# Properties

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

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## 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: $$1$$ Twist minimal: no (minimal twist has level 196) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 2 \beta q^{5} -25 q^{9} +O(q^{10})$$ $$q + \beta q^{3} + 2 \beta q^{5} -25 q^{9} + 26 q^{11} + 24 \beta q^{13} + 4 q^{15} -73 \beta q^{17} -67 \beta q^{19} + 148 q^{23} -117 q^{25} -52 \beta q^{27} -118 q^{29} + 210 \beta q^{31} + 26 \beta q^{33} -254 q^{37} + 48 q^{39} + 65 \beta q^{41} -122 q^{43} -50 \beta q^{45} -218 \beta q^{47} -146 q^{51} -170 q^{53} + 52 \beta q^{55} -134 q^{57} -215 \beta q^{59} -430 \beta q^{61} + 96 q^{65} -420 q^{67} + 148 \beta q^{69} -420 q^{71} + 575 \beta q^{73} -117 \beta q^{75} -1052 q^{79} + 571 q^{81} + 1025 \beta q^{83} -292 q^{85} -118 \beta q^{87} -725 \beta q^{89} + 420 q^{93} -268 q^{95} + 223 \beta q^{97} -650 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 50q^{9} + O(q^{10})$$ $$2q - 50q^{9} + 52q^{11} + 8q^{15} + 296q^{23} - 234q^{25} - 236q^{29} - 508q^{37} + 96q^{39} - 244q^{43} - 292q^{51} - 340q^{53} - 268q^{57} + 192q^{65} - 840q^{67} - 840q^{71} - 2104q^{79} + 1142q^{81} - 584q^{85} + 840q^{93} - 536q^{95} - 1300q^{99} + O(q^{100})$$

## 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 −25.0000 0
1.2 0 1.41421 0 2.82843 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.w 2
4.b odd 2 1 196.4.a.f 2
7.b odd 2 1 inner 784.4.a.w 2
12.b even 2 1 1764.4.a.v 2
28.d even 2 1 196.4.a.f 2
28.f even 6 2 196.4.e.h 4
28.g odd 6 2 196.4.e.h 4
84.h odd 2 1 1764.4.a.v 2
84.j odd 6 2 1764.4.k.s 4
84.n even 6 2 1764.4.k.s 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.4.a.f 2 4.b odd 2 1
196.4.a.f 2 28.d even 2 1
196.4.e.h 4 28.f even 6 2
196.4.e.h 4 28.g odd 6 2
784.4.a.w 2 1.a even 1 1 trivial
784.4.a.w 2 7.b odd 2 1 inner
1764.4.a.v 2 12.b even 2 1
1764.4.a.v 2 84.h odd 2 1
1764.4.k.s 4 84.j odd 6 2
1764.4.k.s 4 84.n even 6 2

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-2 + T^{2}$$
$5$ $$-8 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -26 + T )^{2}$$
$13$ $$-1152 + T^{2}$$
$17$ $$-10658 + T^{2}$$
$19$ $$-8978 + T^{2}$$
$23$ $$( -148 + T )^{2}$$
$29$ $$( 118 + T )^{2}$$
$31$ $$-88200 + T^{2}$$
$37$ $$( 254 + T )^{2}$$
$41$ $$-8450 + T^{2}$$
$43$ $$( 122 + T )^{2}$$
$47$ $$-95048 + T^{2}$$
$53$ $$( 170 + T )^{2}$$
$59$ $$-92450 + T^{2}$$
$61$ $$-369800 + T^{2}$$
$67$ $$( 420 + T )^{2}$$
$71$ $$( 420 + T )^{2}$$
$73$ $$-661250 + T^{2}$$
$79$ $$( 1052 + T )^{2}$$
$83$ $$-2101250 + T^{2}$$
$89$ $$-1051250 + T^{2}$$
$97$ $$-99458 + T^{2}$$
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