# Properties

 Label 784.4.a.r 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 7) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 7q^{3} - 7q^{5} + 22q^{9} + O(q^{10})$$ $$q + 7q^{3} - 7q^{5} + 22q^{9} + 5q^{11} + 14q^{13} - 49q^{15} + 21q^{17} + 49q^{19} + 159q^{23} - 76q^{25} - 35q^{27} + 58q^{29} + 147q^{31} + 35q^{33} + 219q^{37} + 98q^{39} - 350q^{41} + 124q^{43} - 154q^{45} + 525q^{47} + 147q^{51} + 303q^{53} - 35q^{55} + 343q^{57} - 105q^{59} + 413q^{61} - 98q^{65} - 415q^{67} + 1113q^{69} + 432q^{71} + 1113q^{73} - 532q^{75} + 103q^{79} - 839q^{81} + 1092q^{83} - 147q^{85} + 406q^{87} + 329q^{89} + 1029q^{93} - 343q^{95} + 882q^{97} + 110q^{99} + O(q^{100})$$

## 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 7.00000 0 −7.00000 0 0 0 22.0000 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

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.r 1
4.b odd 2 1 49.4.a.c 1
7.b odd 2 1 784.4.a.b 1
7.d odd 6 2 112.4.i.c 2
12.b even 2 1 441.4.a.e 1
20.d odd 2 1 1225.4.a.d 1
28.d even 2 1 49.4.a.d 1
28.f even 6 2 7.4.c.a 2
28.g odd 6 2 49.4.c.a 2
56.j odd 6 2 448.4.i.a 2
56.m even 6 2 448.4.i.f 2
84.h odd 2 1 441.4.a.d 1
84.j odd 6 2 63.4.e.b 2
84.n even 6 2 441.4.e.k 2
140.c even 2 1 1225.4.a.c 1
140.s even 6 2 175.4.e.a 2
140.x odd 12 4 175.4.k.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 28.f even 6 2
49.4.a.c 1 4.b odd 2 1
49.4.a.d 1 28.d even 2 1
49.4.c.a 2 28.g odd 6 2
63.4.e.b 2 84.j odd 6 2
112.4.i.c 2 7.d odd 6 2
175.4.e.a 2 140.s even 6 2
175.4.k.a 4 140.x odd 12 4
441.4.a.d 1 84.h odd 2 1
441.4.a.e 1 12.b even 2 1
441.4.e.k 2 84.n even 6 2
448.4.i.a 2 56.j odd 6 2
448.4.i.f 2 56.m even 6 2
784.4.a.b 1 7.b odd 2 1
784.4.a.r 1 1.a even 1 1 trivial
1225.4.a.c 1 140.c even 2 1
1225.4.a.d 1 20.d odd 2 1

## Hecke kernels

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

 $$T_{3} - 7$$ $$T_{5} + 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-7 + T$$
$5$ $$7 + T$$
$7$ $$T$$
$11$ $$-5 + T$$
$13$ $$-14 + T$$
$17$ $$-21 + T$$
$19$ $$-49 + T$$
$23$ $$-159 + T$$
$29$ $$-58 + T$$
$31$ $$-147 + T$$
$37$ $$-219 + T$$
$41$ $$350 + T$$
$43$ $$-124 + T$$
$47$ $$-525 + T$$
$53$ $$-303 + T$$
$59$ $$105 + T$$
$61$ $$-413 + T$$
$67$ $$415 + T$$
$71$ $$-432 + T$$
$73$ $$-1113 + T$$
$79$ $$-103 + T$$
$83$ $$-1092 + T$$
$89$ $$-329 + T$$
$97$ $$-882 + T$$