# Properties

 Label 784.4.a.bf Level $784$ Weight $4$ Character orbit 784.a Self dual yes Analytic conductor $46.257$ Analytic rank $1$ 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: $$1$$ 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: $$2\cdot 7$$ Twist minimal: no (minimal twist has level 49) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + \beta_{3} q^{5} + ( 10 - 3 \beta_{1} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + \beta_{3} q^{5} + ( 10 - 3 \beta_{1} ) q^{9} + ( -25 + 3 \beta_{1} ) q^{11} + ( 6 \beta_{2} - \beta_{3} ) q^{13} + ( 16 + 4 \beta_{1} ) q^{15} + ( -\beta_{2} - 9 \beta_{3} ) q^{17} + ( 11 \beta_{2} - 4 \beta_{3} ) q^{19} + ( -88 - 4 \beta_{1} ) q^{23} + ( -32 + 11 \beta_{1} ) q^{25} + ( -4 \beta_{2} - 12 \beta_{3} ) q^{27} + ( 65 - 7 \beta_{1} ) q^{29} + ( 38 \beta_{2} + 4 \beta_{3} ) q^{31} + ( 46 \beta_{2} + 12 \beta_{3} ) q^{33} + ( 53 - 3 \beta_{1} ) q^{37} + ( -238 + 14 \beta_{1} ) q^{39} + ( -31 \beta_{2} - 3 \beta_{3} ) q^{41} + ( -135 - 35 \beta_{1} ) q^{43} + ( 12 \beta_{2} - 11 \beta_{3} ) q^{45} + ( 26 \beta_{2} + 32 \beta_{3} ) q^{47} + ( -107 - 39 \beta_{1} ) q^{51} + ( 4 + 18 \beta_{1} ) q^{53} + ( -12 \beta_{2} - 4 \beta_{3} ) q^{55} + ( -471 + 17 \beta_{1} ) q^{57} + ( -49 \beta_{2} - 20 \beta_{3} ) q^{59} + ( 68 \beta_{2} + 27 \beta_{3} ) q^{61} + ( -189 - 35 \beta_{1} ) q^{65} + ( 486 + 38 \beta_{1} ) q^{67} + ( 60 \beta_{2} - 16 \beta_{3} ) q^{69} + ( -562 + 14 \beta_{1} ) q^{71} + ( -37 \beta_{2} + 25 \beta_{3} ) q^{73} + ( 109 \beta_{2} + 44 \beta_{3} ) q^{75} + ( 262 + 94 \beta_{1} ) q^{79} + ( -314 + 21 \beta_{1} ) q^{81} + ( -\beta_{2} - 8 \beta_{3} ) q^{83} + ( -821 - 95 \beta_{1} ) q^{85} + ( -114 \beta_{2} - 28 \beta_{3} ) q^{87} + ( -157 \beta_{2} + 75 \beta_{3} ) q^{89} + ( -1342 + 130 \beta_{1} ) q^{93} + ( -548 - 88 \beta_{1} ) q^{95} + ( -189 \beta_{2} - 91 \beta_{3} ) q^{97} + ( -835 + 105 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 40q^{9} + O(q^{10})$$ $$4q + 40q^{9} - 100q^{11} + 64q^{15} - 352q^{23} - 128q^{25} + 260q^{29} + 212q^{37} - 952q^{39} - 540q^{43} - 428q^{51} + 16q^{53} - 1884q^{57} - 756q^{65} + 1944q^{67} - 2248q^{71} + 1048q^{79} - 1256q^{81} - 3284q^{85} - 5368q^{93} - 2192q^{95} - 3340q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-4 \nu^{3} + 6 \nu^{2} + 200 \nu - 101$$$$)/57$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 11 \nu^{2} - 12 \nu + 182$$$$)/19$$ $$\beta_{3}$$ $$=$$ $$($$$$11 \nu^{3} + 12 \nu^{2} - 265 \nu - 392$$$$)/57$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{3} + 2 \beta_{2} + 7 \beta_{1} + 7$$$$)/14$$ $$\nu^{2}$$ $$=$$ $$($$$$8 \beta_{3} - 20 \beta_{2} + 7 \beta_{1} + 259$$$$)/14$$ $$\nu^{3}$$ $$=$$ $$($$$$16 \beta_{3} + 10 \beta_{2} + 23 \beta_{1} + 55$$$$)/2$$

## 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 2.07730 0 0 0 34.1868 0
1.2 0 −3.57956 0 −13.4791 0 0 0 −14.1868 0
1.3 0 3.57956 0 13.4791 0 0 0 −14.1868 0
1.4 0 7.82220 0 −2.07730 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.bf 4
4.b odd 2 1 49.4.a.e 4
7.b odd 2 1 inner 784.4.a.bf 4
12.b even 2 1 441.4.a.u 4
20.d odd 2 1 1225.4.a.bb 4
28.d even 2 1 49.4.a.e 4
28.f even 6 2 49.4.c.e 8
28.g odd 6 2 49.4.c.e 8
84.h odd 2 1 441.4.a.u 4
84.j odd 6 2 441.4.e.y 8
84.n even 6 2 441.4.e.y 8
140.c even 2 1 1225.4.a.bb 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.a.e 4 4.b odd 2 1
49.4.a.e 4 28.d even 2 1
49.4.c.e 8 28.f even 6 2
49.4.c.e 8 28.g odd 6 2
441.4.a.u 4 12.b even 2 1
441.4.a.u 4 84.h odd 2 1
441.4.e.y 8 84.j odd 6 2
441.4.e.y 8 84.n even 6 2
784.4.a.bf 4 1.a even 1 1 trivial
784.4.a.bf 4 7.b odd 2 1 inner
1225.4.a.bb 4 20.d odd 2 1
1225.4.a.bb 4 140.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(784))$$:

 $$T_{3}^{4} - 74 T_{3}^{2} + 784$$ $$T_{5}^{4} - 186 T_{5}^{2} + 784$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$784 - 74 T^{2} + T^{4}$$
$5$ $$784 - 186 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$( 40 + 50 T + T^{2} )^{2}$$
$13$ $$2458624 - 3234 T^{2} + T^{4}$$
$17$ $$9746884 - 14564 T^{2} + T^{4}$$
$19$ $$52591504 - 14746 T^{2} + T^{4}$$
$23$ $$( 6704 + 176 T + T^{2} )^{2}$$
$29$ $$( 1040 - 130 T + T^{2} )^{2}$$
$31$ $$629407744 - 100104 T^{2} + T^{4}$$
$37$ $$( 2224 - 106 T + T^{2} )^{2}$$
$41$ $$307721764 - 66836 T^{2} + T^{4}$$
$43$ $$( -61400 + 270 T + T^{2} )^{2}$$
$47$ $$8332038400 - 187240 T^{2} + T^{4}$$
$53$ $$( -21044 - 8 T + T^{2} )^{2}$$
$59$ $$1600960144 - 189354 T^{2} + T^{4}$$
$61$ $$5022273424 - 360266 T^{2} + T^{4}$$
$67$ $$( 142336 - 972 T + T^{2} )^{2}$$
$71$ $$( 303104 + 1124 T + T^{2} )^{2}$$
$73$ $$12428236324 - 276756 T^{2} + T^{4}$$
$79$ $$( -505696 - 524 T + T^{2} )^{2}$$
$83$ $$6492304 - 11466 T^{2} + T^{4}$$
$89$ $$2844693770884 - 3623876 T^{2} + T^{4}$$
$97$ $$841222821124 - 3082884 T^{2} + T^{4}$$