# Properties

 Label 784.4.a.l 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

 $$f(q)$$ $$=$$ $$q + q^{3} + 7 q^{5} - 26 q^{9}+O(q^{10})$$ q + q^3 + 7 * q^5 - 26 * q^9 $$q + q^{3} + 7 q^{5} - 26 q^{9} - 35 q^{11} + 66 q^{13} + 7 q^{15} + 59 q^{17} - 137 q^{19} + 7 q^{23} - 76 q^{25} - 53 q^{27} + 106 q^{29} - 75 q^{31} - 35 q^{33} + 11 q^{37} + 66 q^{39} - 498 q^{41} - 260 q^{43} - 182 q^{45} + 171 q^{47} + 59 q^{51} - 417 q^{53} - 245 q^{55} - 137 q^{57} + 17 q^{59} + 51 q^{61} + 462 q^{65} - 439 q^{67} + 7 q^{69} + 784 q^{71} + 295 q^{73} - 76 q^{75} + 495 q^{79} + 649 q^{81} - 932 q^{83} + 413 q^{85} + 106 q^{87} - 873 q^{89} - 75 q^{93} - 959 q^{95} - 290 q^{97} + 910 q^{99}+O(q^{100})$$ q + q^3 + 7 * q^5 - 26 * q^9 - 35 * q^11 + 66 * q^13 + 7 * q^15 + 59 * q^17 - 137 * q^19 + 7 * q^23 - 76 * q^25 - 53 * q^27 + 106 * q^29 - 75 * q^31 - 35 * q^33 + 11 * q^37 + 66 * q^39 - 498 * q^41 - 260 * q^43 - 182 * q^45 + 171 * q^47 + 59 * q^51 - 417 * q^53 - 245 * q^55 - 137 * q^57 + 17 * q^59 + 51 * q^61 + 462 * q^65 - 439 * q^67 + 7 * q^69 + 784 * q^71 + 295 * q^73 - 76 * q^75 + 495 * q^79 + 649 * q^81 - 932 * q^83 + 413 * q^85 + 106 * q^87 - 873 * q^89 - 75 * q^93 - 959 * q^95 - 290 * q^97 + 910 * 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 1.00000 0 7.00000 0 0 0 −26.0000 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.l 1
4.b odd 2 1 98.4.a.b 1
7.b odd 2 1 784.4.a.j 1
7.c even 3 2 112.4.i.b 2
12.b even 2 1 882.4.a.k 1
20.d odd 2 1 2450.4.a.bh 1
28.d even 2 1 98.4.a.c 1
28.f even 6 2 98.4.c.e 2
28.g odd 6 2 14.4.c.b 2
56.k odd 6 2 448.4.i.c 2
56.p even 6 2 448.4.i.d 2
84.h odd 2 1 882.4.a.p 1
84.j odd 6 2 882.4.g.d 2
84.n even 6 2 126.4.g.c 2
140.c even 2 1 2450.4.a.bf 1
140.p odd 6 2 350.4.e.b 2
140.w even 12 4 350.4.j.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b 2 28.g odd 6 2
98.4.a.b 1 4.b odd 2 1
98.4.a.c 1 28.d even 2 1
98.4.c.e 2 28.f even 6 2
112.4.i.b 2 7.c even 3 2
126.4.g.c 2 84.n even 6 2
350.4.e.b 2 140.p odd 6 2
350.4.j.d 4 140.w even 12 4
448.4.i.c 2 56.k odd 6 2
448.4.i.d 2 56.p even 6 2
784.4.a.j 1 7.b odd 2 1
784.4.a.l 1 1.a even 1 1 trivial
882.4.a.k 1 12.b even 2 1
882.4.a.p 1 84.h odd 2 1
882.4.g.d 2 84.j odd 6 2
2450.4.a.bf 1 140.c even 2 1
2450.4.a.bh 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(784))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{5} - 7$$ T5 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 7$$
$7$ $$T$$
$11$ $$T + 35$$
$13$ $$T - 66$$
$17$ $$T - 59$$
$19$ $$T + 137$$
$23$ $$T - 7$$
$29$ $$T - 106$$
$31$ $$T + 75$$
$37$ $$T - 11$$
$41$ $$T + 498$$
$43$ $$T + 260$$
$47$ $$T - 171$$
$53$ $$T + 417$$
$59$ $$T - 17$$
$61$ $$T - 51$$
$67$ $$T + 439$$
$71$ $$T - 784$$
$73$ $$T - 295$$
$79$ $$T - 495$$
$83$ $$T + 932$$
$89$ $$T + 873$$
$97$ $$T + 290$$