# Properties

 Label 784.4.a.v Level $784$ Weight $4$ Character orbit 784.a Self dual yes Analytic conductor $46.257$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 10 \beta q^{5} -25 q^{9} +O(q^{10})$$ $$q + \beta q^{3} + 10 \beta q^{5} -25 q^{9} -54 q^{11} + 16 \beta q^{13} + 20 q^{15} + 23 \beta q^{17} + 45 \beta q^{19} -28 q^{23} + 75 q^{25} -52 \beta q^{27} + 282 q^{29} + 194 \beta q^{31} -54 \beta q^{33} + 146 q^{37} + 32 q^{39} + 241 \beta q^{41} -10 q^{43} -250 \beta q^{45} + 358 \beta q^{47} + 46 q^{51} + 598 q^{53} -540 \beta q^{55} + 90 q^{57} -407 \beta q^{59} + 330 \beta q^{61} + 320 q^{65} -916 q^{67} -28 \beta q^{69} -420 q^{71} -497 \beta q^{73} + 75 \beta q^{75} + 292 q^{79} + 571 q^{81} -815 \beta q^{83} + 460 q^{85} + 282 \beta q^{87} + 315 \beta q^{89} + 388 q^{93} + 900 q^{95} -417 \beta q^{97} + 1350 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 50q^{9} + O(q^{10})$$ $$2q - 50q^{9} - 108q^{11} + 40q^{15} - 56q^{23} + 150q^{25} + 564q^{29} + 292q^{37} + 64q^{39} - 20q^{43} + 92q^{51} + 1196q^{53} + 180q^{57} + 640q^{65} - 1832q^{67} - 840q^{71} + 584q^{79} + 1142q^{81} + 920q^{85} + 776q^{93} + 1800q^{95} + 2700q^{99} + O(q^{100})$$

## 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 −14.1421 0 0 0 −25.0000 0
1.2 0 1.41421 0 14.1421 0 0 0 −25.0000 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.v 2
4.b odd 2 1 392.4.a.f 2
7.b odd 2 1 inner 784.4.a.v 2
28.d even 2 1 392.4.a.f 2
28.f even 6 2 392.4.i.k 4
28.g odd 6 2 392.4.i.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.4.a.f 2 4.b odd 2 1
392.4.a.f 2 28.d even 2 1
392.4.i.k 4 28.f even 6 2
392.4.i.k 4 28.g odd 6 2
784.4.a.v 2 1.a even 1 1 trivial
784.4.a.v 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(784))$$:

 $$T_{3}^{2} - 2$$ $$T_{5}^{2} - 200$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-2 + T^{2}$$
$5$ $$-200 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 54 + T )^{2}$$
$13$ $$-512 + T^{2}$$
$17$ $$-1058 + T^{2}$$
$19$ $$-4050 + T^{2}$$
$23$ $$( 28 + T )^{2}$$
$29$ $$( -282 + T )^{2}$$
$31$ $$-75272 + T^{2}$$
$37$ $$( -146 + T )^{2}$$
$41$ $$-116162 + T^{2}$$
$43$ $$( 10 + T )^{2}$$
$47$ $$-256328 + T^{2}$$
$53$ $$( -598 + T )^{2}$$
$59$ $$-331298 + T^{2}$$
$61$ $$-217800 + T^{2}$$
$67$ $$( 916 + T )^{2}$$
$71$ $$( 420 + T )^{2}$$
$73$ $$-494018 + T^{2}$$
$79$ $$( -292 + T )^{2}$$
$83$ $$-1328450 + T^{2}$$
$89$ $$-198450 + T^{2}$$
$97$ $$-347778 + T^{2}$$