# Properties

 Label 7728.2.a.bg Level $7728$ Weight $2$ Character orbit 7728.a Self dual yes Analytic conductor $61.708$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 483) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( -1 - \beta ) q^{5} - q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( -1 - \beta ) q^{5} - q^{7} + q^{9} + ( -1 + 2 \beta ) q^{11} + ( -2 - 3 \beta ) q^{13} + ( -1 - \beta ) q^{15} + ( -3 + 4 \beta ) q^{17} + ( 7 - 2 \beta ) q^{19} - q^{21} - q^{23} + ( -3 + 3 \beta ) q^{25} + q^{27} + ( 1 + 6 \beta ) q^{29} + 5 q^{31} + ( -1 + 2 \beta ) q^{33} + ( 1 + \beta ) q^{35} + ( -3 - 4 \beta ) q^{37} + ( -2 - 3 \beta ) q^{39} + ( 5 - 6 \beta ) q^{41} + ( 4 - 7 \beta ) q^{43} + ( -1 - \beta ) q^{45} + ( -4 + 2 \beta ) q^{47} + q^{49} + ( -3 + 4 \beta ) q^{51} + ( -3 + \beta ) q^{53} + ( -1 - 3 \beta ) q^{55} + ( 7 - 2 \beta ) q^{57} + ( 3 - 3 \beta ) q^{59} + ( -6 + 5 \beta ) q^{61} - q^{63} + ( 5 + 8 \beta ) q^{65} + ( 5 - 3 \beta ) q^{67} - q^{69} + ( -10 - \beta ) q^{71} + ( -1 + 4 \beta ) q^{73} + ( -3 + 3 \beta ) q^{75} + ( 1 - 2 \beta ) q^{77} + ( -1 + 8 \beta ) q^{79} + q^{81} + ( -3 + 8 \beta ) q^{83} + ( -1 - 5 \beta ) q^{85} + ( 1 + 6 \beta ) q^{87} + ( -12 + 7 \beta ) q^{89} + ( 2 + 3 \beta ) q^{91} + 5 q^{93} + ( -5 - 3 \beta ) q^{95} + ( -11 + 10 \beta ) q^{97} + ( -1 + 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 3q^{5} - 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 3q^{5} - 2q^{7} + 2q^{9} - 7q^{13} - 3q^{15} - 2q^{17} + 12q^{19} - 2q^{21} - 2q^{23} - 3q^{25} + 2q^{27} + 8q^{29} + 10q^{31} + 3q^{35} - 10q^{37} - 7q^{39} + 4q^{41} + q^{43} - 3q^{45} - 6q^{47} + 2q^{49} - 2q^{51} - 5q^{53} - 5q^{55} + 12q^{57} + 3q^{59} - 7q^{61} - 2q^{63} + 18q^{65} + 7q^{67} - 2q^{69} - 21q^{71} + 2q^{73} - 3q^{75} + 6q^{79} + 2q^{81} + 2q^{83} - 7q^{85} + 8q^{87} - 17q^{89} + 7q^{91} + 10q^{93} - 13q^{95} - 12q^{97} + O(q^{100})$$

## 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
 1.61803 −0.618034
0 1.00000 0 −2.61803 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −0.381966 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.bg 2
4.b odd 2 1 483.2.a.g 2
12.b even 2 1 1449.2.a.f 2
28.d even 2 1 3381.2.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.a.g 2 4.b odd 2 1
1449.2.a.f 2 12.b even 2 1
3381.2.a.u 2 28.d even 2 1
7728.2.a.bg 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7728))$$:

 $$T_{5}^{2} + 3 T_{5} + 1$$ $$T_{11}^{2} - 5$$ $$T_{13}^{2} + 7 T_{13} + 1$$ $$T_{17}^{2} + 2 T_{17} - 19$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$1 + 3 T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-5 + T^{2}$$
$13$ $$1 + 7 T + T^{2}$$
$17$ $$-19 + 2 T + T^{2}$$
$19$ $$31 - 12 T + T^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$-29 - 8 T + T^{2}$$
$31$ $$( -5 + T )^{2}$$
$37$ $$5 + 10 T + T^{2}$$
$41$ $$-41 - 4 T + T^{2}$$
$43$ $$-61 - T + T^{2}$$
$47$ $$4 + 6 T + T^{2}$$
$53$ $$5 + 5 T + T^{2}$$
$59$ $$-9 - 3 T + T^{2}$$
$61$ $$-19 + 7 T + T^{2}$$
$67$ $$1 - 7 T + T^{2}$$
$71$ $$109 + 21 T + T^{2}$$
$73$ $$-19 - 2 T + T^{2}$$
$79$ $$-71 - 6 T + T^{2}$$
$83$ $$-79 - 2 T + T^{2}$$
$89$ $$11 + 17 T + T^{2}$$
$97$ $$-89 + 12 T + T^{2}$$