# Properties

 Label 1792.2.a.k 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: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 448) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{3} + ( 1 - \beta ) q^{5} + q^{7} + ( 1 - 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{3} + ( 1 - \beta ) q^{5} + q^{7} + ( 1 - 2 \beta ) q^{9} + ( 2 - 2 \beta ) q^{11} + ( 5 - \beta ) q^{13} + ( -4 + 2 \beta ) q^{15} + 2 q^{17} + ( 3 + \beta ) q^{19} + ( -1 + \beta ) q^{21} + 2 \beta q^{23} + ( -1 - 2 \beta ) q^{25} -4 q^{27} + ( 2 + 2 \beta ) q^{29} -4 q^{31} + ( -8 + 4 \beta ) q^{33} + ( 1 - \beta ) q^{35} + ( 2 + 2 \beta ) q^{37} + ( -8 + 6 \beta ) q^{39} + 2 q^{41} + ( 2 - 2 \beta ) q^{43} + ( 7 - 3 \beta ) q^{45} + ( 4 + 4 \beta ) q^{47} + q^{49} + ( -2 + 2 \beta ) q^{51} + 12 q^{53} + ( 8 - 4 \beta ) q^{55} + 2 \beta q^{57} + ( -1 + 5 \beta ) q^{59} + ( 1 + 7 \beta ) q^{61} + ( 1 - 2 \beta ) q^{63} + ( 8 - 6 \beta ) q^{65} -8 q^{67} + ( 6 - 2 \beta ) q^{69} + ( -4 - 4 \beta ) q^{71} + ( -6 + 4 \beta ) q^{73} + ( -5 + \beta ) q^{75} + ( 2 - 2 \beta ) q^{77} + ( 4 - 4 \beta ) q^{79} + ( 1 + 2 \beta ) q^{81} + ( -3 - 5 \beta ) q^{83} + ( 2 - 2 \beta ) q^{85} + 4 q^{87} + ( 2 + 8 \beta ) q^{89} + ( 5 - \beta ) q^{91} + ( 4 - 4 \beta ) q^{93} -2 \beta q^{95} + ( 2 - 4 \beta ) q^{97} + ( 14 - 6 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + O(q^{10})$$ $$2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{11} + 10 q^{13} - 8 q^{15} + 4 q^{17} + 6 q^{19} - 2 q^{21} - 2 q^{25} - 8 q^{27} + 4 q^{29} - 8 q^{31} - 16 q^{33} + 2 q^{35} + 4 q^{37} - 16 q^{39} + 4 q^{41} + 4 q^{43} + 14 q^{45} + 8 q^{47} + 2 q^{49} - 4 q^{51} + 24 q^{53} + 16 q^{55} - 2 q^{59} + 2 q^{61} + 2 q^{63} + 16 q^{65} - 16 q^{67} + 12 q^{69} - 8 q^{71} - 12 q^{73} - 10 q^{75} + 4 q^{77} + 8 q^{79} + 2 q^{81} - 6 q^{83} + 4 q^{85} + 8 q^{87} + 4 q^{89} + 10 q^{91} + 8 q^{93} + 4 q^{97} + 28 q^{99} + O(q^{100})$$

## 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
 −1.73205 1.73205
0 −2.73205 0 2.73205 0 1.00000 0 4.46410 0
1.2 0 0.732051 0 −0.732051 0 1.00000 0 −2.46410 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.k 2
4.b odd 2 1 1792.2.a.s 2
8.b even 2 1 1792.2.a.q 2
8.d odd 2 1 1792.2.a.i 2
16.e even 4 2 448.2.b.c 4
16.f odd 4 2 448.2.b.d yes 4
48.i odd 4 2 4032.2.c.k 4
48.k even 4 2 4032.2.c.n 4
112.j even 4 2 3136.2.b.h 4
112.l odd 4 2 3136.2.b.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.b.c 4 16.e even 4 2
448.2.b.d yes 4 16.f odd 4 2
1792.2.a.i 2 8.d odd 2 1
1792.2.a.k 2 1.a even 1 1 trivial
1792.2.a.q 2 8.b even 2 1
1792.2.a.s 2 4.b odd 2 1
3136.2.b.g 4 112.l odd 4 2
3136.2.b.h 4 112.j even 4 2
4032.2.c.k 4 48.i odd 4 2
4032.2.c.n 4 48.k even 4 2

## Hecke kernels

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

 $$T_{3}^{2} + 2 T_{3} - 2$$ $$T_{5}^{2} - 2 T_{5} - 2$$ $$T_{23}^{2} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-2 + 2 T + T^{2}$$
$5$ $$-2 - 2 T + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-8 - 4 T + T^{2}$$
$13$ $$22 - 10 T + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$6 - 6 T + T^{2}$$
$23$ $$-12 + T^{2}$$
$29$ $$-8 - 4 T + T^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$-8 - 4 T + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$-8 - 4 T + T^{2}$$
$47$ $$-32 - 8 T + T^{2}$$
$53$ $$( -12 + T )^{2}$$
$59$ $$-74 + 2 T + T^{2}$$
$61$ $$-146 - 2 T + T^{2}$$
$67$ $$( 8 + T )^{2}$$
$71$ $$-32 + 8 T + T^{2}$$
$73$ $$-12 + 12 T + T^{2}$$
$79$ $$-32 - 8 T + T^{2}$$
$83$ $$-66 + 6 T + T^{2}$$
$89$ $$-188 - 4 T + T^{2}$$
$97$ $$-44 - 4 T + T^{2}$$