# Properties

 Label 1792.2.a.r Level $1792$ Weight $2$ Character orbit 1792.a Self dual yes Analytic conductor $14.309$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1792 = 2^{8} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1792.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.3091920422$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 896) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{3} + ( -1 - \beta ) q^{5} + q^{7} + ( 3 + 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{3} + ( -1 - \beta ) q^{5} + q^{7} + ( 3 + 2 \beta ) q^{9} + 4 q^{11} + ( 1 + \beta ) q^{13} + ( -6 - 2 \beta ) q^{15} -2 q^{17} + ( -1 - \beta ) q^{19} + ( 1 + \beta ) q^{21} + ( 2 - 2 \beta ) q^{23} + ( 1 + 2 \beta ) q^{25} + ( 10 + 2 \beta ) q^{27} + ( 6 + 2 \beta ) q^{29} + ( 4 + 4 \beta ) q^{33} + ( -1 - \beta ) q^{35} + ( 2 - 2 \beta ) q^{37} + ( 6 + 2 \beta ) q^{39} -2 q^{41} -4 \beta q^{43} + ( -13 - 5 \beta ) q^{45} + ( -4 + 4 \beta ) q^{47} + q^{49} + ( -2 - 2 \beta ) q^{51} + 4 \beta q^{53} + ( -4 - 4 \beta ) q^{55} + ( -6 - 2 \beta ) q^{57} + ( 7 - \beta ) q^{59} + ( -1 - \beta ) q^{61} + ( 3 + 2 \beta ) q^{63} + ( -6 - 2 \beta ) q^{65} + ( 6 + 2 \beta ) q^{67} -8 q^{69} + ( 4 + 4 \beta ) q^{71} + ( 6 - 4 \beta ) q^{73} + ( 11 + 3 \beta ) q^{75} + 4 q^{77} + ( -4 - 4 \beta ) q^{79} + ( 11 + 6 \beta ) q^{81} + ( 7 - \beta ) q^{83} + ( 2 + 2 \beta ) q^{85} + ( 16 + 8 \beta ) q^{87} + 2 q^{89} + ( 1 + \beta ) q^{91} + ( 6 + 2 \beta ) q^{95} + ( 2 - 4 \beta ) q^{97} + ( 12 + 8 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 2q^{5} + 2q^{7} + 6q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 2q^{5} + 2q^{7} + 6q^{9} + 8q^{11} + 2q^{13} - 12q^{15} - 4q^{17} - 2q^{19} + 2q^{21} + 4q^{23} + 2q^{25} + 20q^{27} + 12q^{29} + 8q^{33} - 2q^{35} + 4q^{37} + 12q^{39} - 4q^{41} - 26q^{45} - 8q^{47} + 2q^{49} - 4q^{51} - 8q^{55} - 12q^{57} + 14q^{59} - 2q^{61} + 6q^{63} - 12q^{65} + 12q^{67} - 16q^{69} + 8q^{71} + 12q^{73} + 22q^{75} + 8q^{77} - 8q^{79} + 22q^{81} + 14q^{83} + 4q^{85} + 32q^{87} + 4q^{89} + 2q^{91} + 12q^{95} + 4q^{97} + 24q^{99} + O(q^{100})$$

## 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
 −0.618034 1.61803
0 −1.23607 0 1.23607 0 1.00000 0 −1.47214 0
1.2 0 3.23607 0 −3.23607 0 1.00000 0 7.47214 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
$$2$$ $$1$$
$$7$$ $$-1$$

## 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 1792.2.a.r 2
4.b odd 2 1 1792.2.a.j 2
8.b even 2 1 1792.2.a.l 2
8.d odd 2 1 1792.2.a.t 2
16.e even 4 2 896.2.b.e 4
16.f odd 4 2 896.2.b.g yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.b.e 4 16.e even 4 2
896.2.b.g yes 4 16.f odd 4 2
1792.2.a.j 2 4.b odd 2 1
1792.2.a.l 2 8.b even 2 1
1792.2.a.r 2 1.a even 1 1 trivial
1792.2.a.t 2 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1792))$$:

 $$T_{3}^{2} - 2 T_{3} - 4$$ $$T_{5}^{2} + 2 T_{5} - 4$$ $$T_{23}^{2} - 4 T_{23} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-4 - 2 T + T^{2}$$
$5$ $$-4 + 2 T + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$-4 - 2 T + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$-4 + 2 T + T^{2}$$
$23$ $$-16 - 4 T + T^{2}$$
$29$ $$16 - 12 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$-16 - 4 T + T^{2}$$
$41$ $$( 2 + T )^{2}$$
$43$ $$-80 + T^{2}$$
$47$ $$-64 + 8 T + T^{2}$$
$53$ $$-80 + T^{2}$$
$59$ $$44 - 14 T + T^{2}$$
$61$ $$-4 + 2 T + T^{2}$$
$67$ $$16 - 12 T + T^{2}$$
$71$ $$-64 - 8 T + T^{2}$$
$73$ $$-44 - 12 T + T^{2}$$
$79$ $$-64 + 8 T + T^{2}$$
$83$ $$44 - 14 T + T^{2}$$
$89$ $$( -2 + T )^{2}$$
$97$ $$-76 - 4 T + T^{2}$$