# Properties

 Label 1792.2.a.x Level $1792$ Weight $2$ Character orbit 1792.a Self dual yes Analytic conductor $14.309$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1792 = 2^{8} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1792.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.3091920422$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.9248.1 Defining polynomial: $$x^{4} - 5 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 56) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + \beta_{1} q^{5} + q^{7} + ( 2 - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + \beta_{1} q^{5} + q^{7} + ( 2 - \beta_{3} ) q^{9} + ( -\beta_{1} + \beta_{2} ) q^{11} + \beta_{1} q^{13} + ( 1 - \beta_{3} ) q^{15} + 2 q^{17} + \beta_{2} q^{19} + \beta_{2} q^{21} + ( -1 + \beta_{3} ) q^{23} + ( 2 + \beta_{3} ) q^{25} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{27} -2 \beta_{2} q^{29} + ( -2 + 2 \beta_{3} ) q^{31} + 4 q^{33} + \beta_{1} q^{35} -2 \beta_{2} q^{37} + ( 1 - \beta_{3} ) q^{39} + ( 4 + 2 \beta_{3} ) q^{41} + ( -\beta_{1} + \beta_{2} ) q^{43} + ( -\beta_{1} + 4 \beta_{2} ) q^{45} + q^{49} + 2 \beta_{2} q^{51} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{53} + ( -6 - 2 \beta_{3} ) q^{55} + ( 5 - \beta_{3} ) q^{57} + ( 2 \beta_{1} - \beta_{2} ) q^{59} + \beta_{1} q^{61} + ( 2 - \beta_{3} ) q^{63} + ( 7 + \beta_{3} ) q^{65} + ( -5 \beta_{1} - \beta_{2} ) q^{67} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{69} + 8 q^{71} + 6 q^{73} + ( -2 \beta_{1} - \beta_{2} ) q^{75} + ( -\beta_{1} + \beta_{2} ) q^{77} + ( 6 - \beta_{3} ) q^{81} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{83} + 2 \beta_{1} q^{85} + ( -10 + 2 \beta_{3} ) q^{87} + ( 8 - 2 \beta_{3} ) q^{89} + \beta_{1} q^{91} + ( -4 \beta_{1} - 8 \beta_{2} ) q^{93} + ( 1 - \beta_{3} ) q^{95} + ( 4 - 2 \beta_{3} ) q^{97} + ( 3 \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{7} + 8q^{9} + O(q^{10})$$ $$4q + 4q^{7} + 8q^{9} + 4q^{15} + 8q^{17} - 4q^{23} + 8q^{25} - 8q^{31} + 16q^{33} + 4q^{39} + 16q^{41} + 4q^{49} - 24q^{55} + 20q^{57} + 8q^{63} + 28q^{65} + 32q^{71} + 24q^{73} + 24q^{81} - 40q^{87} + 32q^{89} + 4q^{95} + 16q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - 3 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 5 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} - 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{2} + 5 \beta_{1}$$$$)/2$$

## 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
 0.662153 2.13578 −2.13578 −0.662153
0 −3.02045 0 −1.69614 0 1.00000 0 6.12311 0
1.2 0 −0.936426 0 3.33513 0 1.00000 0 −2.12311 0
1.3 0 0.936426 0 −3.33513 0 1.00000 0 −2.12311 0
1.4 0 3.02045 0 1.69614 0 1.00000 0 6.12311 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.a.x 4
4.b odd 2 1 1792.2.a.v 4
8.b even 2 1 inner 1792.2.a.x 4
8.d odd 2 1 1792.2.a.v 4
16.e even 4 2 56.2.b.b 4
16.f odd 4 2 224.2.b.b 4
48.i odd 4 2 504.2.c.d 4
48.k even 4 2 2016.2.c.c 4
112.j even 4 2 1568.2.b.d 4
112.l odd 4 2 392.2.b.c 4
112.u odd 12 4 1568.2.t.d 8
112.v even 12 4 1568.2.t.e 8
112.w even 12 4 392.2.p.f 8
112.x odd 12 4 392.2.p.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.b.b 4 16.e even 4 2
224.2.b.b 4 16.f odd 4 2
392.2.b.c 4 112.l odd 4 2
392.2.p.e 8 112.x odd 12 4
392.2.p.f 8 112.w even 12 4
504.2.c.d 4 48.i odd 4 2
1568.2.b.d 4 112.j even 4 2
1568.2.t.d 8 112.u odd 12 4
1568.2.t.e 8 112.v even 12 4
1792.2.a.v 4 4.b odd 2 1
1792.2.a.v 4 8.d odd 2 1
1792.2.a.x 4 1.a even 1 1 trivial
1792.2.a.x 4 8.b even 2 1 inner
2016.2.c.c 4 48.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1792))$$:

 $$T_{3}^{4} - 10 T_{3}^{2} + 8$$ $$T_{5}^{4} - 14 T_{5}^{2} + 32$$ $$T_{23}^{2} + 2 T_{23} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$8 - 10 T^{2} + T^{4}$$
$5$ $$32 - 14 T^{2} + T^{4}$$
$7$ $$( -1 + T )^{4}$$
$11$ $$32 - 20 T^{2} + T^{4}$$
$13$ $$32 - 14 T^{2} + T^{4}$$
$17$ $$( -2 + T )^{4}$$
$19$ $$8 - 10 T^{2} + T^{4}$$
$23$ $$( -16 + 2 T + T^{2} )^{2}$$
$29$ $$128 - 40 T^{2} + T^{4}$$
$31$ $$( -64 + 4 T + T^{2} )^{2}$$
$37$ $$128 - 40 T^{2} + T^{4}$$
$41$ $$( -52 - 8 T + T^{2} )^{2}$$
$43$ $$32 - 20 T^{2} + T^{4}$$
$47$ $$T^{4}$$
$53$ $$512 - 80 T^{2} + T^{4}$$
$59$ $$8 - 58 T^{2} + T^{4}$$
$61$ $$32 - 14 T^{2} + T^{4}$$
$67$ $$32768 - 380 T^{2} + T^{4}$$
$71$ $$( -8 + T )^{4}$$
$73$ $$( -6 + T )^{4}$$
$79$ $$T^{4}$$
$83$ $$2888 - 122 T^{2} + T^{4}$$
$89$ $$( -4 - 16 T + T^{2} )^{2}$$
$97$ $$( -52 - 8 T + T^{2} )^{2}$$