# Properties

 Label 784.4.a.bg Level $784$ Weight $4$ Character orbit 784.a Self dual yes Analytic conductor $46.257$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{65})$$ Defining polynomial: $$x^{4} - 2 x^{3} - 35 x^{2} + 36 x + 194$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 392) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \beta_{1} + \beta_{3} ) q^{3} + ( 3 \beta_{1} + \beta_{3} ) q^{5} + ( 10 - 3 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 2 \beta_{1} + \beta_{3} ) q^{3} + ( 3 \beta_{1} + \beta_{3} ) q^{5} + ( 10 - 3 \beta_{2} ) q^{9} + ( 31 - 5 \beta_{2} ) q^{11} + ( \beta_{1} + 9 \beta_{3} ) q^{13} + ( 40 - 4 \beta_{2} ) q^{15} + ( 31 \beta_{1} + 8 \beta_{3} ) q^{17} + ( -2 \beta_{1} + \beta_{3} ) q^{19} + ( -32 + 20 \beta_{2} ) q^{23} + ( -80 - 5 \beta_{2} ) q^{25} + ( 68 \beta_{1} - 8 \beta_{3} ) q^{27} + ( -63 + 9 \beta_{2} ) q^{29} + ( -64 \beta_{1} + 14 \beta_{3} ) q^{31} + ( 232 \beta_{1} + 46 \beta_{3} ) q^{33} + ( -91 + 29 \beta_{2} ) q^{37} + ( 282 - 10 \beta_{2} ) q^{39} + ( -291 \beta_{1} - 4 \beta_{3} ) q^{41} + ( 305 + 21 \beta_{2} ) q^{43} + ( 135 \beta_{1} + 25 \beta_{3} ) q^{45} + ( 172 \beta_{1} + 70 \beta_{3} ) q^{47} + ( 341 - 39 \beta_{2} ) q^{51} + ( -188 - 46 \beta_{2} ) q^{53} + ( 268 \beta_{1} + 56 \beta_{3} ) q^{55} + ( 25 + \beta_{2} ) q^{57} + ( -250 \beta_{1} + 29 \beta_{3} ) q^{59} + ( 201 \beta_{1} - 73 \beta_{3} ) q^{61} + ( 275 - 19 \beta_{2} ) q^{65} + ( 686 - 18 \beta_{2} ) q^{67} + ( -744 \beta_{1} - 92 \beta_{3} ) q^{69} + ( 430 + 46 \beta_{2} ) q^{71} + ( -83 \beta_{1} - 50 \beta_{3} ) q^{73} + ( 10 \beta_{1} - 65 \beta_{3} ) q^{75} + ( 422 - 2 \beta_{2} ) q^{79} + ( -314 + 21 \beta_{2} ) q^{81} + ( 106 \beta_{1} - 23 \beta_{3} ) q^{83} + ( 395 - 47 \beta_{2} ) q^{85} + ( -432 \beta_{1} - 90 \beta_{3} ) q^{87} + ( 611 \beta_{1} - 8 \beta_{3} ) q^{89} + ( 242 + 50 \beta_{2} ) q^{93} + 20 q^{95} + ( 177 \beta_{1} + 210 \beta_{3} ) q^{97} + ( 1285 - 143 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 40q^{9} + O(q^{10})$$ $$4q + 40q^{9} + 124q^{11} + 160q^{15} - 128q^{23} - 320q^{25} - 252q^{29} - 364q^{37} + 1128q^{39} + 1220q^{43} + 1364q^{51} - 752q^{53} + 100q^{57} + 1100q^{65} + 2744q^{67} + 1720q^{71} + 1688q^{79} - 1256q^{81} + 1580q^{85} + 968q^{93} + 80q^{95} + 5140q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 35 x^{2} + 36 x + 194$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{3} + 3 \nu^{2} + 43 \nu - 22$$$$)/57$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{3} + 6 \nu^{2} + 200 \nu - 101$$$$)/57$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 27 \nu^{2} - 50 \nu - 502$$$$)/57$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 2 \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{3} + \beta_{2} + 37$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$6 \beta_{3} + 23 \beta_{2} - 100 \beta_{1} + 55$$$$)/2$$

## 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
 −2.11692 3.11692 5.94534 −4.94534
0 −7.82220 0 −9.23641 0 0 0 34.1868 0
1.2 0 −3.57956 0 −2.16534 0 0 0 −14.1868 0
1.3 0 3.57956 0 2.16534 0 0 0 −14.1868 0
1.4 0 7.82220 0 9.23641 0 0 0 34.1868 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.bg 4
4.b odd 2 1 392.4.a.m 4
7.b odd 2 1 inner 784.4.a.bg 4
28.d even 2 1 392.4.a.m 4
28.f even 6 2 392.4.i.p 8
28.g odd 6 2 392.4.i.p 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.4.a.m 4 4.b odd 2 1
392.4.a.m 4 28.d even 2 1
392.4.i.p 8 28.f even 6 2
392.4.i.p 8 28.g odd 6 2
784.4.a.bg 4 1.a even 1 1 trivial
784.4.a.bg 4 7.b odd 2 1 inner

## Hecke kernels

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

 $$T_{3}^{4} - 74 T_{3}^{2} + 784$$ $$T_{5}^{4} - 90 T_{5}^{2} + 400$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$784 - 74 T^{2} + T^{4}$$
$5$ $$400 - 90 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -664 - 62 T + T^{2} )^{2}$$
$13$ $$6801664 - 5314 T^{2} + T^{4}$$
$17$ $$386884 - 7076 T^{2} + T^{4}$$
$19$ $$400 - 90 T^{2} + T^{4}$$
$23$ $$( -24976 + 64 T + T^{2} )^{2}$$
$29$ $$( -1296 + 126 T + T^{2} )^{2}$$
$31$ $$13778944 - 32904 T^{2} + T^{4}$$
$37$ $$( -46384 + 182 T + T^{2} )^{2}$$
$41$ $$27729576484 - 335124 T^{2} + T^{4}$$
$43$ $$( 64360 - 610 T + T^{2} )^{2}$$
$47$ $$14813810944 - 393576 T^{2} + T^{4}$$
$53$ $$( -102196 + 376 T + T^{2} )^{2}$$
$59$ $$12676057744 - 334506 T^{2} + T^{4}$$
$61$ $$3645744400 - 572010 T^{2} + T^{4}$$
$67$ $$( 449536 - 1372 T + T^{2} )^{2}$$
$71$ $$( 47360 - 860 T + T^{2} )^{2}$$
$73$ $$5553528484 - 175956 T^{2} + T^{4}$$
$79$ $$( 177824 - 844 T + T^{2} )^{2}$$
$83$ $$108576400 - 89610 T^{2} + T^{4}$$
$89$ $$569074096900 - 1517060 T^{2} + T^{4}$$
$97$ $$2024593185924 - 2887236 T^{2} + T^{4}$$