# Properties

 Label 7728.2.a.ba Level $7728$ Weight $2$ Character orbit 7728.a Self dual yes Analytic conductor $61.708$ Analytic rank $0$ 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: $$0$$ 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 1932) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} -\beta q^{5} + q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} -\beta q^{5} + q^{7} + q^{9} + 3 q^{11} + ( -1 + \beta ) q^{13} + \beta q^{15} + ( -3 + 6 \beta ) q^{17} + ( 1 + 2 \beta ) q^{19} - q^{21} + q^{23} + ( -4 + \beta ) q^{25} - q^{27} + ( -5 + 2 \beta ) q^{29} + 3 q^{31} -3 q^{33} -\beta q^{35} + ( 3 + 4 \beta ) q^{37} + ( 1 - \beta ) q^{39} + ( 1 - 8 \beta ) q^{41} + ( 1 + 5 \beta ) q^{43} -\beta q^{45} + ( 6 + 2 \beta ) q^{47} + q^{49} + ( 3 - 6 \beta ) q^{51} + ( 2 - 7 \beta ) q^{53} -3 \beta q^{55} + ( -1 - 2 \beta ) q^{57} + ( 8 + \beta ) q^{59} + ( 1 + 3 \beta ) q^{61} + q^{63} - q^{65} + ( 8 - 9 \beta ) q^{67} - q^{69} + ( 3 + \beta ) q^{71} + ( 5 - 2 \beta ) q^{73} + ( 4 - \beta ) q^{75} + 3 q^{77} + ( -3 + 6 \beta ) q^{79} + q^{81} + ( 1 + 4 \beta ) q^{83} + ( -6 - 3 \beta ) q^{85} + ( 5 - 2 \beta ) q^{87} + ( 1 - 11 \beta ) q^{89} + ( -1 + \beta ) q^{91} -3 q^{93} + ( -2 - 3 \beta ) q^{95} + ( 7 - 12 \beta ) q^{97} + 3 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - q^{5} + 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - q^{5} + 2q^{7} + 2q^{9} + 6q^{11} - q^{13} + q^{15} + 4q^{19} - 2q^{21} + 2q^{23} - 7q^{25} - 2q^{27} - 8q^{29} + 6q^{31} - 6q^{33} - q^{35} + 10q^{37} + q^{39} - 6q^{41} + 7q^{43} - q^{45} + 14q^{47} + 2q^{49} - 3q^{53} - 3q^{55} - 4q^{57} + 17q^{59} + 5q^{61} + 2q^{63} - 2q^{65} + 7q^{67} - 2q^{69} + 7q^{71} + 8q^{73} + 7q^{75} + 6q^{77} + 2q^{81} + 6q^{83} - 15q^{85} + 8q^{87} - 9q^{89} - q^{91} - 6q^{93} - 7q^{95} + 2q^{97} + 6q^{99} + O(q^{100})$$

## 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
 1.61803 −0.618034
0 −1.00000 0 −1.61803 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 0.618034 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.ba 2
4.b odd 2 1 1932.2.a.g 2
12.b even 2 1 5796.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1932.2.a.g 2 4.b odd 2 1
5796.2.a.l 2 12.b even 2 1
7728.2.a.ba 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7728))$$:

 $$T_{5}^{2} + T_{5} - 1$$ $$T_{11} - 3$$ $$T_{13}^{2} + T_{13} - 1$$ $$T_{17}^{2} - 45$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-1 + T + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$-1 + T + T^{2}$$
$17$ $$-45 + T^{2}$$
$19$ $$-1 - 4 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$11 + 8 T + T^{2}$$
$31$ $$( -3 + T )^{2}$$
$37$ $$5 - 10 T + T^{2}$$
$41$ $$-71 + 6 T + T^{2}$$
$43$ $$-19 - 7 T + T^{2}$$
$47$ $$44 - 14 T + T^{2}$$
$53$ $$-59 + 3 T + T^{2}$$
$59$ $$71 - 17 T + T^{2}$$
$61$ $$-5 - 5 T + T^{2}$$
$67$ $$-89 - 7 T + T^{2}$$
$71$ $$11 - 7 T + T^{2}$$
$73$ $$11 - 8 T + T^{2}$$
$79$ $$-45 + T^{2}$$
$83$ $$-11 - 6 T + T^{2}$$
$89$ $$-131 + 9 T + T^{2}$$
$97$ $$-179 - 2 T + T^{2}$$