# Properties

 Label 7728.2.a.bx Level $7728$ Weight $2$ Character orbit 7728.a Self dual yes Analytic conductor $61.708$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7728.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$61.7083906820$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.837.1 Defining polynomial: $$x^{3} - 6 x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3864) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + q^{3} + \beta_{1} q^{5} - q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + \beta_{1} q^{5} - q^{7} + q^{9} + ( -2 + \beta_{1} - \beta_{2} ) q^{11} + ( -2 \beta_{1} + \beta_{2} ) q^{13} + \beta_{1} q^{15} + ( -\beta_{1} - \beta_{2} ) q^{17} + ( -2 - \beta_{1} + \beta_{2} ) q^{19} - q^{21} + q^{23} + ( -1 + \beta_{2} ) q^{25} + q^{27} + 5 q^{29} + ( -4 + \beta_{1} - \beta_{2} ) q^{31} + ( -2 + \beta_{1} - \beta_{2} ) q^{33} -\beta_{1} q^{35} + ( -3 + 2 \beta_{1} ) q^{37} + ( -2 \beta_{1} + \beta_{2} ) q^{39} + ( 3 + 2 \beta_{2} ) q^{41} + ( 2 - 4 \beta_{1} + \beta_{2} ) q^{43} + \beta_{1} q^{45} + ( -1 - \beta_{1} - \beta_{2} ) q^{47} + q^{49} + ( -\beta_{1} - \beta_{2} ) q^{51} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{53} + ( 3 - 4 \beta_{1} + \beta_{2} ) q^{55} + ( -2 - \beta_{1} + \beta_{2} ) q^{57} + ( -7 - 3 \beta_{2} ) q^{59} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{61} - q^{63} + ( -7 + 2 \beta_{1} - 2 \beta_{2} ) q^{65} + ( -1 + 2 \beta_{1} + 3 \beta_{2} ) q^{67} + q^{69} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{71} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{73} + ( -1 + \beta_{2} ) q^{75} + ( 2 - \beta_{1} + \beta_{2} ) q^{77} + ( -6 - 3 \beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( -2 + 5 \beta_{1} + 3 \beta_{2} ) q^{83} + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{85} + 5 q^{87} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{89} + ( 2 \beta_{1} - \beta_{2} ) q^{91} + ( -4 + \beta_{1} - \beta_{2} ) q^{93} + ( -3 - \beta_{2} ) q^{95} + ( -7 - 4 \beta_{1} + 2 \beta_{2} ) q^{97} + ( -2 + \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{3} - 3q^{7} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{3} - 3q^{7} + 3q^{9} - 6q^{11} - 6q^{19} - 3q^{21} + 3q^{23} - 3q^{25} + 3q^{27} + 15q^{29} - 12q^{31} - 6q^{33} - 9q^{37} + 9q^{41} + 6q^{43} - 3q^{47} + 3q^{49} - 3q^{53} + 9q^{55} - 6q^{57} - 21q^{59} + 3q^{61} - 3q^{63} - 21q^{65} - 3q^{67} + 3q^{69} - 3q^{71} - 3q^{75} + 6q^{77} - 18q^{79} + 3q^{81} - 6q^{83} - 15q^{85} + 15q^{87} + 3q^{89} - 12q^{93} - 9q^{95} - 21q^{97} - 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$

## 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
 −2.36147 −0.167449 2.52892
0 1.00000 0 −2.36147 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −0.167449 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 2.52892 0 −1.00000 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$1$$
$$23$$ $$-1$$

## 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 7728.2.a.bx 3
4.b odd 2 1 3864.2.a.m 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.m 3 4.b odd 2 1
7728.2.a.bx 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7728))$$:

 $$T_{5}^{3} - 6 T_{5} - 1$$ $$T_{11}^{3} + 6 T_{11}^{2} - 3 T_{11} - 20$$ $$T_{13}^{3} - 30 T_{13} - 61$$ $$T_{17}^{3} - 21 T_{17} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$-1 - 6 T + T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$-20 - 3 T + 6 T^{2} + T^{3}$$
$13$ $$-61 - 30 T + T^{3}$$
$17$ $$16 - 21 T + T^{3}$$
$19$ $$-24 - 3 T + 6 T^{2} + T^{3}$$
$23$ $$( -1 + T )^{3}$$
$29$ $$( -5 + T )^{3}$$
$31$ $$6 + 33 T + 12 T^{2} + T^{3}$$
$37$ $$-53 + 3 T + 9 T^{2} + T^{3}$$
$41$ $$237 - 21 T - 9 T^{2} + T^{3}$$
$43$ $$-97 - 84 T - 6 T^{2} + T^{3}$$
$47$ $$-4 - 18 T + 3 T^{2} + T^{3}$$
$53$ $$60 - 39 T + 3 T^{2} + T^{3}$$
$59$ $$-818 + 39 T + 21 T^{2} + T^{3}$$
$61$ $$-50 - 45 T - 3 T^{2} + T^{3}$$
$67$ $$-148 - 147 T + 3 T^{2} + T^{3}$$
$71$ $$50 - 45 T + 3 T^{2} + T^{3}$$
$73$ $$-432 - 189 T + T^{3}$$
$79$ $$12 + 33 T + 18 T^{2} + T^{3}$$
$83$ $$-2388 - 291 T + 6 T^{2} + T^{3}$$
$89$ $$-50 - 45 T - 3 T^{2} + T^{3}$$
$97$ $$-985 + 27 T + 21 T^{2} + T^{3}$$