} + 6 \beta_{2} ) q^{95} + ( 8 - 2 \beta_{3} ) q^{97} + ( 2 - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{3} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{3} + 4q^{9} + 8q^{11} + 16q^{19} + 12q^{25} + 4q^{27} + 8q^{33} - 24q^{35} + 32q^{41} - 4q^{49} + 16q^{57} + 24q^{65} + 32q^{67} + 16q^{73} + 12q^{75} + 4q^{81} + 8q^{83} + 8q^{89} + 32q^{97} + 8q^{99} + O(q^{100})$$

## 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
 −1.93185 −0.517638 0.517638 1.93185
0 1.00000 0 −3.86370 0 2.44949 0 1.00000 0
1.2 0 1.00000 0 −1.03528 0 2.44949 0 1.00000 0
1.3 0 1.00000 0 1.03528 0 −2.44949 0 1.00000 0
1.4 0 1.00000 0 3.86370 0 −2.44949 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.r 4
3.b odd 2 1 9216.2.a.bc 4
4.b odd 2 1 3072.2.a.l 4
8.b even 2 1 3072.2.a.l 4
8.d odd 2 1 inner 3072.2.a.r 4
12.b even 2 1 9216.2.a.bi 4
16.e even 4 2 3072.2.d.h 8
16.f odd 4 2 3072.2.d.h 8
24.f even 2 1 9216.2.a.bc 4
24.h odd 2 1 9216.2.a.bi 4
32.g even 8 2 768.2.j.e 8
32.g even 8 2 768.2.j.f yes 8
32.h odd 8 2 768.2.j.e 8
32.h odd 8 2 768.2.j.f yes 8
96.o even 8 2 2304.2.k.e 8
96.o even 8 2 2304.2.k.l 8
96.p odd 8 2 2304.2.k.e 8
96.p odd 8 2 2304.2.k.l 8

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

## Hecke kernels

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

 $$T_{5}^{4} - 16 T_{5}^{2} + 16$$ $$T_{7}^{2} - 6$$ $$T_{11}^{2} - 4 T_{11} - 8$$ $$T_{19}^{2} - 8 T_{19} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$16 - 16 T^{2} + T^{4}$$
$7$ $$( -6 + T^{2} )^{2}$$
$11$ $$( -8 - 4 T + T^{2} )^{2}$$
$13$ $$( -18 + T^{2} )^{2}$$
$17$ $$( -12 + T^{2} )^{2}$$
$19$ $$( 4 - 8 T + T^{2} )^{2}$$
$23$ $$( -8 + T^{2} )^{2}$$
$29$ $$2704 - 112 T^{2} + T^{4}$$
$31$ $$( -54 + T^{2} )^{2}$$
$37$ $$36 - 84 T^{2} + T^{4}$$
$41$ $$( 52 - 16 T + T^{2} )^{2}$$
$43$ $$( -12 + T^{2} )^{2}$$
$47$ $$( -8 + T^{2} )^{2}$$
$53$ $$1936 - 112 T^{2} + T^{4}$$
$59$ $$( -192 + T^{2} )^{2}$$
$61$ $$36 - 84 T^{2} + T^{4}$$
$67$ $$( 16 - 16 T + T^{2} )^{2}$$
$71$ $$10816 - 304 T^{2} + T^{4}$$
$73$ $$( -4 + T )^{4}$$
$79$ $$( -6 + T^{2} )^{2}$$
$83$ $$( -8 - 4 T + T^{2} )^{2}$$
$89$ $$( -44 - 4 T + T^{2} )^{2}$$
$97$ $$( 16 - 16 T + T^{2} )^{2}$$