# Properties

 Label 1792.1.o.a Level $1792$ Weight $1$ Character orbit 1792.o Analytic conductor $0.894$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1792 = 2^{8} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1792.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.894324502638$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 224) Projective image: $$A_{4}$$ Projective field: Galois closure of 4.0.3136.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{4} q^{3} + \zeta_{12}^{5} q^{5} -\zeta_{12}^{3} q^{7} +O(q^{10})$$ $$q + \zeta_{12}^{4} q^{3} + \zeta_{12}^{5} q^{5} -\zeta_{12}^{3} q^{7} + \zeta_{12}^{4} q^{11} -\zeta_{12}^{3} q^{15} + \zeta_{12}^{4} q^{17} -\zeta_{12}^{2} q^{19} + \zeta_{12} q^{21} + \zeta_{12}^{5} q^{23} - q^{27} -\zeta_{12} q^{31} -\zeta_{12}^{2} q^{33} + \zeta_{12}^{2} q^{35} -\zeta_{12}^{5} q^{37} + \zeta_{12}^{5} q^{47} - q^{49} -\zeta_{12}^{2} q^{51} -\zeta_{12} q^{53} -\zeta_{12}^{3} q^{55} + q^{57} -\zeta_{12}^{4} q^{59} + \zeta_{12}^{5} q^{61} + \zeta_{12}^{4} q^{67} -\zeta_{12}^{3} q^{69} + \zeta_{12}^{4} q^{73} + \zeta_{12} q^{77} -\zeta_{12}^{5} q^{79} -\zeta_{12}^{4} q^{81} -\zeta_{12}^{3} q^{85} -\zeta_{12}^{2} q^{89} -\zeta_{12}^{5} q^{93} + \zeta_{12} q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + O(q^{10})$$ $$4 q - 2 q^{3} - 2 q^{11} - 2 q^{17} - 2 q^{19} - 4 q^{27} - 2 q^{33} + 2 q^{35} - 4 q^{49} - 2 q^{51} + 4 q^{57} + 2 q^{59} - 2 q^{67} - 2 q^{73} + 2 q^{81} - 2 q^{89} + O(q^{100})$$

## Character values

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

 $$n$$ $$1023$$ $$1025$$ $$1541$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
639.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
0 −0.500000 0.866025i 0 −0.866025 0.500000i 0 1.00000i 0 0 0
639.2 0 −0.500000 0.866025i 0 0.866025 + 0.500000i 0 1.00000i 0 0 0
1663.1 0 −0.500000 + 0.866025i 0 −0.866025 + 0.500000i 0 1.00000i 0 0 0
1663.2 0 −0.500000 + 0.866025i 0 0.866025 0.500000i 0 1.00000i 0 0 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
7.c even 3 1 inner
8.d odd 2 1 inner
56.k odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.1.o.a 4
4.b odd 2 1 1792.1.o.b 4
7.c even 3 1 inner 1792.1.o.a 4
8.b even 2 1 1792.1.o.b 4
8.d odd 2 1 inner 1792.1.o.a 4
16.e even 4 1 224.1.r.a 4
16.e even 4 1 448.1.r.a 4
16.f odd 4 1 224.1.r.a 4
16.f odd 4 1 448.1.r.a 4
28.g odd 6 1 1792.1.o.b 4
48.i odd 4 1 2016.1.cd.a 4
48.k even 4 1 2016.1.cd.a 4
56.k odd 6 1 inner 1792.1.o.a 4
56.p even 6 1 1792.1.o.b 4
112.j even 4 1 1568.1.r.a 4
112.j even 4 1 3136.1.r.b 4
112.l odd 4 1 1568.1.r.a 4
112.l odd 4 1 3136.1.r.b 4
112.u odd 12 1 224.1.r.a 4
112.u odd 12 1 448.1.r.a 4
112.u odd 12 1 1568.1.d.b 2
112.u odd 12 1 3136.1.d.b 2
112.v even 12 1 1568.1.d.a 2
112.v even 12 1 1568.1.r.a 4
112.v even 12 1 3136.1.d.d 2
112.v even 12 1 3136.1.r.b 4
112.w even 12 1 224.1.r.a 4
112.w even 12 1 448.1.r.a 4
112.w even 12 1 1568.1.d.b 2
112.w even 12 1 3136.1.d.b 2
112.x odd 12 1 1568.1.d.a 2
112.x odd 12 1 1568.1.r.a 4
112.x odd 12 1 3136.1.d.d 2
112.x odd 12 1 3136.1.r.b 4
336.bt odd 12 1 2016.1.cd.a 4
336.bu even 12 1 2016.1.cd.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.1.r.a 4 16.e even 4 1
224.1.r.a 4 16.f odd 4 1
224.1.r.a 4 112.u odd 12 1
224.1.r.a 4 112.w even 12 1
448.1.r.a 4 16.e even 4 1
448.1.r.a 4 16.f odd 4 1
448.1.r.a 4 112.u odd 12 1
448.1.r.a 4 112.w even 12 1
1568.1.d.a 2 112.v even 12 1
1568.1.d.a 2 112.x odd 12 1
1568.1.d.b 2 112.u odd 12 1
1568.1.d.b 2 112.w even 12 1
1568.1.r.a 4 112.j even 4 1
1568.1.r.a 4 112.l odd 4 1
1568.1.r.a 4 112.v even 12 1
1568.1.r.a 4 112.x odd 12 1
1792.1.o.a 4 1.a even 1 1 trivial
1792.1.o.a 4 7.c even 3 1 inner
1792.1.o.a 4 8.d odd 2 1 inner
1792.1.o.a 4 56.k odd 6 1 inner
1792.1.o.b 4 4.b odd 2 1
1792.1.o.b 4 8.b even 2 1
1792.1.o.b 4 28.g odd 6 1
1792.1.o.b 4 56.p even 6 1
2016.1.cd.a 4 48.i odd 4 1
2016.1.cd.a 4 48.k even 4 1
2016.1.cd.a 4 336.bt odd 12 1
2016.1.cd.a 4 336.bu even 12 1
3136.1.d.b 2 112.u odd 12 1
3136.1.d.b 2 112.w even 12 1
3136.1.d.d 2 112.v even 12 1
3136.1.d.d 2 112.x odd 12 1
3136.1.r.b 4 112.j even 4 1
3136.1.r.b 4 112.l odd 4 1
3136.1.r.b 4 112.v even 12 1
3136.1.r.b 4 112.x odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1792, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$1 - T^{2} + T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( 1 + T + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( 1 + T + T^{2} )^{2}$$
$19$ $$( 1 + T + T^{2} )^{2}$$
$23$ $$1 - T^{2} + T^{4}$$
$29$ $$T^{4}$$
$31$ $$1 - T^{2} + T^{4}$$
$37$ $$1 - T^{2} + T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$1 - T^{2} + T^{4}$$
$53$ $$1 - T^{2} + T^{4}$$
$59$ $$( 1 - T + T^{2} )^{2}$$
$61$ $$1 - T^{2} + T^{4}$$
$67$ $$( 1 + T + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$( 1 + T + T^{2} )^{2}$$
$79$ $$1 - T^{2} + T^{4}$$
$83$ $$T^{4}$$
$89$ $$( 1 + T + T^{2} )^{2}$$
$97$ $$T^{4}$$