# Properties

 Label 784.4.a.c Level $784$ Weight $4$ Character orbit 784.a Self dual yes Analytic conductor $46.257$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 5q^{3} + 9q^{5} - 2q^{9} + O(q^{10})$$ $$q - 5q^{3} + 9q^{5} - 2q^{9} + 57q^{11} + 70q^{13} - 45q^{15} - 51q^{17} + 5q^{19} - 69q^{23} - 44q^{25} + 145q^{27} + 114q^{29} + 23q^{31} - 285q^{33} - 253q^{37} - 350q^{39} + 42q^{41} + 124q^{43} - 18q^{45} + 201q^{47} + 255q^{51} - 393q^{53} + 513q^{55} - 25q^{57} + 219q^{59} + 709q^{61} + 630q^{65} - 419q^{67} + 345q^{69} + 96q^{71} + 313q^{73} + 220q^{75} - 461q^{79} - 671q^{81} - 588q^{83} - 459q^{85} - 570q^{87} + 1017q^{89} - 115q^{93} + 45q^{95} + 1834q^{97} - 114q^{99} + O(q^{100})$$

## 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
 0
0 −5.00000 0 9.00000 0 0 0 −2.00000 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.4.a.c 1
4.b odd 2 1 98.4.a.f 1
7.b odd 2 1 784.4.a.p 1
7.d odd 6 2 112.4.i.a 2
12.b even 2 1 882.4.a.c 1
20.d odd 2 1 2450.4.a.d 1
28.d even 2 1 98.4.a.d 1
28.f even 6 2 14.4.c.a 2
28.g odd 6 2 98.4.c.a 2
56.j odd 6 2 448.4.i.e 2
56.m even 6 2 448.4.i.b 2
84.h odd 2 1 882.4.a.f 1
84.j odd 6 2 126.4.g.d 2
84.n even 6 2 882.4.g.u 2
140.c even 2 1 2450.4.a.q 1
140.s even 6 2 350.4.e.e 2
140.x odd 12 4 350.4.j.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 28.f even 6 2
98.4.a.d 1 28.d even 2 1
98.4.a.f 1 4.b odd 2 1
98.4.c.a 2 28.g odd 6 2
112.4.i.a 2 7.d odd 6 2
126.4.g.d 2 84.j odd 6 2
350.4.e.e 2 140.s even 6 2
350.4.j.b 4 140.x odd 12 4
448.4.i.b 2 56.m even 6 2
448.4.i.e 2 56.j odd 6 2
784.4.a.c 1 1.a even 1 1 trivial
784.4.a.p 1 7.b odd 2 1
882.4.a.c 1 12.b even 2 1
882.4.a.f 1 84.h odd 2 1
882.4.g.u 2 84.n even 6 2
2450.4.a.d 1 20.d odd 2 1
2450.4.a.q 1 140.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(784))$$:

 $$T_{3} + 5$$ $$T_{5} - 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$5 + T$$
$5$ $$-9 + T$$
$7$ $$T$$
$11$ $$-57 + T$$
$13$ $$-70 + T$$
$17$ $$51 + T$$
$19$ $$-5 + T$$
$23$ $$69 + T$$
$29$ $$-114 + T$$
$31$ $$-23 + T$$
$37$ $$253 + T$$
$41$ $$-42 + T$$
$43$ $$-124 + T$$
$47$ $$-201 + T$$
$53$ $$393 + T$$
$59$ $$-219 + T$$
$61$ $$-709 + T$$
$67$ $$419 + T$$
$71$ $$-96 + T$$
$73$ $$-313 + T$$
$79$ $$461 + T$$
$83$ $$588 + T$$
$89$ $$-1017 + T$$
$97$ $$-1834 + T$$