# Properties

 Label 784.2.a.n Level $784$ Weight $2$ Character orbit 784.a Self dual yes Analytic conductor $6.260$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} -\beta q^{5} + 5 q^{9} +O(q^{10})$$ $$q + \beta q^{3} -\beta q^{5} + 5 q^{9} + 4 q^{11} + \beta q^{13} -8 q^{15} + 2 \beta q^{17} -\beta q^{19} + 3 q^{25} + 2 \beta q^{27} + 2 q^{29} -2 \beta q^{31} + 4 \beta q^{33} + 10 q^{37} + 8 q^{39} -2 \beta q^{41} + 4 q^{43} -5 \beta q^{45} + 2 \beta q^{47} + 16 q^{51} + 6 q^{53} -4 \beta q^{55} -8 q^{57} -\beta q^{59} -5 \beta q^{61} -8 q^{65} -12 q^{67} + 3 \beta q^{75} -8 q^{79} + q^{81} -5 \beta q^{83} -16 q^{85} + 2 \beta q^{87} -16 q^{93} + 8 q^{95} -2 \beta q^{97} + 20 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 10q^{9} + O(q^{10})$$ $$2q + 10q^{9} + 8q^{11} - 16q^{15} + 6q^{25} + 4q^{29} + 20q^{37} + 16q^{39} + 8q^{43} + 32q^{51} + 12q^{53} - 16q^{57} - 16q^{65} - 24q^{67} - 16q^{79} + 2q^{81} - 32q^{85} - 32q^{93} + 16q^{95} + 40q^{99} + O(q^{100})$$

## 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 −2.82843 0 2.82843 0 0 0 5.00000 0
1.2 0 2.82843 0 −2.82843 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 784.2.a.n 2
3.b odd 2 1 7056.2.a.cj 2
4.b odd 2 1 392.2.a.h 2
7.b odd 2 1 inner 784.2.a.n 2
7.c even 3 2 784.2.i.k 4
7.d odd 6 2 784.2.i.k 4
8.b even 2 1 3136.2.a.bq 2
8.d odd 2 1 3136.2.a.bt 2
12.b even 2 1 3528.2.a.bj 2
20.d odd 2 1 9800.2.a.bw 2
21.c even 2 1 7056.2.a.cj 2
28.d even 2 1 392.2.a.h 2
28.f even 6 2 392.2.i.g 4
28.g odd 6 2 392.2.i.g 4
56.e even 2 1 3136.2.a.bt 2
56.h odd 2 1 3136.2.a.bq 2
84.h odd 2 1 3528.2.a.bj 2
84.j odd 6 2 3528.2.s.be 4
84.n even 6 2 3528.2.s.be 4
140.c even 2 1 9800.2.a.bw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.a.h 2 4.b odd 2 1
392.2.a.h 2 28.d even 2 1
392.2.i.g 4 28.f even 6 2
392.2.i.g 4 28.g odd 6 2
784.2.a.n 2 1.a even 1 1 trivial
784.2.a.n 2 7.b odd 2 1 inner
784.2.i.k 4 7.c even 3 2
784.2.i.k 4 7.d odd 6 2
3136.2.a.bq 2 8.b even 2 1
3136.2.a.bq 2 56.h odd 2 1
3136.2.a.bt 2 8.d odd 2 1
3136.2.a.bt 2 56.e even 2 1
3528.2.a.bj 2 12.b even 2 1
3528.2.a.bj 2 84.h odd 2 1
3528.2.s.be 4 84.j odd 6 2
3528.2.s.be 4 84.n even 6 2
7056.2.a.cj 2 3.b odd 2 1
7056.2.a.cj 2 21.c even 2 1
9800.2.a.bw 2 20.d odd 2 1
9800.2.a.bw 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(784))$$:

 $$T_{3}^{2} - 8$$ $$T_{5}^{2} - 8$$ $$T_{11} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-8 + T^{2}$$
$5$ $$-8 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -4 + T )^{2}$$
$13$ $$-8 + T^{2}$$
$17$ $$-32 + T^{2}$$
$19$ $$-8 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$-32 + T^{2}$$
$37$ $$( -10 + T )^{2}$$
$41$ $$-32 + T^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$-32 + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$-8 + T^{2}$$
$61$ $$-200 + T^{2}$$
$67$ $$( 12 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$-200 + T^{2}$$
$89$ $$T^{2}$$
$97$ $$-32 + T^{2}$$