# Properties

 Label 1792.2.b.m 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: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 224) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} + q^{7} + ( -3 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( \beta_{1} + \beta_{2} ) q^{5} + q^{7} + ( -3 + \beta_{3} ) q^{9} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{11} + ( \beta_{1} - \beta_{2} ) q^{13} -4 q^{15} + \beta_{3} q^{17} -\beta_{1} q^{19} + \beta_{1} q^{21} + 4 q^{23} + ( -1 - \beta_{3} ) q^{25} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{27} + ( 2 \beta_{1} + \beta_{2} ) q^{29} + ( -2 + \beta_{3} ) q^{31} -8 q^{33} + ( \beta_{1} + \beta_{2} ) q^{35} + ( -2 \beta_{1} - \beta_{2} ) q^{37} + ( -8 + 2 \beta_{3} ) q^{39} + ( 4 + \beta_{3} ) q^{41} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -\beta_{1} + 3 \beta_{2} ) q^{45} + ( -6 - \beta_{3} ) q^{47} + q^{49} + ( -2 \beta_{1} + 4 \beta_{2} ) q^{51} -5 \beta_{2} q^{53} + ( -12 - 2 \beta_{3} ) q^{55} + ( 6 - \beta_{3} ) q^{57} + ( \beta_{1} + 4 \beta_{2} ) q^{59} + ( -\beta_{1} - 5 \beta_{2} ) q^{61} + ( -3 + \beta_{3} ) q^{63} + ( -2 + \beta_{3} ) q^{65} + 2 \beta_{2} q^{67} + 4 \beta_{1} q^{69} + ( 4 - 2 \beta_{3} ) q^{71} + ( -6 + 2 \beta_{3} ) q^{73} + ( \beta_{1} - 4 \beta_{2} ) q^{75} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{77} + ( -4 - 2 \beta_{3} ) q^{79} + ( 11 - 3 \beta_{3} ) q^{81} + ( -\beta_{1} - 4 \beta_{2} ) q^{83} + ( 2 \beta_{1} + 6 \beta_{2} ) q^{85} + ( -10 + \beta_{3} ) q^{87} + 6 q^{89} + ( \beta_{1} - \beta_{2} ) q^{91} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{93} + 4 q^{95} + ( 8 + \beta_{3} ) q^{97} + ( -2 \beta_{1} + 6 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{7} - 12q^{9} + O(q^{10})$$ $$4q + 4q^{7} - 12q^{9} - 16q^{15} + 16q^{23} - 4q^{25} - 8q^{31} - 32q^{33} - 32q^{39} + 16q^{41} - 24q^{47} + 4q^{49} - 48q^{55} + 24q^{57} - 12q^{63} - 8q^{65} + 16q^{71} - 24q^{73} - 16q^{79} + 44q^{81} - 40q^{87} + 24q^{89} + 16q^{95} + 32q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} + 4 \nu$$ $$\beta_{3}$$ $$=$$ $$4 \nu^{2} + 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 6$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{2} - 2 \beta_{1}$$$$)/2$$

## 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
 − 1.61803i − 0.618034i 0.618034i 1.61803i
0 3.23607i 0 1.23607i 0 1.00000 0 −7.47214 0
897.2 0 1.23607i 0 3.23607i 0 1.00000 0 1.47214 0
897.3 0 1.23607i 0 3.23607i 0 1.00000 0 1.47214 0
897.4 0 3.23607i 0 1.23607i 0 1.00000 0 −7.47214 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.m 4
4.b odd 2 1 1792.2.b.k 4
8.b even 2 1 inner 1792.2.b.m 4
8.d odd 2 1 1792.2.b.k 4
16.e even 4 1 224.2.a.d yes 2
16.e even 4 1 448.2.a.i 2
16.f odd 4 1 224.2.a.c 2
16.f odd 4 1 448.2.a.j 2
48.i odd 4 1 2016.2.a.o 2
48.i odd 4 1 4032.2.a.bv 2
48.k even 4 1 2016.2.a.r 2
48.k even 4 1 4032.2.a.bw 2
80.k odd 4 1 5600.2.a.bk 2
80.q even 4 1 5600.2.a.z 2
112.j even 4 1 1568.2.a.v 2
112.j even 4 1 3136.2.a.bf 2
112.l odd 4 1 1568.2.a.k 2
112.l odd 4 1 3136.2.a.by 2
112.u odd 12 2 1568.2.i.v 4
112.v even 12 2 1568.2.i.n 4
112.w even 12 2 1568.2.i.m 4
112.x odd 12 2 1568.2.i.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.c 2 16.f odd 4 1
224.2.a.d yes 2 16.e even 4 1
448.2.a.i 2 16.e even 4 1
448.2.a.j 2 16.f odd 4 1
1568.2.a.k 2 112.l odd 4 1
1568.2.a.v 2 112.j even 4 1
1568.2.i.m 4 112.w even 12 2
1568.2.i.n 4 112.v even 12 2
1568.2.i.v 4 112.u odd 12 2
1568.2.i.w 4 112.x odd 12 2
1792.2.b.k 4 4.b odd 2 1
1792.2.b.k 4 8.d odd 2 1
1792.2.b.m 4 1.a even 1 1 trivial
1792.2.b.m 4 8.b even 2 1 inner
2016.2.a.o 2 48.i odd 4 1
2016.2.a.r 2 48.k even 4 1
3136.2.a.bf 2 112.j even 4 1
3136.2.a.by 2 112.l odd 4 1
4032.2.a.bv 2 48.i odd 4 1
4032.2.a.bw 2 48.k even 4 1
5600.2.a.z 2 80.q even 4 1
5600.2.a.bk 2 80.k odd 4 1

## Hecke kernels

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

 $$T_{3}^{4} + 12 T_{3}^{2} + 16$$ $$T_{5}^{4} + 12 T_{5}^{2} + 16$$ $$T_{11}^{4} + 48 T_{11}^{2} + 256$$ $$T_{23} - 4$$ $$T_{31}^{2} + 4 T_{31} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 + 12 T^{2} + T^{4}$$
$5$ $$16 + 12 T^{2} + T^{4}$$
$7$ $$( -1 + T )^{4}$$
$11$ $$256 + 48 T^{2} + T^{4}$$
$13$ $$16 + 28 T^{2} + T^{4}$$
$17$ $$( -20 + T^{2} )^{2}$$
$19$ $$16 + 12 T^{2} + T^{4}$$
$23$ $$( -4 + T )^{4}$$
$29$ $$( 20 + T^{2} )^{2}$$
$31$ $$( -16 + 4 T + T^{2} )^{2}$$
$37$ $$( 20 + T^{2} )^{2}$$
$41$ $$( -4 - 8 T + T^{2} )^{2}$$
$43$ $$256 + 48 T^{2} + T^{4}$$
$47$ $$( 16 + 12 T + T^{2} )^{2}$$
$53$ $$( 100 + T^{2} )^{2}$$
$59$ $$1936 + 108 T^{2} + T^{4}$$
$61$ $$5776 + 172 T^{2} + T^{4}$$
$67$ $$( 16 + T^{2} )^{2}$$
$71$ $$( -64 - 8 T + T^{2} )^{2}$$
$73$ $$( -44 + 12 T + T^{2} )^{2}$$
$79$ $$( -64 + 8 T + T^{2} )^{2}$$
$83$ $$1936 + 108 T^{2} + T^{4}$$
$89$ $$( -6 + T )^{4}$$
$97$ $$( 44 - 16 T + T^{2} )^{2}$$