# Properties

 Label 784.4.a.x Level $784$ Weight $4$ Character orbit 784.a Self dual yes Analytic conductor $46.257$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 784.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$46.2574974445$$ 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: $$2^{2}$$ Twist minimal: no (minimal twist has level 392) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 4\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} -\beta q^{5} + 5 q^{9} +O(q^{10})$$ $$q + \beta q^{3} -\beta q^{5} + 5 q^{9} + 4 q^{11} + \beta q^{13} -32 q^{15} -22 \beta q^{17} + 23 \beta q^{19} -120 q^{23} -93 q^{25} -22 \beta q^{27} + 218 q^{29} -26 \beta q^{31} + 4 \beta q^{33} + 130 q^{37} + 32 q^{39} -26 \beta q^{41} -332 q^{43} -5 \beta q^{45} -22 \beta q^{47} -704 q^{51} -498 q^{53} -4 \beta q^{55} + 736 q^{57} -97 \beta q^{59} + 115 \beta q^{61} -32 q^{65} -156 q^{67} -120 \beta q^{69} -240 q^{71} -93 \beta q^{75} -1112 q^{79} -839 q^{81} -5 \beta q^{83} + 704 q^{85} + 218 \beta q^{87} -240 \beta q^{89} -832 q^{93} -736 q^{95} + 262 \beta q^{97} + 20 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 10q^{9} + O(q^{10})$$ $$2q + 10q^{9} + 8q^{11} - 64q^{15} - 240q^{23} - 186q^{25} + 436q^{29} + 260q^{37} + 64q^{39} - 664q^{43} - 1408q^{51} - 996q^{53} + 1472q^{57} - 64q^{65} - 312q^{67} - 480q^{71} - 2224q^{79} - 1678q^{81} + 1408q^{85} - 1664q^{93} - 1472q^{95} + 40q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 −5.65685 0 5.65685 0 0 0 5.00000 0
1.2 0 5.65685 0 −5.65685 0 0 0 5.00000 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
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.4.a.x 2
4.b odd 2 1 392.4.a.g 2
7.b odd 2 1 inner 784.4.a.x 2
28.d even 2 1 392.4.a.g 2
28.f even 6 2 392.4.i.j 4
28.g odd 6 2 392.4.i.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.4.a.g 2 4.b odd 2 1
392.4.a.g 2 28.d even 2 1
392.4.i.j 4 28.f even 6 2
392.4.i.j 4 28.g odd 6 2
784.4.a.x 2 1.a even 1 1 trivial
784.4.a.x 2 7.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(784))$$:

 $$T_{3}^{2} - 32$$ $$T_{5}^{2} - 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-32 + T^{2}$$
$5$ $$-32 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$-32 + T^{2}$$
$17$ $$-15488 + T^{2}$$
$19$ $$-16928 + T^{2}$$
$23$ $$( 120 + T )^{2}$$
$29$ $$( -218 + T )^{2}$$
$31$ $$-21632 + T^{2}$$
$37$ $$( -130 + T )^{2}$$
$41$ $$-21632 + T^{2}$$
$43$ $$( 332 + T )^{2}$$
$47$ $$-15488 + T^{2}$$
$53$ $$( 498 + T )^{2}$$
$59$ $$-301088 + T^{2}$$
$61$ $$-423200 + T^{2}$$
$67$ $$( 156 + T )^{2}$$
$71$ $$( 240 + T )^{2}$$
$73$ $$T^{2}$$
$79$ $$( 1112 + T )^{2}$$
$83$ $$-800 + T^{2}$$
$89$ $$-1843200 + T^{2}$$
$97$ $$-2196608 + T^{2}$$