Properties

 Label 784.4.a.a 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 28) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - 10q^{3} + 8q^{5} + 73q^{9} + O(q^{10})$$ $$q - 10q^{3} + 8q^{5} + 73q^{9} + 40q^{11} + 12q^{13} - 80q^{15} + 58q^{17} + 26q^{19} + 64q^{23} - 61q^{25} - 460q^{27} - 62q^{29} + 252q^{31} - 400q^{33} + 26q^{37} - 120q^{39} - 6q^{41} - 416q^{43} + 584q^{45} - 396q^{47} - 580q^{51} - 450q^{53} + 320q^{55} - 260q^{57} + 274q^{59} + 576q^{61} + 96q^{65} + 476q^{67} - 640q^{69} + 448q^{71} + 158q^{73} + 610q^{75} + 936q^{79} + 2629q^{81} + 530q^{83} + 464q^{85} + 620q^{87} + 390q^{89} - 2520q^{93} + 208q^{95} - 214q^{97} + 2920q^{99} + O(q^{100})$$

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 −10.0000 0 8.00000 0 0 0 73.0000 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.a 1
4.b odd 2 1 196.4.a.d 1
7.b odd 2 1 112.4.a.g 1
12.b even 2 1 1764.4.a.c 1
21.c even 2 1 1008.4.a.o 1
28.d even 2 1 28.4.a.a 1
28.f even 6 2 196.4.e.f 2
28.g odd 6 2 196.4.e.a 2
56.e even 2 1 448.4.a.p 1
56.h odd 2 1 448.4.a.a 1
84.h odd 2 1 252.4.a.d 1
84.j odd 6 2 1764.4.k.d 2
84.n even 6 2 1764.4.k.m 2
140.c even 2 1 700.4.a.n 1
140.j odd 4 2 700.4.e.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.a 1 28.d even 2 1
112.4.a.g 1 7.b odd 2 1
196.4.a.d 1 4.b odd 2 1
196.4.e.a 2 28.g odd 6 2
196.4.e.f 2 28.f even 6 2
252.4.a.d 1 84.h odd 2 1
448.4.a.a 1 56.h odd 2 1
448.4.a.p 1 56.e even 2 1
700.4.a.n 1 140.c even 2 1
700.4.e.a 2 140.j odd 4 2
784.4.a.a 1 1.a even 1 1 trivial
1008.4.a.o 1 21.c even 2 1
1764.4.a.c 1 12.b even 2 1
1764.4.k.d 2 84.j odd 6 2
1764.4.k.m 2 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(784))$$:

 $$T_{3} + 10$$ $$T_{5} - 8$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$10 + T$$
$5$ $$-8 + T$$
$7$ $$T$$
$11$ $$-40 + T$$
$13$ $$-12 + T$$
$17$ $$-58 + T$$
$19$ $$-26 + T$$
$23$ $$-64 + T$$
$29$ $$62 + T$$
$31$ $$-252 + T$$
$37$ $$-26 + T$$
$41$ $$6 + T$$
$43$ $$416 + T$$
$47$ $$396 + T$$
$53$ $$450 + T$$
$59$ $$-274 + T$$
$61$ $$-576 + T$$
$67$ $$-476 + T$$
$71$ $$-448 + T$$
$73$ $$-158 + T$$
$79$ $$-936 + T$$
$83$ $$-530 + T$$
$89$ $$-390 + T$$
$97$ $$214 + T$$