# Properties

 Label 1792.2.b.o Level $1792$ Weight $2$ Character orbit 1792.b Analytic conductor $14.309$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.399424.1 Defining polynomial: $$x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ 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 basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{3} + \beta_{4} q^{5} - q^{7} + ( -2 - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{5} q^{3} + \beta_{4} q^{5} - q^{7} + ( -2 - \beta_{3} ) q^{9} + ( \beta_{2} + \beta_{4} + \beta_{5} ) q^{11} + ( 2 \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{13} + ( 2 - \beta_{1} - 2 \beta_{3} ) q^{15} + ( -1 + \beta_{1} + \beta_{3} ) q^{17} + ( -2 \beta_{2} + \beta_{5} ) q^{19} -\beta_{5} q^{21} -\beta_{1} q^{23} + ( -6 + \beta_{3} ) q^{25} + ( -2 \beta_{4} - 2 \beta_{5} ) q^{27} + ( \beta_{2} - 2 \beta_{5} ) q^{29} + ( 1 - \beta_{1} + \beta_{3} ) q^{31} + ( -2 - 2 \beta_{3} ) q^{33} -\beta_{4} q^{35} + ( -\beta_{2} - 2 \beta_{4} ) q^{37} + ( -6 + \beta_{1} - 2 \beta_{3} ) q^{39} + ( -5 + \beta_{1} + \beta_{3} ) q^{41} + ( \beta_{2} - \beta_{4} + 3 \beta_{5} ) q^{43} + ( 4 \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{45} + ( 1 + \beta_{1} + \beta_{3} ) q^{47} + q^{49} + ( -4 \beta_{2} + 2 \beta_{4} ) q^{51} + \beta_{2} q^{53} + ( -8 - 2 \beta_{1} ) q^{55} + ( -7 - 2 \beta_{1} - 3 \beta_{3} ) q^{57} + ( 4 \beta_{2} + 2 \beta_{4} + 5 \beta_{5} ) q^{59} + ( 4 \beta_{2} - \beta_{4} ) q^{61} + ( 2 + \beta_{3} ) q^{63} + ( -5 - 4 \beta_{1} - \beta_{3} ) q^{65} + ( 3 \beta_{2} + \beta_{4} - \beta_{5} ) q^{67} + ( 4 \beta_{2} + 2 \beta_{5} ) q^{69} + 2 \beta_{3} q^{73} + ( 2 \beta_{4} - 3 \beta_{5} ) q^{75} + ( -\beta_{2} - \beta_{4} - \beta_{5} ) q^{77} + ( 4 + 2 \beta_{1} ) q^{79} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{81} + ( 2 \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{83} + ( 4 \beta_{2} + 6 \beta_{5} ) q^{85} + ( 11 + \beta_{1} + 3 \beta_{3} ) q^{87} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{89} + ( -2 \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{91} + ( 4 \beta_{2} + 2 \beta_{4} + 6 \beta_{5} ) q^{93} + ( \beta_{1} - 4 \beta_{3} ) q^{95} + ( -3 - 3 \beta_{1} - \beta_{3} ) q^{97} + ( 3 \beta_{2} - \beta_{4} - 5 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{7} - 14q^{9} + O(q^{10})$$ $$6q - 6q^{7} - 14q^{9} + 8q^{15} - 4q^{17} - 34q^{25} + 8q^{31} - 16q^{33} - 40q^{39} - 28q^{41} + 8q^{47} + 6q^{49} - 48q^{55} - 48q^{57} + 14q^{63} - 32q^{65} + 4q^{73} + 24q^{79} + 6q^{81} + 72q^{87} + 4q^{89} - 8q^{95} - 20q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 3 x^{4} - 6 x^{3} + 6 x^{2} - 8 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{4} + 2 \nu^{3} - \nu^{2} + 2 \nu - 2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} - 3 \nu^{3} + 4 \nu^{2} - 2 \nu + 8$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} + 2 \nu^{4} - 3 \nu^{3} + 6 \nu^{2} - 2 \nu + 6$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{5} + 5 \nu^{3} - 4 \nu^{2} - 6 \nu - 12$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{5} - 4 \nu^{4} + 11 \nu^{3} - 16 \nu^{2} + 14 \nu - 28$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + \beta_{3} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_{1} - 2$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_{1} + 5$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{5} - \beta_{4} + 2 \beta_{3} - 6 \beta_{2} - \beta_{1} + 6$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 6 \beta_{2} - 2 \beta_{1} + 7$$$$)/4$$

## 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
 1.40680 − 0.144584i 0.264658 − 1.38923i −0.671462 − 1.24464i −0.671462 + 1.24464i 0.264658 + 1.38923i 1.40680 + 0.144584i
0 3.10278i 0 2.52444i 0 −1.00000 0 −6.62721 0
897.2 0 2.24914i 0 3.30777i 0 −1.00000 0 −2.05863 0
897.3 0 1.14637i 0 3.83221i 0 −1.00000 0 1.68585 0
897.4 0 1.14637i 0 3.83221i 0 −1.00000 0 1.68585 0
897.5 0 2.24914i 0 3.30777i 0 −1.00000 0 −2.05863 0
897.6 0 3.10278i 0 2.52444i 0 −1.00000 0 −6.62721 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 897.6 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.o 6
4.b odd 2 1 1792.2.b.p 6
8.b even 2 1 inner 1792.2.b.o 6
8.d odd 2 1 1792.2.b.p 6
16.e even 4 1 896.2.a.j yes 3
16.e even 4 1 896.2.a.k yes 3
16.f odd 4 1 896.2.a.i 3
16.f odd 4 1 896.2.a.l yes 3
48.i odd 4 1 8064.2.a.cb 3
48.i odd 4 1 8064.2.a.ch 3
48.k even 4 1 8064.2.a.bu 3
48.k even 4 1 8064.2.a.ce 3
112.j even 4 1 6272.2.a.u 3
112.j even 4 1 6272.2.a.x 3
112.l odd 4 1 6272.2.a.v 3
112.l odd 4 1 6272.2.a.w 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.a.i 3 16.f odd 4 1
896.2.a.j yes 3 16.e even 4 1
896.2.a.k yes 3 16.e even 4 1
896.2.a.l yes 3 16.f odd 4 1
1792.2.b.o 6 1.a even 1 1 trivial
1792.2.b.o 6 8.b even 2 1 inner
1792.2.b.p 6 4.b odd 2 1
1792.2.b.p 6 8.d odd 2 1
6272.2.a.u 3 112.j even 4 1
6272.2.a.v 3 112.l odd 4 1
6272.2.a.w 3 112.l odd 4 1
6272.2.a.x 3 112.j even 4 1
8064.2.a.bu 3 48.k even 4 1
8064.2.a.cb 3 48.i odd 4 1
8064.2.a.ce 3 48.k even 4 1
8064.2.a.ch 3 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 $$S_{2}^{\mathrm{new}}(1792, [\chi])$$:

 $$T_{3}^{6} + 16 T_{3}^{4} + 68 T_{3}^{2} + 64$$ $$T_{5}^{6} + 32 T_{5}^{4} + 324 T_{5}^{2} + 1024$$ $$T_{11}^{6} + 36 T_{11}^{4} + 320 T_{11}^{2} + 256$$ $$T_{23}^{3} - 28 T_{23} - 16$$ $$T_{31}^{3} - 4 T_{31}^{2} - 56 T_{31} + 256$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$64 + 68 T^{2} + 16 T^{4} + T^{6}$$
$5$ $$1024 + 324 T^{2} + 32 T^{4} + T^{6}$$
$7$ $$( 1 + T )^{6}$$
$11$ $$256 + 320 T^{2} + 36 T^{4} + T^{6}$$
$13$ $$18496 + 2116 T^{2} + 80 T^{4} + T^{6}$$
$17$ $$( 8 - 28 T + 2 T^{2} + T^{3} )^{2}$$
$19$ $$4096 + 1572 T^{2} + 80 T^{4} + T^{6}$$
$23$ $$( -16 - 28 T + T^{3} )^{2}$$
$29$ $$7744 + 1776 T^{2} + 92 T^{4} + T^{6}$$
$31$ $$( 256 - 56 T - 4 T^{2} + T^{3} )^{2}$$
$37$ $$18496 + 3056 T^{2} + 124 T^{4} + T^{6}$$
$41$ $$( -8 + 36 T + 14 T^{2} + T^{3} )^{2}$$
$43$ $$92416 + 8000 T^{2} + 196 T^{4} + T^{6}$$
$47$ $$( 64 - 24 T - 4 T^{2} + T^{3} )^{2}$$
$53$ $$( 4 + T^{2} )^{3}$$
$59$ $$2483776 + 56484 T^{2} + 416 T^{4} + T^{6}$$
$61$ $$246016 + 18308 T^{2} + 256 T^{4} + T^{6}$$
$67$ $$1024 + 5248 T^{2} + 164 T^{4} + T^{6}$$
$71$ $$T^{6}$$
$73$ $$( 8 - 68 T - 2 T^{2} + T^{3} )^{2}$$
$79$ $$( 512 - 64 T - 12 T^{2} + T^{3} )^{2}$$
$83$ $$135424 + 10500 T^{2} + 224 T^{4} + T^{6}$$
$89$ $$( 296 - 116 T - 2 T^{2} + T^{3} )^{2}$$
$97$ $$( -1816 - 188 T + 10 T^{2} + T^{3} )^{2}$$