# Properties

 Label 1792.2.b.l Level $1792$ Weight $2$ Character orbit 1792.b 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.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.3091920422$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 896) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( 1 - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{5} - q^{7} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{3} + ( 1 - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{5} - q^{7} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{9} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{11} + ( -1 + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{13} -2 q^{15} + ( 4 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{17} + ( -3 + 6 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{19} + ( -1 + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{21} + ( 2 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{23} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{25} -4 \zeta_{12}^{3} q^{27} + ( -2 + 4 \zeta_{12}^{2} ) q^{29} + ( -6 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{31} + ( 8 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{33} + ( -1 + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{35} + ( -2 + 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{37} + 2 q^{39} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{41} + ( -2 + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{43} + ( 1 - 2 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{45} + ( -2 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{47} + q^{49} + ( 2 - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{51} -6 \zeta_{12}^{3} q^{53} + 4 q^{55} + ( 8 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{57} + ( 3 - 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{59} + ( 7 - 14 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{61} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{63} + ( 4 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{65} + ( -4 + 8 \zeta_{12}^{2} + 8 \zeta_{12}^{3} ) q^{67} + ( -2 + 4 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{69} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{71} + ( -2 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{73} + ( 3 - 6 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{75} + ( 2 - 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{77} -8 q^{79} + ( 1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{81} + ( -1 + 2 \zeta_{12}^{2} - 9 \zeta_{12}^{3} ) q^{83} + ( 6 - 12 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{85} + ( 6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{87} + ( 2 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{89} + ( 1 - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{91} + ( -8 + 16 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{93} + ( 10 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{95} + ( 4 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{97} + ( 6 - 12 \zeta_{12}^{2} + 14 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{7} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{7} - 4q^{9} - 8q^{15} + 16q^{17} + 8q^{23} + 4q^{25} - 24q^{31} + 32q^{33} + 8q^{39} - 8q^{47} + 4q^{49} + 16q^{55} + 32q^{57} + 4q^{63} + 16q^{65} - 8q^{73} - 32q^{79} + 4q^{81} + 24q^{87} + 8q^{89} + 40q^{95} + 16q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$$.

 $$n$$ $$1023$$ $$1025$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
897.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 2.73205i 0 0.732051i 0 −1.00000 0 −4.46410 0
897.2 0 0.732051i 0 2.73205i 0 −1.00000 0 2.46410 0
897.3 0 0.732051i 0 2.73205i 0 −1.00000 0 2.46410 0
897.4 0 2.73205i 0 0.732051i 0 −1.00000 0 −4.46410 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.b.l 4
4.b odd 2 1 1792.2.b.n 4
8.b even 2 1 inner 1792.2.b.l 4
8.d odd 2 1 1792.2.b.n 4
16.e even 4 1 896.2.a.e 2
16.e even 4 1 896.2.a.h yes 2
16.f odd 4 1 896.2.a.f yes 2
16.f odd 4 1 896.2.a.g yes 2
48.i odd 4 1 8064.2.a.bf 2
48.i odd 4 1 8064.2.a.br 2
48.k even 4 1 8064.2.a.be 2
48.k even 4 1 8064.2.a.bm 2
112.j even 4 1 6272.2.a.j 2
112.j even 4 1 6272.2.a.s 2
112.l odd 4 1 6272.2.a.i 2
112.l odd 4 1 6272.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.a.e 2 16.e even 4 1
896.2.a.f yes 2 16.f odd 4 1
896.2.a.g yes 2 16.f odd 4 1
896.2.a.h yes 2 16.e even 4 1
1792.2.b.l 4 1.a even 1 1 trivial
1792.2.b.l 4 8.b even 2 1 inner
1792.2.b.n 4 4.b odd 2 1
1792.2.b.n 4 8.d odd 2 1
6272.2.a.i 2 112.l odd 4 1
6272.2.a.j 2 112.j even 4 1
6272.2.a.s 2 112.j even 4 1
6272.2.a.t 2 112.l odd 4 1
8064.2.a.be 2 48.k even 4 1
8064.2.a.bf 2 48.i odd 4 1
8064.2.a.bm 2 48.k even 4 1
8064.2.a.br 2 48.i odd 4 1

## Hecke kernels

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

 $$T_{3}^{4} + 8 T_{3}^{2} + 4$$ $$T_{5}^{4} + 8 T_{5}^{2} + 4$$ $$T_{11}^{4} + 32 T_{11}^{2} + 64$$ $$T_{23}^{2} - 4 T_{23} - 44$$ $$T_{31}^{2} + 12 T_{31} + 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$4 + 8 T^{2} + T^{4}$$
$5$ $$4 + 8 T^{2} + T^{4}$$
$7$ $$( 1 + T )^{4}$$
$11$ $$64 + 32 T^{2} + T^{4}$$
$13$ $$4 + 8 T^{2} + T^{4}$$
$17$ $$( 4 - 8 T + T^{2} )^{2}$$
$19$ $$676 + 56 T^{2} + T^{4}$$
$23$ $$( -44 - 4 T + T^{2} )^{2}$$
$29$ $$( 12 + T^{2} )^{2}$$
$31$ $$( 24 + 12 T + T^{2} )^{2}$$
$37$ $$2704 + 152 T^{2} + T^{4}$$
$41$ $$( -12 + T^{2} )^{2}$$
$43$ $$576 + 96 T^{2} + T^{4}$$
$47$ $$( -8 + 4 T + T^{2} )^{2}$$
$53$ $$( 36 + T^{2} )^{2}$$
$59$ $$676 + 56 T^{2} + T^{4}$$
$61$ $$21316 + 296 T^{2} + T^{4}$$
$67$ $$256 + 224 T^{2} + T^{4}$$
$71$ $$( -192 + T^{2} )^{2}$$
$73$ $$( -44 + 4 T + T^{2} )^{2}$$
$79$ $$( 8 + T )^{4}$$
$83$ $$6084 + 168 T^{2} + T^{4}$$
$89$ $$( -44 - 4 T + T^{2} )^{2}$$
$97$ $$( 4 - 8 T + T^{2} )^{2}$$