# Properties

 Label 784.4.a.s 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

 $$f(q)$$ $$=$$ $$q + 8 q^{3} + 14 q^{5} + 37 q^{9}+O(q^{10})$$ q + 8 * q^3 + 14 * q^5 + 37 * q^9 $$q + 8 q^{3} + 14 q^{5} + 37 q^{9} + 28 q^{11} - 18 q^{13} + 112 q^{15} - 74 q^{17} + 80 q^{19} + 112 q^{23} + 71 q^{25} + 80 q^{27} + 190 q^{29} + 72 q^{31} + 224 q^{33} - 346 q^{37} - 144 q^{39} - 162 q^{41} + 412 q^{43} + 518 q^{45} + 24 q^{47} - 592 q^{51} + 318 q^{53} + 392 q^{55} + 640 q^{57} - 200 q^{59} + 198 q^{61} - 252 q^{65} + 716 q^{67} + 896 q^{69} - 392 q^{71} - 538 q^{73} + 568 q^{75} - 240 q^{79} - 359 q^{81} - 1072 q^{83} - 1036 q^{85} + 1520 q^{87} - 810 q^{89} + 576 q^{93} + 1120 q^{95} - 1354 q^{97} + 1036 q^{99}+O(q^{100})$$ q + 8 * q^3 + 14 * q^5 + 37 * q^9 + 28 * q^11 - 18 * q^13 + 112 * q^15 - 74 * q^17 + 80 * q^19 + 112 * q^23 + 71 * q^25 + 80 * q^27 + 190 * q^29 + 72 * q^31 + 224 * q^33 - 346 * q^37 - 144 * q^39 - 162 * q^41 + 412 * q^43 + 518 * q^45 + 24 * q^47 - 592 * q^51 + 318 * q^53 + 392 * q^55 + 640 * q^57 - 200 * q^59 + 198 * q^61 - 252 * q^65 + 716 * q^67 + 896 * q^69 - 392 * q^71 - 538 * q^73 + 568 * q^75 - 240 * q^79 - 359 * q^81 - 1072 * q^83 - 1036 * q^85 + 1520 * q^87 - 810 * q^89 + 576 * q^93 + 1120 * q^95 - 1354 * q^97 + 1036 * 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 8.00000 0 14.0000 0 0 0 37.0000 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.s 1
4.b odd 2 1 98.4.a.a 1
7.b odd 2 1 112.4.a.a 1
12.b even 2 1 882.4.a.i 1
20.d odd 2 1 2450.4.a.bo 1
21.c even 2 1 1008.4.a.s 1
28.d even 2 1 14.4.a.a 1
28.f even 6 2 98.4.c.d 2
28.g odd 6 2 98.4.c.f 2
56.e even 2 1 448.4.a.b 1
56.h odd 2 1 448.4.a.o 1
84.h odd 2 1 126.4.a.h 1
84.j odd 6 2 882.4.g.b 2
84.n even 6 2 882.4.g.k 2
140.c even 2 1 350.4.a.l 1
140.j odd 4 2 350.4.c.b 2
308.g odd 2 1 1694.4.a.g 1
364.h even 2 1 2366.4.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.a.a 1 28.d even 2 1
98.4.a.a 1 4.b odd 2 1
98.4.c.d 2 28.f even 6 2
98.4.c.f 2 28.g odd 6 2
112.4.a.a 1 7.b odd 2 1
126.4.a.h 1 84.h odd 2 1
350.4.a.l 1 140.c even 2 1
350.4.c.b 2 140.j odd 4 2
448.4.a.b 1 56.e even 2 1
448.4.a.o 1 56.h odd 2 1
784.4.a.s 1 1.a even 1 1 trivial
882.4.a.i 1 12.b even 2 1
882.4.g.b 2 84.j odd 6 2
882.4.g.k 2 84.n even 6 2
1008.4.a.s 1 21.c even 2 1
1694.4.a.g 1 308.g odd 2 1
2366.4.a.h 1 364.h even 2 1
2450.4.a.bo 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(784))$$:

 $$T_{3} - 8$$ T3 - 8 $$T_{5} - 14$$ T5 - 14

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 8$$
$5$ $$T - 14$$
$7$ $$T$$
$11$ $$T - 28$$
$13$ $$T + 18$$
$17$ $$T + 74$$
$19$ $$T - 80$$
$23$ $$T - 112$$
$29$ $$T - 190$$
$31$ $$T - 72$$
$37$ $$T + 346$$
$41$ $$T + 162$$
$43$ $$T - 412$$
$47$ $$T - 24$$
$53$ $$T - 318$$
$59$ $$T + 200$$
$61$ $$T - 198$$
$67$ $$T - 716$$
$71$ $$T + 392$$
$73$ $$T + 538$$
$79$ $$T + 240$$
$83$ $$T + 1072$$
$89$ $$T + 810$$
$97$ $$T + 1354$$