# Properties

 Label 1792.2.f.j Level $1792$ Weight $2$ Character orbit 1792.f Analytic conductor $14.309$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$14.3091920422$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 448) 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 $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{3} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{7} - q^{9} +O(q^{10})$$ $$q + ( \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{3} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} ) q^{5} + ( -\zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{7} - q^{9} + ( -2 + 4 \zeta_{24}^{4} ) q^{11} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{13} -2 \zeta_{24}^{6} q^{15} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{17} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{19} + ( \zeta_{24} - 4 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{21} + 3 q^{25} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{27} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{31} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{33} + ( -2 - \zeta_{24} - \zeta_{24}^{3} + 4 \zeta_{24}^{4} + \zeta_{24}^{5} ) q^{35} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{37} + 6 \zeta_{24}^{6} q^{39} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{41} + ( -2 + 4 \zeta_{24}^{4} ) q^{43} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{45} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{47} + ( 5 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{49} + ( -4 + 8 \zeta_{24}^{4} ) q^{51} + ( 16 \zeta_{24}^{2} - 8 \zeta_{24}^{6} ) q^{53} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{55} + 6 q^{57} + ( 5 \zeta_{24} + 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} ) q^{59} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{61} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{63} + 6 q^{65} + ( 6 - 12 \zeta_{24}^{4} ) q^{67} -6 \zeta_{24}^{6} q^{71} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{73} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{75} + ( 6 \zeta_{24} + 4 \zeta_{24}^{2} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{77} + 14 \zeta_{24}^{6} q^{79} -5 q^{81} + ( -7 \zeta_{24} - 7 \zeta_{24}^{3} + 7 \zeta_{24}^{5} ) q^{83} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{85} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{89} + ( 6 + 3 \zeta_{24} + 3 \zeta_{24}^{3} - 12 \zeta_{24}^{4} - 3 \zeta_{24}^{5} ) q^{91} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{93} -6 \zeta_{24}^{6} q^{95} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{97} + ( 2 - 4 \zeta_{24}^{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{9} + O(q^{10})$$ $$8q - 8q^{9} + 24q^{25} + 40q^{49} + 48q^{57} + 48q^{65} - 40q^{81} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1791.1
 0.258819 − 0.965926i −0.965926 + 0.258819i 0.258819 + 0.965926i −0.965926 − 0.258819i 0.965926 + 0.258819i −0.258819 − 0.965926i 0.965926 − 0.258819i −0.258819 + 0.965926i
0 −1.41421 0 1.41421i 0 −2.44949 + 1.00000i 0 −1.00000 0
1791.2 0 −1.41421 0 1.41421i 0 2.44949 + 1.00000i 0 −1.00000 0
1791.3 0 −1.41421 0 1.41421i 0 −2.44949 1.00000i 0 −1.00000 0
1791.4 0 −1.41421 0 1.41421i 0 2.44949 1.00000i 0 −1.00000 0
1791.5 0 1.41421 0 1.41421i 0 −2.44949 1.00000i 0 −1.00000 0
1791.6 0 1.41421 0 1.41421i 0 2.44949 1.00000i 0 −1.00000 0
1791.7 0 1.41421 0 1.41421i 0 −2.44949 + 1.00000i 0 −1.00000 0
1791.8 0 1.41421 0 1.41421i 0 2.44949 + 1.00000i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1791.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
28.d even 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.f.j 8
4.b odd 2 1 inner 1792.2.f.j 8
7.b odd 2 1 inner 1792.2.f.j 8
8.b even 2 1 inner 1792.2.f.j 8
8.d odd 2 1 inner 1792.2.f.j 8
16.e even 4 2 448.2.e.b 8
16.f odd 4 2 448.2.e.b 8
28.d even 2 1 inner 1792.2.f.j 8
48.i odd 4 2 4032.2.p.f 8
48.k even 4 2 4032.2.p.f 8
56.e even 2 1 inner 1792.2.f.j 8
56.h odd 2 1 inner 1792.2.f.j 8
112.j even 4 2 448.2.e.b 8
112.l odd 4 2 448.2.e.b 8
336.v odd 4 2 4032.2.p.f 8
336.y even 4 2 4032.2.p.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.e.b 8 16.e even 4 2
448.2.e.b 8 16.f odd 4 2
448.2.e.b 8 112.j even 4 2
448.2.e.b 8 112.l odd 4 2
1792.2.f.j 8 1.a even 1 1 trivial
1792.2.f.j 8 4.b odd 2 1 inner
1792.2.f.j 8 7.b odd 2 1 inner
1792.2.f.j 8 8.b even 2 1 inner
1792.2.f.j 8 8.d odd 2 1 inner
1792.2.f.j 8 28.d even 2 1 inner
1792.2.f.j 8 56.e even 2 1 inner
1792.2.f.j 8 56.h odd 2 1 inner
4032.2.p.f 8 48.i odd 4 2
4032.2.p.f 8 48.k even 4 2
4032.2.p.f 8 336.v odd 4 2
4032.2.p.f 8 336.y even 4 2

## Hecke kernels

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

 $$T_{3}^{2} - 2$$ $$T_{5}^{2} + 2$$ $$T_{29}$$ $$T_{31}^{2} - 24$$ $$T_{37}^{2} - 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( -2 + T^{2} )^{4}$$
$5$ $$( 2 + T^{2} )^{4}$$
$7$ $$( 49 - 10 T^{2} + T^{4} )^{2}$$
$11$ $$( 12 + T^{2} )^{4}$$
$13$ $$( 18 + T^{2} )^{4}$$
$17$ $$( 24 + T^{2} )^{4}$$
$19$ $$( -18 + T^{2} )^{4}$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$( -24 + T^{2} )^{4}$$
$37$ $$( -48 + T^{2} )^{4}$$
$41$ $$( 96 + T^{2} )^{4}$$
$43$ $$( 12 + T^{2} )^{4}$$
$47$ $$( -24 + T^{2} )^{4}$$
$53$ $$( -192 + T^{2} )^{4}$$
$59$ $$( -50 + T^{2} )^{4}$$
$61$ $$( 18 + T^{2} )^{4}$$
$67$ $$( 108 + T^{2} )^{4}$$
$71$ $$( 36 + T^{2} )^{4}$$
$73$ $$( 24 + T^{2} )^{4}$$
$79$ $$( 196 + T^{2} )^{4}$$
$83$ $$( -98 + T^{2} )^{4}$$
$89$ $$( 216 + T^{2} )^{4}$$
$97$ $$( 24 + T^{2} )^{4}$$