# Properties

 Label 1792.2.e.e Level $1792$ Weight $2$ Character orbit 1792.e Analytic conductor $14.309$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$14.3091920422$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.12745506816.1 Defining polynomial: $$x^{8} + 23 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( \beta_{2} - \beta_{4} - \beta_{5} ) q^{7} + ( -2 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( \beta_{2} - \beta_{4} - \beta_{5} ) q^{7} + ( -2 + \beta_{3} ) q^{9} + ( 1 - \beta_{3} ) q^{11} + ( -\beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{13} + ( -3 \beta_{4} + \beta_{7} ) q^{15} + ( -\beta_{1} - \beta_{6} ) q^{17} + ( -2 \beta_{1} - \beta_{6} ) q^{19} + ( -\beta_{2} - 2 \beta_{4} + \beta_{7} ) q^{21} + \beta_{7} q^{23} -\beta_{3} q^{25} + ( -3 \beta_{1} + \beta_{6} ) q^{27} -3 \beta_{4} q^{29} + ( \beta_{4} + 2 \beta_{5} ) q^{31} + ( 5 \beta_{1} - \beta_{6} ) q^{33} + ( 3 + \beta_{1} - \beta_{3} ) q^{35} + \beta_{4} q^{37} + ( \beta_{4} - \beta_{7} ) q^{39} + 2 \beta_{1} q^{41} + ( -1 + \beta_{3} ) q^{43} + ( -7 \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{45} + ( \beta_{4} + 2 \beta_{5} ) q^{47} + ( 5 + \beta_{1} - \beta_{6} ) q^{49} + 4 q^{51} + ( \beta_{4} - 2 \beta_{7} ) q^{53} + ( 6 \beta_{2} - \beta_{4} - 2 \beta_{5} ) q^{55} + ( 9 - \beta_{3} ) q^{57} + \beta_{6} q^{59} + ( -3 \beta_{2} + 2 \beta_{4} + 4 \beta_{5} ) q^{61} + ( -5 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{63} + ( -1 + \beta_{3} ) q^{65} + ( 9 - \beta_{3} ) q^{67} + ( -4 \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{69} + ( -\beta_{4} + 2 \beta_{7} ) q^{71} + ( -3 \beta_{1} - \beta_{6} ) q^{73} + ( 4 \beta_{1} - \beta_{6} ) q^{75} + ( 4 \beta_{2} + 2 \beta_{5} + \beta_{7} ) q^{77} + \beta_{4} q^{79} + ( 10 - \beta_{3} ) q^{81} + \beta_{1} q^{83} + 2 \beta_{4} q^{85} -6 \beta_{2} q^{87} + ( -3 \beta_{1} - \beta_{6} ) q^{89} + ( -9 - \beta_{3} + \beta_{6} ) q^{91} -2 \beta_{4} q^{93} + ( 5 \beta_{4} - \beta_{7} ) q^{95} + ( -3 \beta_{1} - 3 \beta_{6} ) q^{97} + ( -23 + 3 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 16q^{9} + O(q^{10})$$ $$8q - 16q^{9} + 8q^{11} + 24q^{35} - 8q^{43} + 40q^{49} + 32q^{51} + 72q^{57} - 8q^{65} + 72q^{67} + 80q^{81} - 72q^{91} - 184q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 23 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + 24 \nu^{3} + 5 \nu$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} - 24 \nu^{3} + 5 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{4} + 23$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{6} - 48 \nu^{2}$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{7} + \nu^{6} + \nu^{5} - 115 \nu^{3} + 24 \nu^{2} + 24 \nu$$$$)/5$$ $$\beta_{6}$$ $$=$$ $$($$$$9 \nu^{7} + 2 \nu^{5} + 206 \nu^{3} + 43 \nu$$$$)/5$$ $$\beta_{7}$$ $$=$$ $$($$$$-6 \nu^{6} - 134 \nu^{2}$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - 3 \beta_{4}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{6} + 2 \beta_{5} + \beta_{4} - 10 \beta_{2} + 9 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{3} - 23$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{6} + 10 \beta_{5} + 5 \beta_{4} - 48 \beta_{2} - 43 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-24 \beta_{7} + 67 \beta_{4}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$12 \beta_{6} - 24 \beta_{5} - 12 \beta_{4} + 115 \beta_{2} - 103 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
895.1
 −1.54779 − 1.54779i 1.54779 − 1.54779i −0.323042 − 0.323042i 0.323042 − 0.323042i −0.323042 + 0.323042i 0.323042 + 0.323042i −1.54779 + 1.54779i 1.54779 + 1.54779i
0 3.09557i 0 −3.09557 0 −2.44949 + 1.00000i 0 −6.58258 0
895.2 0 3.09557i 0 3.09557 0 2.44949 1.00000i 0 −6.58258 0
895.3 0 0.646084i 0 −0.646084 0 2.44949 + 1.00000i 0 2.58258 0
895.4 0 0.646084i 0 0.646084 0 −2.44949 1.00000i 0 2.58258 0
895.5 0 0.646084i 0 −0.646084 0 2.44949 1.00000i 0 2.58258 0
895.6 0 0.646084i 0 0.646084 0 −2.44949 + 1.00000i 0 2.58258 0
895.7 0 3.09557i 0 −3.09557 0 −2.44949 1.00000i 0 −6.58258 0
895.8 0 3.09557i 0 3.09557 0 2.44949 + 1.00000i 0 −6.58258 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 895.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.d odd 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.e.e 8
4.b odd 2 1 1792.2.e.d 8
7.b odd 2 1 inner 1792.2.e.e 8
8.b even 2 1 1792.2.e.d 8
8.d odd 2 1 inner 1792.2.e.e 8
16.e even 4 1 896.2.f.c 8
16.e even 4 1 896.2.f.d yes 8
16.f odd 4 1 896.2.f.c 8
16.f odd 4 1 896.2.f.d yes 8
28.d even 2 1 1792.2.e.d 8
56.e even 2 1 inner 1792.2.e.e 8
56.h odd 2 1 1792.2.e.d 8
112.j even 4 1 896.2.f.c 8
112.j even 4 1 896.2.f.d yes 8
112.l odd 4 1 896.2.f.c 8
112.l odd 4 1 896.2.f.d yes 8

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

## Hecke kernels

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

 $$T_{3}^{4} + 10 T_{3}^{2} + 4$$ $$T_{11}^{2} - 2 T_{11} - 20$$ $$T_{31}^{4} - 40 T_{31}^{2} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 4 + 10 T^{2} + T^{4} )^{2}$$
$5$ $$( 4 - 10 T^{2} + T^{4} )^{2}$$
$7$ $$( 49 - 10 T^{2} + T^{4} )^{2}$$
$11$ $$( -20 - 2 T + T^{2} )^{4}$$
$13$ $$( 100 - 34 T^{2} + T^{4} )^{2}$$
$17$ $$( 64 + 40 T^{2} + T^{4} )^{2}$$
$19$ $$( 900 + 66 T^{2} + T^{4} )^{2}$$
$23$ $$( 400 + 44 T^{2} + T^{4} )^{2}$$
$29$ $$( 36 + T^{2} )^{4}$$
$31$ $$( 64 - 40 T^{2} + T^{4} )^{2}$$
$37$ $$( 4 + T^{2} )^{4}$$
$41$ $$( 64 + 40 T^{2} + T^{4} )^{2}$$
$43$ $$( -20 + 2 T + T^{2} )^{4}$$
$47$ $$( 64 - 40 T^{2} + T^{4} )^{2}$$
$53$ $$( 84 + T^{2} )^{4}$$
$59$ $$( 100 + 34 T^{2} + T^{4} )^{2}$$
$61$ $$( 4900 - 154 T^{2} + T^{4} )^{2}$$
$67$ $$( 60 - 18 T + T^{2} )^{4}$$
$71$ $$( 84 + T^{2} )^{4}$$
$73$ $$( 56 + T^{2} )^{4}$$
$79$ $$( 4 + T^{2} )^{4}$$
$83$ $$( 4 + 10 T^{2} + T^{4} )^{2}$$
$89$ $$( 56 + T^{2} )^{4}$$
$97$ $$( 5184 + 360 T^{2} + T^{4} )^{2}$$