# Properties

 Label 1792.2.f.l Level $1792$ Weight $2$ Character orbit 1792.f Analytic conductor $14.309$ Analytic rank $0$ Dimension $8$ CM discriminant -56 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: 8.0.2517630976.5 Defining polynomial: $$x^{8} - 2 x^{6} + 11 x^{4} + 4 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: no (minimal twist has level 896) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + \beta_{5} q^{5} -\beta_{2} q^{7} + ( 3 + \beta_{1} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{3} + \beta_{5} q^{5} -\beta_{2} q^{7} + ( 3 + \beta_{1} ) q^{9} + ( -\beta_{4} - \beta_{5} ) q^{13} + ( -2 \beta_{2} + \beta_{6} ) q^{15} -\beta_{7} q^{19} + ( -\beta_{4} + \beta_{5} ) q^{21} -\beta_{6} q^{23} + ( -5 - 3 \beta_{1} ) q^{25} + ( \beta_{3} - \beta_{7} ) q^{27} + ( -2 \beta_{3} + \beta_{7} ) q^{35} + ( -2 \beta_{2} - \beta_{6} ) q^{39} + ( -\beta_{4} + 5 \beta_{5} ) q^{45} -7 q^{49} + ( 2 + 5 \beta_{1} ) q^{57} + ( 4 \beta_{3} + \beta_{7} ) q^{59} + ( 2 \beta_{4} - \beta_{5} ) q^{61} + ( -3 \beta_{2} + \beta_{6} ) q^{63} + ( 6 - \beta_{1} ) q^{65} + ( -\beta_{4} - 6 \beta_{5} ) q^{69} -6 \beta_{2} q^{71} + ( -8 \beta_{3} + 3 \beta_{7} ) q^{75} -2 \beta_{2} q^{79} + ( -1 + 3 \beta_{1} ) q^{81} + ( -\beta_{3} + 2 \beta_{7} ) q^{83} + ( -3 \beta_{3} - 2 \beta_{7} ) q^{91} + ( -6 \beta_{2} + \beta_{6} ) q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 24q^{9} + O(q^{10})$$ $$8q + 24q^{9} - 40q^{25} - 56q^{49} + 16q^{57} + 48q^{65} - 8q^{81} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{6} + 11 x^{4} + 4 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 2 \nu^{4} + \nu^{2} + 26$$$$)/9$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{6} + 10 \nu^{4} - 40 \nu^{2} - 5$$$$)/9$$ $$\beta_{3}$$ $$=$$ $$($$$$-4 \nu^{7} + 10 \nu^{5} - 49 \nu^{3} + 13 \nu$$$$)/9$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{7} + 8 \nu^{5} - 41 \nu^{3} - 40 \nu$$$$)/9$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{7} - 8 \nu^{5} + 50 \nu^{3} + 49 \nu$$$$)/9$$ $$\beta_{6}$$ $$=$$ $$\nu^{6} - 2 \nu^{4} + 13 \nu^{2} + 2$$ $$\beta_{7}$$ $$=$$ $$($$$$7 \nu^{7} - 22 \nu^{5} + 88 \nu^{3} - 25 \nu$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + 2 \beta_{5} + \beta_{4} + 3 \beta_{3}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + 2 \beta_{2} - \beta_{1} + 2$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + 6 \beta_{5} + 7 \beta_{4} - 3 \beta_{3}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{6} + 3 \beta_{2} + 3 \beta_{1} - 9$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-17 \beta_{7} + 2 \beta_{5} + 3 \beta_{4} - 31 \beta_{3}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{6} - 14 \beta_{2} + 25 \beta_{1} - 70$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-27 \beta_{7} - 62 \beta_{5} - 75 \beta_{4} - 49 \beta_{3}$$$$)/8$$

## 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
 −0.435132 − 0.629640i −0.435132 + 0.629640i −1.52009 − 1.05050i −1.52009 + 1.05050i 1.52009 − 1.05050i 1.52009 + 1.05050i 0.435132 − 0.629640i 0.435132 + 0.629640i
0 −2.97127 0 4.29945i 0 2.64575i 0 5.82843 0
1791.2 0 −2.97127 0 4.29945i 0 2.64575i 0 5.82843 0
1791.3 0 −1.78089 0 1.23074i 0 2.64575i 0 0.171573 0
1791.4 0 −1.78089 0 1.23074i 0 2.64575i 0 0.171573 0
1791.5 0 1.78089 0 1.23074i 0 2.64575i 0 0.171573 0
1791.6 0 1.78089 0 1.23074i 0 2.64575i 0 0.171573 0
1791.7 0 2.97127 0 4.29945i 0 2.64575i 0 5.82843 0
1791.8 0 2.97127 0 4.29945i 0 2.64575i 0 5.82843 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
56.h odd 2 1 CM by $$\Q(\sqrt{-14})$$
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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.2.f.l 8
4.b odd 2 1 inner 1792.2.f.l 8
7.b odd 2 1 inner 1792.2.f.l 8
8.b even 2 1 inner 1792.2.f.l 8
8.d odd 2 1 inner 1792.2.f.l 8
16.e even 4 2 896.2.e.g 8
16.f odd 4 2 896.2.e.g 8
28.d even 2 1 inner 1792.2.f.l 8
56.e even 2 1 inner 1792.2.f.l 8
56.h odd 2 1 CM 1792.2.f.l 8
112.j even 4 2 896.2.e.g 8
112.l odd 4 2 896.2.e.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.e.g 8 16.e even 4 2
896.2.e.g 8 16.f odd 4 2
896.2.e.g 8 112.j even 4 2
896.2.e.g 8 112.l odd 4 2
1792.2.f.l 8 1.a even 1 1 trivial
1792.2.f.l 8 4.b odd 2 1 inner
1792.2.f.l 8 7.b odd 2 1 inner
1792.2.f.l 8 8.b even 2 1 inner
1792.2.f.l 8 8.d odd 2 1 inner
1792.2.f.l 8 28.d even 2 1 inner
1792.2.f.l 8 56.e even 2 1 inner
1792.2.f.l 8 56.h odd 2 1 CM

## Hecke kernels

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

 $$T_{3}^{4} - 12 T_{3}^{2} + 28$$ $$T_{5}^{4} + 20 T_{5}^{2} + 28$$ $$T_{29}$$ $$T_{31}$$ $$T_{37}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 28 - 12 T^{2} + T^{4} )^{2}$$
$5$ $$( 28 + 20 T^{2} + T^{4} )^{2}$$
$7$ $$( 7 + T^{2} )^{4}$$
$11$ $$T^{8}$$
$13$ $$( 28 + 52 T^{2} + T^{4} )^{2}$$
$17$ $$T^{8}$$
$19$ $$( 1372 - 76 T^{2} + T^{4} )^{2}$$
$23$ $$( 56 + T^{2} )^{4}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$T^{8}$$
$41$ $$T^{8}$$
$43$ $$T^{8}$$
$47$ $$T^{8}$$
$53$ $$T^{8}$$
$59$ $$( 8092 - 236 T^{2} + T^{4} )^{2}$$
$61$ $$( 14812 + 244 T^{2} + T^{4} )^{2}$$
$67$ $$T^{8}$$
$71$ $$( 252 + T^{2} )^{4}$$
$73$ $$T^{8}$$
$79$ $$( 28 + T^{2} )^{4}$$
$83$ $$( 26908 - 332 T^{2} + T^{4} )^{2}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$