# Properties

 Label 784.4.a.p Level $784$ Weight $4$ Character orbit 784.a Self dual yes Analytic conductor $46.257$ Analytic rank $1$ 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: $$1$$ 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

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

## 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.p 1
4.b odd 2 1 98.4.a.d 1
7.b odd 2 1 784.4.a.c 1
7.c even 3 2 112.4.i.a 2
12.b even 2 1 882.4.a.f 1
20.d odd 2 1 2450.4.a.q 1
28.d even 2 1 98.4.a.f 1
28.f even 6 2 98.4.c.a 2
28.g odd 6 2 14.4.c.a 2
56.k odd 6 2 448.4.i.b 2
56.p even 6 2 448.4.i.e 2
84.h odd 2 1 882.4.a.c 1
84.j odd 6 2 882.4.g.u 2
84.n even 6 2 126.4.g.d 2
140.c even 2 1 2450.4.a.d 1
140.p odd 6 2 350.4.e.e 2
140.w even 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.g odd 6 2
98.4.a.d 1 4.b odd 2 1
98.4.a.f 1 28.d even 2 1
98.4.c.a 2 28.f even 6 2
112.4.i.a 2 7.c even 3 2
126.4.g.d 2 84.n even 6 2
350.4.e.e 2 140.p odd 6 2
350.4.j.b 4 140.w even 12 4
448.4.i.b 2 56.k odd 6 2
448.4.i.e 2 56.p even 6 2
784.4.a.c 1 7.b odd 2 1
784.4.a.p 1 1.a even 1 1 trivial
882.4.a.c 1 84.h odd 2 1
882.4.a.f 1 12.b even 2 1
882.4.g.u 2 84.j odd 6 2
2450.4.a.d 1 140.c even 2 1
2450.4.a.q 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(784))$$:

 $$T_{3} - 5$$ T3 - 5 $$T_{5} + 9$$ T5 + 9

## Hecke characteristic polynomials

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