# Properties

 Label 784.4.a.z Level $784$ Weight $4$ Character orbit 784.a Self dual yes Analytic conductor $46.257$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{22})$$ Defining polynomial: $$x^{2} - 22$$ x^2 - 22 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 98) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{22}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + \beta q^{5} + 61 q^{9}+O(q^{10})$$ q + b * q^3 + b * q^5 + 61 * q^9 $$q + \beta q^{3} + \beta q^{5} + 61 q^{9} - 20 q^{11} + 7 \beta q^{13} + 88 q^{15} + 6 \beta q^{17} - \beta q^{19} - 48 q^{23} - 37 q^{25} + 34 \beta q^{27} - 166 q^{29} + 22 \beta q^{31} - 20 \beta q^{33} - 78 q^{37} + 616 q^{39} + 42 \beta q^{41} - 436 q^{43} + 61 \beta q^{45} - 22 \beta q^{47} + 528 q^{51} + 62 q^{53} - 20 \beta q^{55} - 88 q^{57} + 71 \beta q^{59} + 29 \beta q^{61} + 616 q^{65} - 580 q^{67} - 48 \beta q^{69} + 544 q^{71} - 64 \beta q^{73} - 37 \beta q^{75} + 680 q^{79} + 1345 q^{81} - 21 \beta q^{83} + 528 q^{85} - 166 \beta q^{87} - 160 \beta q^{89} + 1936 q^{93} - 88 q^{95} - 70 \beta q^{97} - 1220 q^{99} +O(q^{100})$$ q + b * q^3 + b * q^5 + 61 * q^9 - 20 * q^11 + 7*b * q^13 + 88 * q^15 + 6*b * q^17 - b * q^19 - 48 * q^23 - 37 * q^25 + 34*b * q^27 - 166 * q^29 + 22*b * q^31 - 20*b * q^33 - 78 * q^37 + 616 * q^39 + 42*b * q^41 - 436 * q^43 + 61*b * q^45 - 22*b * q^47 + 528 * q^51 + 62 * q^53 - 20*b * q^55 - 88 * q^57 + 71*b * q^59 + 29*b * q^61 + 616 * q^65 - 580 * q^67 - 48*b * q^69 + 544 * q^71 - 64*b * q^73 - 37*b * q^75 + 680 * q^79 + 1345 * q^81 - 21*b * q^83 + 528 * q^85 - 166*b * q^87 - 160*b * q^89 + 1936 * q^93 - 88 * q^95 - 70*b * q^97 - 1220 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 122 q^{9}+O(q^{10})$$ 2 * q + 122 * q^9 $$2 q + 122 q^{9} - 40 q^{11} + 176 q^{15} - 96 q^{23} - 74 q^{25} - 332 q^{29} - 156 q^{37} + 1232 q^{39} - 872 q^{43} + 1056 q^{51} + 124 q^{53} - 176 q^{57} + 1232 q^{65} - 1160 q^{67} + 1088 q^{71} + 1360 q^{79} + 2690 q^{81} + 1056 q^{85} + 3872 q^{93} - 176 q^{95} - 2440 q^{99}+O(q^{100})$$ 2 * q + 122 * q^9 - 40 * q^11 + 176 * q^15 - 96 * q^23 - 74 * q^25 - 332 * q^29 - 156 * q^37 + 1232 * q^39 - 872 * q^43 + 1056 * q^51 + 124 * q^53 - 176 * q^57 + 1232 * q^65 - 1160 * q^67 + 1088 * q^71 + 1360 * q^79 + 2690 * q^81 + 1056 * q^85 + 3872 * q^93 - 176 * q^95 - 2440 * 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
 −4.69042 4.69042
0 −9.38083 0 −9.38083 0 0 0 61.0000 0
1.2 0 9.38083 0 9.38083 0 0 0 61.0000 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.4.a.z 2
4.b odd 2 1 98.4.a.h 2
7.b odd 2 1 inner 784.4.a.z 2
12.b even 2 1 882.4.a.w 2
20.d odd 2 1 2450.4.a.bs 2
28.d even 2 1 98.4.a.h 2
28.f even 6 2 98.4.c.g 4
28.g odd 6 2 98.4.c.g 4
84.h odd 2 1 882.4.a.w 2
84.j odd 6 2 882.4.g.bi 4
84.n even 6 2 882.4.g.bi 4
140.c even 2 1 2450.4.a.bs 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
98.4.a.h 2 4.b odd 2 1
98.4.a.h 2 28.d even 2 1
98.4.c.g 4 28.f even 6 2
98.4.c.g 4 28.g odd 6 2
784.4.a.z 2 1.a even 1 1 trivial
784.4.a.z 2 7.b odd 2 1 inner
882.4.a.w 2 12.b even 2 1
882.4.a.w 2 84.h odd 2 1
882.4.g.bi 4 84.j odd 6 2
882.4.g.bi 4 84.n even 6 2
2450.4.a.bs 2 20.d odd 2 1
2450.4.a.bs 2 140.c even 2 1

## Hecke kernels

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

 $$T_{3}^{2} - 88$$ T3^2 - 88 $$T_{5}^{2} - 88$$ T5^2 - 88

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 88$$
$5$ $$T^{2} - 88$$
$7$ $$T^{2}$$
$11$ $$(T + 20)^{2}$$
$13$ $$T^{2} - 4312$$
$17$ $$T^{2} - 3168$$
$19$ $$T^{2} - 88$$
$23$ $$(T + 48)^{2}$$
$29$ $$(T + 166)^{2}$$
$31$ $$T^{2} - 42592$$
$37$ $$(T + 78)^{2}$$
$41$ $$T^{2} - 155232$$
$43$ $$(T + 436)^{2}$$
$47$ $$T^{2} - 42592$$
$53$ $$(T - 62)^{2}$$
$59$ $$T^{2} - 443608$$
$61$ $$T^{2} - 74008$$
$67$ $$(T + 580)^{2}$$
$71$ $$(T - 544)^{2}$$
$73$ $$T^{2} - 360448$$
$79$ $$(T - 680)^{2}$$
$83$ $$T^{2} - 38808$$
$89$ $$T^{2} - 2252800$$
$97$ $$T^{2} - 431200$$