# Properties

 Label 7728.2.a.bc 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: $$\Q(\sqrt{41})$$ Defining polynomial: $$x^{2} - x - 10$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 966) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{41})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + \beta q^{5} - q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + \beta q^{5} - q^{7} + q^{9} + 4 q^{11} + ( 2 + \beta ) q^{13} -\beta q^{15} + 4 q^{17} + ( -2 + 2 \beta ) q^{19} + q^{21} - q^{23} + ( 5 + \beta ) q^{25} - q^{27} + ( 4 - \beta ) q^{29} + 2 q^{31} -4 q^{33} -\beta q^{35} + ( 8 - \beta ) q^{37} + ( -2 - \beta ) q^{39} + ( -4 + \beta ) q^{41} + ( -2 + \beta ) q^{43} + \beta q^{45} + 3 \beta q^{47} + q^{49} -4 q^{51} + ( -2 - 2 \beta ) q^{53} + 4 \beta q^{55} + ( 2 - 2 \beta ) q^{57} + 2 \beta q^{59} -2 q^{61} - q^{63} + ( 10 + 3 \beta ) q^{65} -4 \beta q^{67} + q^{69} + 2 \beta q^{71} + ( -6 + 2 \beta ) q^{73} + ( -5 - \beta ) q^{75} -4 q^{77} -8 q^{79} + q^{81} + ( -6 - 2 \beta ) q^{83} + 4 \beta q^{85} + ( -4 + \beta ) q^{87} + ( -4 - 2 \beta ) q^{89} + ( -2 - \beta ) q^{91} -2 q^{93} + 20 q^{95} + ( 10 - \beta ) q^{97} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + q^{5} - 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + q^{5} - 2q^{7} + 2q^{9} + 8q^{11} + 5q^{13} - q^{15} + 8q^{17} - 2q^{19} + 2q^{21} - 2q^{23} + 11q^{25} - 2q^{27} + 7q^{29} + 4q^{31} - 8q^{33} - q^{35} + 15q^{37} - 5q^{39} - 7q^{41} - 3q^{43} + q^{45} + 3q^{47} + 2q^{49} - 8q^{51} - 6q^{53} + 4q^{55} + 2q^{57} + 2q^{59} - 4q^{61} - 2q^{63} + 23q^{65} - 4q^{67} + 2q^{69} + 2q^{71} - 10q^{73} - 11q^{75} - 8q^{77} - 16q^{79} + 2q^{81} - 14q^{83} + 4q^{85} - 7q^{87} - 10q^{89} - 5q^{91} - 4q^{93} + 40q^{95} + 19q^{97} + 8q^{99} + O(q^{100})$$

## 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
 −2.70156 3.70156
0 −1.00000 0 −2.70156 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 3.70156 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.bc 2
4.b odd 2 1 966.2.a.n 2
12.b even 2 1 2898.2.a.bb 2
28.d even 2 1 6762.2.a.bo 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.n 2 4.b odd 2 1
2898.2.a.bb 2 12.b even 2 1
6762.2.a.bo 2 28.d even 2 1
7728.2.a.bc 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7728))$$:

 $$T_{5}^{2} - T_{5} - 10$$ $$T_{11} - 4$$ $$T_{13}^{2} - 5 T_{13} - 4$$ $$T_{17} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-10 - T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$-4 - 5 T + T^{2}$$
$17$ $$( -4 + T )^{2}$$
$19$ $$-40 + 2 T + T^{2}$$
$23$ $$( 1 + T )^{2}$$
$29$ $$2 - 7 T + T^{2}$$
$31$ $$( -2 + T )^{2}$$
$37$ $$46 - 15 T + T^{2}$$
$41$ $$2 + 7 T + T^{2}$$
$43$ $$-8 + 3 T + T^{2}$$
$47$ $$-90 - 3 T + T^{2}$$
$53$ $$-32 + 6 T + T^{2}$$
$59$ $$-40 - 2 T + T^{2}$$
$61$ $$( 2 + T )^{2}$$
$67$ $$-160 + 4 T + T^{2}$$
$71$ $$-40 - 2 T + T^{2}$$
$73$ $$-16 + 10 T + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$8 + 14 T + T^{2}$$
$89$ $$-16 + 10 T + T^{2}$$
$97$ $$80 - 19 T + T^{2}$$