# Properties

 Label 784.2.a.k Level $784$ Weight $2$ Character orbit 784.a Self dual yes Analytic conductor $6.260$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$6.26027151847$$ 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 392) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} -2 \beta q^{5} - q^{9} +O(q^{10})$$ $$q + \beta q^{3} -2 \beta q^{5} - q^{9} -6 q^{11} + 4 \beta q^{13} -4 q^{15} -\beta q^{17} -3 \beta q^{19} -4 q^{23} + 3 q^{25} -4 \beta q^{27} -6 q^{29} + 2 \beta q^{31} -6 \beta q^{33} + 2 q^{37} + 8 q^{39} + \beta q^{41} -10 q^{43} + 2 \beta q^{45} -2 \beta q^{47} -2 q^{51} -2 q^{53} + 12 \beta q^{55} -6 q^{57} + \beta q^{59} + 6 \beta q^{61} -16 q^{65} -4 q^{67} -4 \beta q^{69} + 12 q^{71} + 7 \beta q^{73} + 3 \beta q^{75} + 4 q^{79} -5 q^{81} + \beta q^{83} + 4 q^{85} -6 \beta q^{87} + 3 \beta q^{89} + 4 q^{93} + 12 q^{95} -9 \beta q^{97} + 6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} - 12q^{11} - 8q^{15} - 8q^{23} + 6q^{25} - 12q^{29} + 4q^{37} + 16q^{39} - 20q^{43} - 4q^{51} - 4q^{53} - 12q^{57} - 32q^{65} - 8q^{67} + 24q^{71} + 8q^{79} - 10q^{81} + 8q^{85} + 8q^{93} + 24q^{95} + 12q^{99} + O(q^{100})$$

## 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 −1.41421 0 2.82843 0 0 0 −1.00000 0
1.2 0 1.41421 0 −2.82843 0 0 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
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 784.2.a.k 2
3.b odd 2 1 7056.2.a.ct 2
4.b odd 2 1 392.2.a.g 2
7.b odd 2 1 inner 784.2.a.k 2
7.c even 3 2 784.2.i.n 4
7.d odd 6 2 784.2.i.n 4
8.b even 2 1 3136.2.a.bp 2
8.d odd 2 1 3136.2.a.bk 2
12.b even 2 1 3528.2.a.be 2
20.d odd 2 1 9800.2.a.bv 2
21.c even 2 1 7056.2.a.ct 2
28.d even 2 1 392.2.a.g 2
28.f even 6 2 392.2.i.h 4
28.g odd 6 2 392.2.i.h 4
56.e even 2 1 3136.2.a.bk 2
56.h odd 2 1 3136.2.a.bp 2
84.h odd 2 1 3528.2.a.be 2
84.j odd 6 2 3528.2.s.bj 4
84.n even 6 2 3528.2.s.bj 4
140.c even 2 1 9800.2.a.bv 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.a.g 2 4.b odd 2 1
392.2.a.g 2 28.d even 2 1
392.2.i.h 4 28.f even 6 2
392.2.i.h 4 28.g odd 6 2
784.2.a.k 2 1.a even 1 1 trivial
784.2.a.k 2 7.b odd 2 1 inner
784.2.i.n 4 7.c even 3 2
784.2.i.n 4 7.d odd 6 2
3136.2.a.bk 2 8.d odd 2 1
3136.2.a.bk 2 56.e even 2 1
3136.2.a.bp 2 8.b even 2 1
3136.2.a.bp 2 56.h odd 2 1
3528.2.a.be 2 12.b even 2 1
3528.2.a.be 2 84.h odd 2 1
3528.2.s.bj 4 84.j odd 6 2
3528.2.s.bj 4 84.n even 6 2
7056.2.a.ct 2 3.b odd 2 1
7056.2.a.ct 2 21.c even 2 1
9800.2.a.bv 2 20.d odd 2 1
9800.2.a.bv 2 140.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(784))$$:

 $$T_{3}^{2} - 2$$ $$T_{5}^{2} - 8$$ $$T_{11} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-2 + T^{2}$$
$5$ $$-8 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 6 + T )^{2}$$
$13$ $$-32 + T^{2}$$
$17$ $$-2 + T^{2}$$
$19$ $$-18 + T^{2}$$
$23$ $$( 4 + T )^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$-8 + T^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$-2 + T^{2}$$
$43$ $$( 10 + T )^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$( 2 + T )^{2}$$
$59$ $$-2 + T^{2}$$
$61$ $$-72 + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$-98 + T^{2}$$
$79$ $$( -4 + T )^{2}$$
$83$ $$-2 + T^{2}$$
$89$ $$-18 + T^{2}$$
$97$ $$-162 + T^{2}$$