# Properties

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

# 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{5} + q^{7} -3 q^{9} +O(q^{10})$$ $$q -\beta q^{5} + q^{7} -3 q^{9} + \beta q^{11} + \beta q^{13} -2 q^{17} + 2 \beta q^{19} -8 q^{23} + 3 q^{25} + 2 \beta q^{29} -8 q^{31} -\beta q^{35} -2 \beta q^{37} + 6 q^{41} + \beta q^{43} + 3 \beta q^{45} -8 q^{47} + q^{49} -4 \beta q^{53} -8 q^{55} -4 \beta q^{59} -\beta q^{61} -3 q^{63} -8 q^{65} -3 \beta q^{67} -8 q^{71} -6 q^{73} + \beta q^{77} + 8 q^{79} + 9 q^{81} -2 \beta q^{83} + 2 \beta q^{85} -6 q^{89} + \beta q^{91} -16 q^{95} + 14 q^{97} -3 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{7} - 6q^{9} + O(q^{10})$$ $$2q + 2q^{7} - 6q^{9} - 4q^{17} - 16q^{23} + 6q^{25} - 16q^{31} + 12q^{41} - 16q^{47} + 2q^{49} - 16q^{55} - 6q^{63} - 16q^{65} - 16q^{71} - 12q^{73} + 16q^{79} + 18q^{81} - 12q^{89} - 32q^{95} + 28q^{97} + O(q^{100})$$

## 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
 1.41421 −1.41421
0 0 0 −2.82843 0 1.00000 0 −3.00000 0
1.2 0 0 0 2.82843 0 1.00000 0 −3.00000 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
$$2$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.a.o 2
4.b odd 2 1 1792.2.a.m 2
8.b even 2 1 inner 1792.2.a.o 2
8.d odd 2 1 1792.2.a.m 2
16.e even 4 2 896.2.b.b 2
16.f odd 4 2 896.2.b.d yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.b.b 2 16.e even 4 2
896.2.b.d yes 2 16.f odd 4 2
1792.2.a.m 2 4.b odd 2 1
1792.2.a.m 2 8.d odd 2 1
1792.2.a.o 2 1.a even 1 1 trivial
1792.2.a.o 2 8.b even 2 1 inner

## Hecke kernels

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

 $$T_{3}$$ $$T_{5}^{2} - 8$$ $$T_{23} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-8 + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-8 + T^{2}$$
$13$ $$-8 + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$-32 + T^{2}$$
$23$ $$( 8 + T )^{2}$$
$29$ $$-32 + T^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$-32 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$-8 + T^{2}$$
$47$ $$( 8 + T )^{2}$$
$53$ $$-128 + T^{2}$$
$59$ $$-128 + T^{2}$$
$61$ $$-8 + T^{2}$$
$67$ $$-72 + T^{2}$$
$71$ $$( 8 + T )^{2}$$
$73$ $$( 6 + T )^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$-32 + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$( -14 + T )^{2}$$