# Properties

 Label 1792.2.a.p 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 56) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta q^{3} -\beta q^{5} + q^{7} - q^{9} +O(q^{10})$$ $$q + \beta q^{3} -\beta q^{5} + q^{7} - q^{9} -2 \beta q^{11} + 3 \beta q^{13} -2 q^{15} -6 q^{17} -3 \beta q^{19} + \beta q^{21} -6 q^{23} -3 q^{25} -4 \beta q^{27} -2 \beta q^{29} + 4 q^{31} -4 q^{33} -\beta q^{35} + 6 \beta q^{37} + 6 q^{39} -6 q^{41} + 6 \beta q^{43} + \beta q^{45} + q^{49} -6 \beta q^{51} -4 \beta q^{53} + 4 q^{55} -6 q^{57} + \beta q^{59} -9 \beta q^{61} - q^{63} -6 q^{65} -6 \beta q^{69} -2 q^{73} -3 \beta q^{75} -2 \beta q^{77} -8 q^{79} -5 q^{81} + 11 \beta q^{83} + 6 \beta q^{85} -4 q^{87} -6 q^{89} + 3 \beta q^{91} + 4 \beta q^{93} + 6 q^{95} -10 q^{97} + 2 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{7} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{7} - 2q^{9} - 4q^{15} - 12q^{17} - 12q^{23} - 6q^{25} + 8q^{31} - 8q^{33} + 12q^{39} - 12q^{41} + 2q^{49} + 8q^{55} - 12q^{57} - 2q^{63} - 12q^{65} - 4q^{73} - 16q^{79} - 10q^{81} - 8q^{87} - 12q^{89} + 12q^{95} - 20q^{97} + O(q^{100})$$

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

## 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.p 2
4.b odd 2 1 1792.2.a.n 2
8.b even 2 1 inner 1792.2.a.p 2
8.d odd 2 1 1792.2.a.n 2
16.e even 4 2 224.2.b.a 2
16.f odd 4 2 56.2.b.a 2
48.i odd 4 2 2016.2.c.a 2
48.k even 4 2 504.2.c.a 2
112.j even 4 2 392.2.b.b 2
112.l odd 4 2 1568.2.b.a 2
112.u odd 12 4 392.2.p.a 4
112.v even 12 4 392.2.p.b 4
112.w even 12 4 1568.2.t.c 4
112.x odd 12 4 1568.2.t.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.b.a 2 16.f odd 4 2
224.2.b.a 2 16.e even 4 2
392.2.b.b 2 112.j even 4 2
392.2.p.a 4 112.u odd 12 4
392.2.p.b 4 112.v even 12 4
504.2.c.a 2 48.k even 4 2
1568.2.b.a 2 112.l odd 4 2
1568.2.t.b 4 112.x odd 12 4
1568.2.t.c 4 112.w even 12 4
1792.2.a.n 2 4.b odd 2 1
1792.2.a.n 2 8.d odd 2 1
1792.2.a.p 2 1.a even 1 1 trivial
1792.2.a.p 2 8.b even 2 1 inner
2016.2.c.a 2 48.i odd 4 2

## Hecke kernels

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

 $$T_{3}^{2} - 2$$ $$T_{5}^{2} - 2$$ $$T_{23} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-2 + T^{2}$$
$5$ $$-2 + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-8 + T^{2}$$
$13$ $$-18 + T^{2}$$
$17$ $$( 6 + T )^{2}$$
$19$ $$-18 + T^{2}$$
$23$ $$( 6 + T )^{2}$$
$29$ $$-8 + T^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$-72 + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$-72 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$-32 + T^{2}$$
$59$ $$-2 + T^{2}$$
$61$ $$-162 + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$( 2 + T )^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$-242 + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$( 10 + T )^{2}$$