# Properties

 Label 784.4.a.y 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 98) 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.

 $$f(q)$$ $$=$$ $$q + 5 \beta q^{3} -14 \beta q^{5} + 23 q^{9} +O(q^{10})$$ $$q + 5 \beta q^{3} -14 \beta q^{5} + 23 q^{9} + 14 q^{11} + 36 \beta q^{13} -140 q^{15} + \beta q^{17} + \beta q^{19} -140 q^{23} + 267 q^{25} -20 \beta q^{27} -286 q^{29} + 66 \beta q^{31} + 70 \beta q^{33} -38 q^{37} + 360 q^{39} -89 \beta q^{41} + 34 q^{43} -322 \beta q^{45} -370 \beta q^{47} + 10 q^{51} -74 q^{53} -196 \beta q^{55} + 10 q^{57} -307 \beta q^{59} + 10 \beta q^{61} -1008 q^{65} -684 q^{67} -700 \beta q^{69} -588 q^{71} -191 \beta q^{73} + 1335 \beta q^{75} -1220 q^{79} -821 q^{81} -299 \beta q^{83} -28 q^{85} -1430 \beta q^{87} + 437 \beta q^{89} + 660 q^{93} -28 q^{95} + 1049 \beta q^{97} + 322 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 46q^{9} + O(q^{10})$$ $$2q + 46q^{9} + 28q^{11} - 280q^{15} - 280q^{23} + 534q^{25} - 572q^{29} - 76q^{37} + 720q^{39} + 68q^{43} + 20q^{51} - 148q^{53} + 20q^{57} - 2016q^{65} - 1368q^{67} - 1176q^{71} - 2440q^{79} - 1642q^{81} - 56q^{85} + 1320q^{93} - 56q^{95} + 644q^{99} + O(q^{100})$$

## 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 −7.07107 0 19.7990 0 0 0 23.0000 0
1.2 0 7.07107 0 −19.7990 0 0 0 23.0000 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.y 2
4.b odd 2 1 98.4.a.g 2
7.b odd 2 1 inner 784.4.a.y 2
12.b even 2 1 882.4.a.bg 2
20.d odd 2 1 2450.4.a.bx 2
28.d even 2 1 98.4.a.g 2
28.f even 6 2 98.4.c.h 4
28.g odd 6 2 98.4.c.h 4
84.h odd 2 1 882.4.a.bg 2
84.j odd 6 2 882.4.g.ba 4
84.n even 6 2 882.4.g.ba 4
140.c even 2 1 2450.4.a.bx 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.g 2 4.b odd 2 1
98.4.a.g 2 28.d even 2 1
98.4.c.h 4 28.f even 6 2
98.4.c.h 4 28.g odd 6 2
784.4.a.y 2 1.a even 1 1 trivial
784.4.a.y 2 7.b odd 2 1 inner
882.4.a.bg 2 12.b even 2 1
882.4.a.bg 2 84.h odd 2 1
882.4.g.ba 4 84.j odd 6 2
882.4.g.ba 4 84.n even 6 2
2450.4.a.bx 2 20.d odd 2 1
2450.4.a.bx 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(784))$$:

 $$T_{3}^{2} - 50$$ $$T_{5}^{2} - 392$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-50 + T^{2}$$
$5$ $$-392 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -14 + T )^{2}$$
$13$ $$-2592 + T^{2}$$
$17$ $$-2 + T^{2}$$
$19$ $$-2 + T^{2}$$
$23$ $$( 140 + T )^{2}$$
$29$ $$( 286 + T )^{2}$$
$31$ $$-8712 + T^{2}$$
$37$ $$( 38 + T )^{2}$$
$41$ $$-15842 + T^{2}$$
$43$ $$( -34 + T )^{2}$$
$47$ $$-273800 + T^{2}$$
$53$ $$( 74 + T )^{2}$$
$59$ $$-188498 + T^{2}$$
$61$ $$-200 + T^{2}$$
$67$ $$( 684 + T )^{2}$$
$71$ $$( 588 + T )^{2}$$
$73$ $$-72962 + T^{2}$$
$79$ $$( 1220 + T )^{2}$$
$83$ $$-178802 + T^{2}$$
$89$ $$-381938 + T^{2}$$
$97$ $$-2200802 + T^{2}$$