# Properties

 Label 3072.2.a.s Level $3072$ Weight $2$ Character orbit 3072.a Self dual yes Analytic conductor $24.530$ Analytic rank $0$ 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{16})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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.

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

## 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
 0.765367 1.84776 −0.765367 −1.84776
0 1.00000 0 −4.02734 0 4.61313 0 1.00000 0
1.2 0 1.00000 0 0.331821 0 3.08239 0 1.00000 0
1.3 0 1.00000 0 1.19891 0 −0.613126 0 1.00000 0
1.4 0 1.00000 0 2.49661 0 0.917608 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.s 4
3.b odd 2 1 9216.2.a.bm 4
4.b odd 2 1 3072.2.a.j 4
8.b even 2 1 3072.2.a.m 4
8.d odd 2 1 3072.2.a.p 4
12.b even 2 1 9216.2.a.ba 4
16.e even 4 2 3072.2.d.e 8
16.f odd 4 2 3072.2.d.j 8
24.f even 2 1 9216.2.a.z 4
24.h odd 2 1 9216.2.a.bl 4
32.g even 8 2 1536.2.j.e 8
32.g even 8 2 1536.2.j.j yes 8
32.h odd 8 2 1536.2.j.f yes 8
32.h odd 8 2 1536.2.j.i yes 8
96.o even 8 2 4608.2.k.bc 8
96.o even 8 2 4608.2.k.bj 8
96.p odd 8 2 4608.2.k.be 8
96.p odd 8 2 4608.2.k.bh 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.j.e 8 32.g even 8 2
1536.2.j.f yes 8 32.h odd 8 2
1536.2.j.i yes 8 32.h odd 8 2
1536.2.j.j yes 8 32.g even 8 2
3072.2.a.j 4 4.b odd 2 1
3072.2.a.m 4 8.b even 2 1
3072.2.a.p 4 8.d odd 2 1
3072.2.a.s 4 1.a even 1 1 trivial
3072.2.d.e 8 16.e even 4 2
3072.2.d.j 8 16.f odd 4 2
4608.2.k.bc 8 96.o even 8 2
4608.2.k.be 8 96.p odd 8 2
4608.2.k.bh 8 96.p odd 8 2
4608.2.k.bj 8 96.o even 8 2
9216.2.a.z 4 24.f even 2 1
9216.2.a.ba 4 12.b even 2 1
9216.2.a.bl 4 24.h odd 2 1
9216.2.a.bm 4 3.b odd 2 1

## Hecke kernels

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

 $$T_{5}^{4} - 12 T_{5}^{2} + 16 T_{5} - 4$$ $$T_{7}^{4} - 8 T_{7}^{3} + 16 T_{7}^{2} - 8$$ $$T_{11}^{4} - 16 T_{11}^{2} + 32$$ $$T_{19}^{4} - 32 T_{19}^{2} - 64 T_{19} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$-4 + 16 T - 12 T^{2} + T^{4}$$
$7$ $$-8 + 16 T^{2} - 8 T^{3} + T^{4}$$
$11$ $$32 - 16 T^{2} + T^{4}$$
$13$ $$-188 + 176 T - 12 T^{2} - 8 T^{3} + T^{4}$$
$17$ $$-32 - 64 T - 32 T^{2} + T^{4}$$
$19$ $$-32 - 64 T - 32 T^{2} + T^{4}$$
$23$ $$( -4 + T )^{4}$$
$29$ $$-68 + 112 T - 44 T^{2} + T^{4}$$
$31$ $$248 + 128 T - 16 T^{2} - 8 T^{3} + T^{4}$$
$37$ $$388 + 112 T - 44 T^{2} - 8 T^{3} + T^{4}$$
$41$ $$992 + 64 T - 96 T^{2} + T^{4}$$
$43$ $$992 + 64 T - 96 T^{2} + T^{4}$$
$47$ $$( -16 - 8 T + T^{2} )^{2}$$
$53$ $$188 + 272 T - 108 T^{2} + T^{4}$$
$59$ $$-4352 - 1280 T - 32 T^{2} + 16 T^{3} + T^{4}$$
$61$ $$4 - 176 T + 148 T^{2} - 24 T^{3} + T^{4}$$
$67$ $$-7936 + 2304 T - 96 T^{2} - 16 T^{3} + T^{4}$$
$71$ $$-2176 + 768 T - 16 T^{3} + T^{4}$$
$73$ $$1552 - 32 T - 120 T^{2} + 8 T^{3} + T^{4}$$
$79$ $$-7688 + 1344 T + 80 T^{2} - 24 T^{3} + T^{4}$$
$83$ $$544 - 128 T - 80 T^{2} + T^{4}$$
$89$ $$272 + 288 T - 168 T^{2} + 8 T^{3} + T^{4}$$
$97$ $$-256 - 256 T - 32 T^{2} + 16 T^{3} + T^{4}$$